diff --git a/dox/FILES.txt b/dox/FILES.txt index 2b3d9ccfc6..9dead2a5e6 100644 --- a/dox/FILES.txt +++ b/dox/FILES.txt @@ -33,14 +33,13 @@ user_guides/ocaf_wp/ocaf_wp.md user_guides/voxels_wp/voxels_wp.md dev_guides/dev_guides.md -dev_guides/contribution/coding_rules.md -dev_guides/cdl/cdl.md -dev_guides/tests/tests.md dev_guides/documentation/documentation.md -dev_guides/wok/wok.md +dev_guides/contribution/coding_rules.md dev_guides/contribution_workflow/contribution_workflow.md dev_guides/git_guide/git_guide.md - +dev_guides/tests/tests.md +dev_guides/cdl/cdl.md +dev_guides/wok/wok.md dev_guides/building/building.md dev_guides/building/3rdparty/3rdparty_windows.md diff --git a/dox/dev_guides/contribution_workflow/contribution_workflow.md b/dox/dev_guides/contribution_workflow/contribution_workflow.md index 8db8e35c85..d8bb29d876 100644 --- a/dox/dev_guides/contribution_workflow/contribution_workflow.md +++ b/dox/dev_guides/contribution_workflow/contribution_workflow.md @@ -1,4 +1,4 @@ -Contribution Workflow {#dev_guides_contribution_workflow} +Contribution Workflow {#dev_guides__contribution_workflow} ==================================== @tableofcontents @@ -143,8 +143,8 @@ The definition of the following attributes is obligatory: This is done to have the master branch history plain and clean. The following picture illustrates the process: -@image html OCCT_ContributionWorkflow_V3_image002.jpg "Integration of several branches" -@image latex OCCT_ContributionWorkflow_V3_image002.jpg "Integration of several branches" +@image html OCCT_ContributionWorkflow_V3_image002.png "Integration of several branches" +@image latex OCCT_ContributionWorkflow_V3_image002.png "Integration of several branches" The new master branch is tested against possible regressions that might appear due to interference between separate changes. When the tests are Ok, the new master is pushed to the official repository and the original branches are removed from it. diff --git a/dox/dev_guides/contribution_workflow/images/OCCT_ContributionWorkflow_V3_image002.jpg b/dox/dev_guides/contribution_workflow/images/OCCT_ContributionWorkflow_V3_image002.jpg deleted file mode 100644 index c362ec4614..0000000000 Binary files a/dox/dev_guides/contribution_workflow/images/OCCT_ContributionWorkflow_V3_image002.jpg and /dev/null differ diff --git a/dox/dev_guides/contribution_workflow/images/OCCT_ContributionWorkflow_V3_image002.png b/dox/dev_guides/contribution_workflow/images/OCCT_ContributionWorkflow_V3_image002.png new file mode 100644 index 0000000000..d4dc06aa10 Binary files /dev/null and b/dox/dev_guides/contribution_workflow/images/OCCT_ContributionWorkflow_V3_image002.png differ diff --git a/dox/dev_guides/dev_guides.md b/dox/dev_guides/dev_guides.md index 327cc39f46..a56dad1c74 100644 --- a/dox/dev_guides/dev_guides.md +++ b/dox/dev_guides/dev_guides.md @@ -6,9 +6,9 @@ The following documents provide information on OCCT building, development and te * @subpage dev_guides__building "Building OCCT from sources" * @subpage dev_guides__documentation "Documentation system" * @subpage dev_guides__coding_rules "Coding Rules" -* Contribution Workflow -* Guide to installing and using Git for OCCT development - @subpage dev_guides__tests "Automatic Testing system" +* @subpage dev_guides__contribution_workflow "Contribution Workflow" +* @subpage dev_guides__git_guide "Guide to installing and using Git for OCCT development" +* @subpage dev_guides__tests "Automatic Testing system" Two other documents provide details on obsolete technologies used by OCCT, to be removed in future releases: diff --git a/dox/dev_guides/documentation/documentation.md b/dox/dev_guides/documentation/documentation.md index da2141e3a5..9ad43e71ed 100644 --- a/dox/dev_guides/documentation/documentation.md +++ b/dox/dev_guides/documentation/documentation.md @@ -460,7 +460,7 @@ Equations can be written by several ways: These formulas should be put between \@f\[ and \@f\] tags. An example: @verbatim -@f$[ +@f[ |I_2|=\left| \int_{0}^T \psi(t) \left\{ u(a,t)- @@ -469,7 +469,7 @@ These formulas should be put between \@f\[ and \@f\] tags. An example: \int_{a}^\theta c(\xi)u_t(\xi,t)\,d\xi \right\} dt \right| -@f$] +@f] @endverbatim gives the following result: diff --git a/dox/dev_guides/git_guide/git_guide.md b/dox/dev_guides/git_guide/git_guide.md index 1daf4f1cdc..03739bd356 100644 --- a/dox/dev_guides/git_guide/git_guide.md +++ b/dox/dev_guides/git_guide/git_guide.md @@ -1,4 +1,4 @@ -Guide to installing and using Git for OCCT development {#dev_guides_git_guide} +Guide to installing and using Git for OCCT development {#dev_guides__git_guide} ================================= @tableofcontents @@ -28,11 +28,11 @@ Guide to installing and using Git for OCCT development {#dev_guides_git_guide} URL of the official OCCT source code Git repository (accessed by SSH protocol) is: - gitolite@git.dev.opencascade.org:occt + gitolite@git.dev.opencascade.org:occt or - ssh://gitolite@git.opencascade.org/occt.git + ssh://gitolite@dev.opencascade.org/occt.git @subsection occt_gitguide_1_3 Content @@ -136,10 +136,12 @@ The official repository contains: * Complete the installation. + TortoiseGit integrates to Windows Explorer, thus it is possible to use context menu in Windows Explorer to access its functionality: + @image html OCCT_GitGuide_V2_image005.png @image latex OCCT_GitGuide_V2_image005.png - TortoiseGit integrates to Windows Explorer, thus it is possible to use popup menu in Windows Explorer to access its functionality: + Note that if you have installed MSysGit or have Git installed in non-default path, on the first time you use TortoiseGit you may get the message demanding to define path to Git. @@ -247,18 +249,17 @@ The official repository contains: @subsection occt_gitguide_3_3 Adding public key in your account - Log in on the portal http://dev.opencascade.org and click on 'My account' link to the right. - If you have a Contributor status, you will see 'SSH keys' tab to the right. - Click on that tab, then click 'Add a public key', and paste the text of the public key - (see above sections on how to generate the key) into the text box. - Click "Save" to input the key to the system. +Log in on the portal http://dev.opencascade.org and click on **My account** link to the right. If you have a Contributor status, you will see **SSH keys** tab to the right. +Click on that tab, then click **Add a public key**, and paste the text of the public key (see above sections on how to generate the key) into the text box. + Click **Save** to input the key to the system. +@image html OCCT_GitGuide_V2_image008.png +@image latex OCCT_GitGuide_V2_image008.png + Note that a user can have several SSH keys. You can distinguish between these keys by the Title field ID; by default it is taken from SSH key comment. It is typical to use your e-mail address or workstation name for this field; no restrictions are set by the portal. - -@image html OCCT_GitGuide_V2_image008.png -@image latex OCCT_GitGuide_V2_image008.png + Please note that some time (5-10 min) is needed for the system to update the configuration after the new key is added. @@ -308,19 +309,14 @@ The official repository contains: where is the path to the new folder which will be created for the repository. - * In TortoiseGit: right-click in the Explorer window, then choose "Git Clone": + * In TortoiseGit: create a new folder, open it and right-click in the Explorer window, then choose **Git Clone** in the context menu: @image html OCCT_GitGuide_V2_image009.png @image latex OCCT_GitGuide_V2_image009.png - If you have chosen Putty as SSH client during TortoiseGit installation, check the “Load Putty Key” option and specify the location of the private key file saved by PuttyGen (see 3.2.1). This shall be done for the first time only. + If you have chosen Putty as SSH client during TortoiseGit installation, check the **Load Putty Key** option and specify the location of the private key file saved by PuttyGen (see 3.2.1). This shall be done for the first time only. - Note that on the first connection to the repository server you may be requested to enter a password for your private SSH key; further you can get a message that the authenticity of the host cannot be established and will be asked if you want to continue connecting or not: - -@image html OCCT_GitGuide_V2_image010.png -@image latex OCCT_GitGuide_V2_image010.png - - Choose “Yes” to continue. Then the host’s key will be stored in $HOME/.ssh/known_hosts file. + Note that on the first connection to the repository server you may be requested to enter a password for your private SSH key; further you can get a message that the authenticity of the host cannot be established and will be asked if you want to continue connecting or not. Choose **Yes** to continue. The host’s key will be stored in $HOME/.ssh/known_hosts file. @subsection occt_gitguide_4_3 Branch creation @@ -338,17 +334,17 @@ In the console: In TortoiseGit: * Go to the local copy of the repository. - * Right-click in the Explorer window, then choose "Git Create Branch". + * Right-click in the Explorer window, then choose **Git Create Branch**. @image html OCCT_GitGuide_V2_image011.png @image latex OCCT_GitGuide_V2_image011.png - * Select “Base On” Branch remotes/origin/master + * Select **Base On** Branch *remotes/origin/master*. @image html OCCT_GitGuide_V2_image012.png @image latex OCCT_GitGuide_V2_image012.png - Check option ‘Switch to new branch’ if you are going to start working with the newly created branch immediately. + Check option **Switch to new branch** if you are going to start working with the newly created branch immediately. @subsection occt_gitguide_4_4 Branch switching @@ -359,14 +355,14 @@ In TortoiseGit: > git checkout CR12345 ~~~~~ - In TortoiseGit: right-click, TortoiseGit -> Checkout/switch + In TortoiseGit: right-click in the explorer window and select in the context menu **TortoiseGit** -> **Switch/Checkout**. @image html OCCT_GitGuide_V2_image013.png @image latex OCCT_GitGuide_V2_image013.png Note that in order to work with the branch locally you need to set option - “Create new branch” when you checkout the branch from the remote repository for the first time. - Option “Track” stores association between the local branch and the original branch in a remote repository. + **Create new branch** when you checkout the branch from the remote repository for the first time. + Option **Track** stores association between the local branch and the original branch in a remote repository. @subsection occt_gitguide_4_5 Committing branch changes @@ -394,7 +390,7 @@ In TortoiseGit: > git add file1.hxx file2.cxx ~~~~~ - * In TortoiseGit: right-click, choose “Git Commit -> CR…”: + * In TortoiseGit: right-click in the explorer window and select in the context menu Git Commit -> CR…: @image html OCCT_GitGuide_V2_image014.png @image latex OCCT_GitGuide_V2_image014.png @@ -413,147 +409,109 @@ In TortoiseGit: > git push "origin" CR12345:CR12345 ~~~~~ - * In TortoiseGit: right-click, TortoiseGit -> Push + * In TortoiseGit: right-click in the explorer window and select in the context menu, TortoiseGit -> **Push** @image html OCCT_GitGuide_V2_image015.png @image latex OCCT_GitGuide_V2_image015.png -Note that Git will forbid pushing a branch if the corresponding remote branch already exists and has some changes, which are not in the history of your local branch. This may happen in different situations: - * You have amended the last commit which is already in the remote repository. - If you are sure that nobody else uses your branch, push again with ‘force’ option. - * You have rebased your branch, so that now it is completely different - from the branch in the remote repository. In this case, push it under a different name (add a suffix): +Note that Git forbids pushing a branch if the corresponding remote branch already exists and has some changes, which are not in the history of your local branch. This may happen in different situations: + * You have amended the last commit which is already in the remote repository. If you are sure that nobody else uses your branch, push again with **Force** option. + * You have rebased your branch, so that now it is completely different from the branch in the remote repository. In this case, push it under a different name (add a suffix): @image html OCCT_GitGuide_V2_image016.png @image latex OCCT_GitGuide_V2_image016.png - Then remove the original remote branch so that other people recognize that it has been replaced by the new one. For that, select TortoiseGit -> Push again, then select an empty line for your local branch name, - and enter the name of the branch to be removed in ‘Remote’ field: + Then remove the original remote branch so that other people recognize that it has been replaced by the new one. For that, select TortoiseGit -> **Push** again, select an empty line for your local branch name, + and enter the name of the branch to be removed in **Remote** field: -@image html OCCT_GitGuide_V2_image017.png -@image latex OCCT_GitGuide_V2_image017.png - - * The other developer has committed some changes in the remote branch. - In this case, pull changes from the remote repository to have them merged - with your version, and push your branch after it is successfully merged. + * The other developer has committed some changes in the remote branch. In this case, **Pull** changes from the remote repository to have them merged with your version, and push your branch after it is successfully merged. @subsection occt_gitguide_4_7 Synchronizing with remote repository Maintain your repository synchronized with the remote one and clean unnecessary stuff regularly. - Use Git command fetch with option prune to get update of all branches - from the remote repository and to clean your local repository from the remote branches that have been deleted. + + Use Git command *fetch* with option *prune* to get the update of all branches from the remote repository and to clean your local repository from the remote branches that have been deleted. * In the console: - ~~~~~ > git fetch --prune ~~~~~ - * In TortoiseGit: + * In TortoiseGit: right-click in the explorer window and select in the context menu **TortoiseGit** -> **Fetch**. Check in **Prune** check-box. @image html OCCT_GitGuide_V2_image018.png @image latex OCCT_GitGuide_V2_image018.png - If some changes have been made to the branch you are working with in the remote repository, - use Git command pull to get the remote changes and merge them with your local branch. + If the branch you are working with has been changed in the remote repository, use Git command *pull* to get the remote changes and merge them with your local branch. + This operation is required in particular to update your local master branch when the remote master changes. * In console: ~~~~~ > git pull ~~~~~ - * In TortoiseGit: + + * In TortoiseGit: right-click in the explorer window and select in the context menu **TortoiseGit** -> **Pull**. @image html OCCT_GitGuide_V2_image019.png @image latex OCCT_GitGuide_V2_image019.png - Note that the local branches of your repository are the primary place - where your changes are stored until they get integrated to the official version - of OCCT (master branch). The branches submitted to official repository - are for collaborative work, review, and integration; - that repository should not be used for long-term storage of incomplete changes. +Note that the local branches of your repository are the primary place, where your changes are stored until they get integrated to the official version of OCCT (master branch). The branches submitted to official repository are for collaborative work, review, and integration - that repository should not be used for long-term storage of incomplete changes. -Remove the local branches that you do not need any more. Note that you cannot delete the current branch, thus you need to switch to another one (e.g. master) if the branch you are going to delete is the current one. +Remove the local branches that you do not need any more. Note that you cannot delete the current branch. It means that you need to switch to another one (e.g. master) if the branch you are going to delete is the current one. * In the console: ~~~~~ > git branch -d CR12345 ~~~~~ - * In TortoiseGit: right-click, select TortoiseGit -> Show log + * In TortoiseGit: right-click in the explorer window and select in the context menu **TortoiseGit** -> **Git Show Log**. @image html OCCT_GitGuide_V2_image020.png @image latex OCCT_GitGuide_V2_image020.png - Select “All branches” to view all branches. - Right-click on the branch you want to delete and select Delete… menu item corresponding to this branch. + Select **All branches** check-box to view all branches. + Right-click on the branch you want to delete and select **Delete** item in the context menu. - Note that the log view in TortoiseGit is a convenient tool - to visualize and manage branches; it provides short-cuts to many functions described above. +Note that many functions described above can be accessed from the Log View, which is a very convenient tool to visualize and manage branches. @subsection occt_gitguide_4_8 Applying a fix made on older version of OCCT - If you have a fix made on a previous version of OCCT, - perform the following sequence of operations to prepare it - for testing and integration to the current development version. - - * Identify the version of OCCT on which the fix has been made. - In most cases, this will be an OCCT release, e.g. OCCT 6.7.0.; - then just find a tag or a commit corresponding to this version in the Git history log of the master branch. +If you have a fix made on a previous version of OCCT, perform the following sequence of operations to prepare it for testing and integration to the current development version: + * Identify the version of OCCT on which the fix has been made. In most cases, this will be an OCCT release, e.g. OCCT 6.7.0. + * Find a tag or a commit corresponding to this version in the Git history log of the master branch. * Create a branch basing on this tag or commit. In TortoiseGit history log: right-click on the base commit, then select **Create branch at this version**. @image html OCCT_GitGuide_V2_image021.png @image latex OCCT_GitGuide_V2_image021.png - Check option **Switch to the new branch** to start working within the new branch immediately, or switch to it separately afterwards. - + * Check option **Switch to the new branch** to start working within the new branch immediately, or switch to it separately afterwards. * Put your fix in the working copy, build and check that it works, then commit to the branch. - * Rebase the branch on the current master. - - In TortoiseGit: right-click on the working directory, - then choose TortoiseGit->Rebase; then select *remotes/origin/master* as UpStream revision, and click **Start**: + * Rebase the branch on the current master. In TortoiseGit: right-click on the working directory, choose **TortoiseGit** -> **Rebase**, select *remotes/origin/master* as UpStream revision, and click **Start**: @image html OCCT_GitGuide_V2_image022.png @image latex OCCT_GitGuide_V2_image022.png - Note that you can get some conflicts during rebase. - To resolve the conflicts double-click on each conflicted file - (highlighted by red in the file list) to open visual merge tool. - Switch between conflicting fragments by red arrows, and for each one - decide if the code of one or both conflicting versions is to be taken. - Use the toolbar. +Note that you can get some conflicts during rebase. To resolve them, double-click on each conflicted file (highlighted by red in the file list) to open visual merge tool. Switch between conflicting fragments by red arrows, and for each one decide if the code of one or both conflicting versions is to be taken. @subsection occt_gitguide_4_9 Rebasing with history clean-up - At some moments you might need to rebase your branch on the latest version of the master. +At some moments you might need to rebase your branch on the latest version of the master. - We recommend rebasing before the first submission of the branch - for review or when the master has diverged substantially from your branch. +We recommend rebasing before the first submission of the branch for review or when the master has diverged substantially from your branch. - Rebasing is a good occasion to clean-up the history of commits in the branch. - Consider collapsing (squashing, in terms of Git) the history of your branch - into a single commit unless you deem that having separate commits is important - for your future work with the branch or its code reviewing. - Git also allows you to change the order of commits, edit commit contents and messages, etc. - - Here is the sequence of actions to rebase your branch into a single commit: +Rebasing is a good occasion to clean-up the history of commits in the branch. Consider collapsing (squashing, in terms of Git) the history of your branch into a single commit unless you deem that having separate commits is important for your future work with the branch or its code reviewing. Git also allows changing the order of commits, edit commit contents and messages, etc. +To rebase your branch into a single commit, you need to do the following: * Switch to your branch (e.g. “CR12345”) - * In TortoiseGit history log, select a branch to rebase on (usually remotes/origin/master) - and in the context menu choose **Rebase “CR12345” onto this**. - * In the Rebase dialog, check **Squash All**. - + * In TortoiseGit history log, select a branch to rebase on *(remotes/origin/master)* and in the context menu choose **Rebase “CR12345” onto this**. + * In the **Rebase** dialog, check **Squash All**. You can also change the order of commits and define for each commit whether it should be kept (**Pick**), edited, or just skipped. + @image html OCCT_GitGuide_V2_image023.png @image latex OCCT_GitGuide_V2_image023.png - **Note** that you can also change the order of commits and define for each commit - whether it should be kept (“pick”), edited, or just skipped. - * Click **Start**. - * The process will stop if a conflict is detected. - In such case, find files with status ‘Conflicted’ in the list (marked by red), - and double-click on them to resolve the conflict. - When all conflicts are resolved, click ‘Continue’. + * The process will stop if a conflict is detected. In that case, find files with status **Conflicted** in the list (marked by red), and double-click on them to resolve the conflict. When all conflicts are resolved, click **Continue**. @image html OCCT_GitGuide_V2_image024.png @image latex OCCT_GitGuide_V2_image024.png @@ -570,32 +528,22 @@ Remove the local branches that you do not need any more. Note that you cannot de The changes made in the branch can be reviewed without direct access to Git, using GitWeb interface: * Open GitWeb in your web browser: http://git.dev.opencascade.org/gitweb/?p=occt.git - * Locate the branch you want to review among heads (click ‘…’ at the bottom of the page to see the full list). - * Click log (or shortlog) to see the history of the branch. + * Locate the branch you want to review among **heads** (click ‘…’ at the bottom of the page to see the full list). + * Click **log** (or **shortlog**) to see the history of the branch. - **Note** that the branch can contain more than one commit, and you need to distinguish commits - that belong to that branch (those to be reviewed) from the commits - corresponding to the previous state of the master branch. - Normally the first commit in the list that starts from the ID - of the other issue indicates the branching point; - commits above it are the ones to be reviewed. + **Note** that the branch can contain more than one commit, and you need to distinguish commits that belong to that branch (those to be reviewed) from the commits corresponding to the previous state of the master branch. Normally the first commit in the list that starts from the ID of the other issue indicates the branching point; commits above it are the ones to be reviewed. - * Click *commitdiff* on each log entry to review the changes (highlighted with color format). + * Click **commitdiff** on each log entry to review the changes (highlighted with color format). @subsection occt_gitguide_5_2 Review branch changes with TortoiseGit Use of TortoiseGit is recommended for convenient code review: - * Fetch the changes from the remote repository as described in 4.7; - * Right-click on the repository, choose TortoiseGit -> Show log; + * Fetch the changes from the remote repository as described in Synchronizing with remote repository section. + * Right-click on the repository, choose **TortoiseGit** -> **Show** log; * Locate the remote branch you need to review; - * To review commits one-by-one, select each commit in the log. - The list of changed files is shown at the bottom of the window; - double-click on the file will open visual compare tool. - * To review all changes made in the branch at once, or to compare - two arbitrary revisions, select the corresponding commits in the log - (e.g. the last commit in the branch and the branching point), - right-click for the context menu, and choose **Compare revisions**. + * To review commits one-by-one, select each commit in the log. The list of changed files is shown at the bottom of the window; double-click on the file will open visual compare tool. + * To review all changes made in the branch at once, or to compare two arbitrary revisions, select the corresponding commits in the log (e.g. the last commit in the branch and the branching point), ight-click for the context menu, and choose **Compare revisions**. @image html OCCT_GitGuide_V2_image026.png @image latex OCCT_GitGuide_V2_image026.png diff --git a/dox/dev_guides/git_guide/images/OCCT_GitGuide_V2_image001.png b/dox/dev_guides/git_guide/images/OCCT_GitGuide_V2_image001.png index 8152cc9dff..674cf90fd1 100644 Binary files a/dox/dev_guides/git_guide/images/OCCT_GitGuide_V2_image001.png and b/dox/dev_guides/git_guide/images/OCCT_GitGuide_V2_image001.png differ diff --git a/dox/dev_guides/git_guide/images/OCCT_GitGuide_V2_image002.png b/dox/dev_guides/git_guide/images/OCCT_GitGuide_V2_image002.png index 33894e21c9..8aed9b096f 100644 Binary files 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b/dox/dev_guides/git_guide/images/OCCT_GitGuide_V2_image026.png differ diff --git a/dox/occdoc.tcl b/dox/occdoc.tcl index 5d05cc4c8e..442d0c71d6 100644 --- a/dox/occdoc.tcl +++ b/dox/occdoc.tcl @@ -6,7 +6,7 @@ # get OCCT version from file Standard_Version.hxx (if available) proc OverviewDoc_DetectCasVersion {theCasRoot} { - set occt_ver 16.7.0 + set occt_ver 6.7.0 set occt_ver_add "" if { [file exist $theCasRoot/src/Standard/Standard_Version.hxx] } { set fh [open $theCasRoot/src/Standard/Standard_Version.hxx] diff --git a/dox/overview/Overview.md b/dox/overview/Overview.md index 03299fb2e6..ff7bd5d095 100644 --- a/dox/overview/Overview.md +++ b/dox/overview/Overview.md @@ -438,8 +438,7 @@ Declaration of available plug-ins is done through special resource file(s). The pload command loads the plug-in in accordance with the specified resource file and activates the commands implemented in the plug-in. -The whole process of using the plug-in mechanism as well as the instructions for extending Test Harness is described in the -@ref user_guides__test_harness. +The whole process of using the plug-in mechanism as well as the instructions for extending Test Harness is described in the @ref user_guides__test_harness. Draw Test Harness provides an environment for OCCT automated testing system. Please, consult its @ref dev_guides__tests "Automated Testing System" for details. @@ -562,11 +561,9 @@ The combination of these resources allows creating substantial applications. Voxel ------ -This is a demonstration application showing OCCT voxel models. -It also includes a set of non-regression tests and other commands -for testing this functionality (accessible only through TEST pre-processor definition). +This is a demonstration application showing OCCT voxel models. It also includes a set of non-regression tests and other commands for testing this functionality (accessible only through TEST pre-processor definition). -**See also:** Voxels User's guide (under construction) +**See also:** Voxels User's guide **Remarks:** @@ -575,8 +572,7 @@ for testing this functionality (accessible only through TEST pre-processor defin @subsubsection OCCT_OVW_SECTION_7_3_3 C# -C# sample demonstrates integration of OCCT 3D Viewer and Import / Export functionality -into .NET applications (using Windows Forms and WPF front ends). +C# sample demonstrates integration of OCCT 3D Viewer and Import / Export functionality into .NET applications (using Windows Forms and WPF front ends). @image html /overview/images/overview_c__ie.png @image latex /overview/images/overview_c__ie.png diff --git a/dox/technical_overview/images/technical_overview_occ_0007.png b/dox/technical_overview/images/technical_overview_occ_0007.png index a7d2856248..cf2c20902a 100644 Binary files a/dox/technical_overview/images/technical_overview_occ_0007.png and b/dox/technical_overview/images/technical_overview_occ_0007.png differ diff --git a/dox/technical_overview/images/technical_overview_occ_0008.png b/dox/technical_overview/images/technical_overview_occ_0008.png index b61acb27c0..612028625b 100644 Binary files a/dox/technical_overview/images/technical_overview_occ_0008.png and b/dox/technical_overview/images/technical_overview_occ_0008.png differ diff --git a/dox/technical_overview/images/technical_overview_shapeattrib.png b/dox/technical_overview/images/technical_overview_shapeattrib.png index 6add718a92..1ff53764eb 100644 Binary files a/dox/technical_overview/images/technical_overview_shapeattrib.png and b/dox/technical_overview/images/technical_overview_shapeattrib.png differ diff --git a/dox/technical_overview/technical_overview.md b/dox/technical_overview/technical_overview.md index a1081e0857..ebec72db31 100644 --- a/dox/technical_overview/technical_overview.md +++ b/dox/technical_overview/technical_overview.md @@ -75,9 +75,9 @@ These services are organized into the following libraries: * Math Utilities * Basic Persistence -The technical overview provides only a basic description of the libraries. Please, refer for more details to Foundation Classes User's guide +The technical overview provides only a basic description of the libraries. Please, refer for more details to Foundation Classes User's guide -See also: our web site at E-learning and Training. +See also: our web site at E-learning and Training. @subsection OCCT_TOVW_SECTION_2_1 Kernel Classes @@ -346,13 +346,12 @@ These services are organized into the following libraries: * Geometry Utilities * Topology -The technical overview provides only a basic description of the libraries. Please, -refer for more details to Modeling Data User's guide +The technical overview provides only a basic description of the libraries. Please, refer for more details to Modeling Data User's guide 3D geometric models are stored in OCCT native BREP format. It is possible to learn -more about it in BREP Format Description White Paper +more about it in BREP Format Description White Paper -See also: our web site at E-learning and Training. +See also: our web site at E-learning and Training. @subsection OCCT_TOVW_SECTION_3_1 2D Geometry Types @@ -952,10 +951,9 @@ These services are organized into the following libraries: @image html /technical_overview/images/technical_overview_ma.png @image latex /technical_overview/images/technical_overview_ma.png -The technical overview provides only a basic description of the libraries. -Please, refer for more details to Modeling Algorithms User's guide +The technical overview provides only a basic description of the libraries. Please, refer for more details to Modeling Algorithms User's guide -See also: our web site at E-learning and Training. +See also: our web site at E-learning and Training. @subsection OCCT_TOVW_SECTION_4_1 Geometric Tools @@ -1177,10 +1175,10 @@ to deformation. Then, the algorithm is called to calculate the final surface. It looks for a solution satisfying constraints and minimizing energy input. @image html /technical_overview/images/technical_overview_occ_0007.png "Surface generated from four curves and a point" -@image latex /technical_overview/images/technical_overview_occ_0007.png "Surface generated from four curves and a point" +@image latex /technical_overview/images/technical_overview_occ_0007.png "Surface generated from four curves and a point" @image html /technical_overview/images/technical_overview_occ_0008.png "Surface generated from two curves and a point" -@image latex /technical_overview/images/technical_overview_occ_0008.png "Surface generated from two curves and a point" +@image latex /technical_overview/images/technical_overview_occ_0008.png "Surface generated from two curves and a point" #### Extension of a 3D curve or surface beyond its original bounds @@ -1395,9 +1393,9 @@ It has several sub-domains, each with its own scope of functionality: * customization - modifying shape representation to fit specific needs (the shape is not modified, only the form of its representation is modified); * processing - mechanism of shape modification via a user-editable resource file. -The technical overview provides only a basic description of the libraries. Please, refer for more details to Shape Healing User's guide +The technical overview provides only a basic description of the libraries. Please, refer for more details to Shape Healing User's guide -See also: our web site at E-learning and Training. +See also: our web site at E-learning and Training. @subsection OCCT_TOVW_SECTION_4_8 Miscellaneous modelling algorithms. @@ -1496,13 +1494,13 @@ Visualization in Open CASCADE Technology is based on the separation of modeling For visualizing data structures, OCCT provides ready-to-use algorithms, which create graphic presentations from geometric models. These data structures may be used with the viewers supplied, and can be customized to take the specificity of your application into account. -Displaying is managed through presentation services, and selection in its turn is managed through selection services. With these services, data structures and algorithms are provided to display objects of an application, and to support graphical selection of these objects. +Display is managed through presentation services, and selection in its turn is managed through selection services. With these services, data structures and algorithms are provided to display objects of an application and to support graphical selection of these objects. Application Interactive Services (AIS) are provided to manage displaying, detection and selection of graphical presentations. These services associate data structures and interactive objects. -Please, refer for more details to Visualization User's guide +The technical overview provides only a basic description of the libraries. Please, refer for more details to Visualization User's guide -See also: our web site at E-learning and Training. +See also: our web site at E-learning and Training. @image html /technical_overview/images/technical_overview_viz.png @image latex /technical_overview/images/technical_overview_viz.png @@ -1914,7 +1912,8 @@ with fast access to the data and optimal allocation of data in memory. Also, a special visualization toolkit allows visualizing voxels as colored or black/white points and cubes, displaying only the voxels visible from the user's point of view. -Please, see for more information Voxels User's Guide white paper. +Please, see for more information Voxels User's Guide. + @subsection OCCT_TOVW_SECTION_5_11 Examples @@ -2119,9 +2118,10 @@ of the loading operation, you use the following enumerations from the IFSele To read and write attributes such as names, colors, layers for IGES and STEP and validation properties and structure of assemblies for STEP, you can use an XDE document. -It is possible to learn more about XDE documents from XDE User's guide +It is possible to learn more about XDE documents from XDE User's guide. + +See also: our web site at E-learning and Training. -See also: our web site at E-learning and Training. @subsection OCCT_TOVW_SECTION_6_2 IGES @@ -2144,27 +2144,21 @@ Please, note: * an IGES entity is an entity in the IGES normal sense. * a root entity is the highest level entity of any given type, e.g. type 144 for surfaces and type 186 for solids. Roots are not referenced by other entities. -It is possible to learn more about the IGES interface from IGES User's guide +It is possible to learn more about the IGES interface from IGES User's guide. -See also: our web site at E-learning and Training. +See also: our web site at E-learning and Training. @subsection OCCT_TOVW_SECTION_6_3 STEP -The STEP interface reads STEP files produced in accordance with STEP Application Protocol 214 -(Conformance Class 2 both CD and DIS versions of schema) and translates them -to Open CASCADE Technology models. STEP Application Protocol 203 is also supported. +The STEP interface reads STEP files produced in accordance with STEP Application Protocol 214 (Conformance Class 2 both CD and DIS versions of schema) and translates them to Open CASCADE Technology models. STEP Application Protocol 203 is also supported. -The STEP interface also translates OCCT models to STEP files. STEP files that are produced -by this interface conform to STEP AP 203 or AP 214 ( -Conformance Class 2, either CD or DIS version of the schema) depending on the user's option. +The STEP interface also translates OCCT models to STEP files. STEP files that are produced by this interface conform to STEP AP 203 or AP 214 (Conformance Class 2, either CD or DIS version of the schema) depending on the user's option. Basic interface reads and writes geometrical, topological STEP data and assembly structures. The interface is able to translate one entity, a group of entities or a whole file. -Other kinds of data such as colors, validation properties, layers, names -and the structure of assemblies can be read or written -with the help of XDE tools - STEPCAFControl_Reader and STEPCAFControl_Writer. +Other kinds of data such as colors, validation properties, layers, names and the structure of assemblies can be read or written with the help of XDE tools - STEPCAFControl_Reader and STEPCAFControl_Writer. To choose a translation mode when exporting to a STEP format, use STEPControl_STEPModelType. @@ -2175,8 +2169,9 @@ Please, note: * all references to shapes indicate OCCT shapes unless otherwise explicitly stated; * a root entity is the highest level entity of any given type, i.e. an entity that is not referenced by any other one. -It is possible to learn more about the STEP interface from STEP User's guide -See also: our web site at E-learning and Training. +It is possible to learn more about the STEP interface from STEP User's guide. + +See also: our web site at E-learning and Training. @subsection OCCT_TOVW_SECTION_6_4 STL @@ -2228,12 +2223,12 @@ in the same way as a number is the value of the Integer attribute and a string i OCAF documents are in their turn managed by an OCAF application. -Please, refer for more details to OCAF User's guide and the OCAF white papers: -* Application Framework -* Distribution of Data through OCAF Tree -* Application Framework Function Mechanism +Please, refer for more details to OCAF User's guide and the OCAF white papers: +* Application Framework +* Distribution of Data through OCAF Tree +* Application Framework Function Mechanism -See also: our web site at E-learning and Training. +See also: our web site at E-learning and Training. @subsection OCCT_TOVW_SECTION_7_1 How to start working with OCAF @@ -2380,23 +2375,11 @@ label depending on necessity: TNaming_NamedShape may contain a few pairs of hooks with the same evolution. In this case topology shape, which belongs to the named shape, is a compound of new shapes. -The data model contains both the topology and the hooks, and functions handle both -topological entities and hooks. Consider the case of a box function, which creates -a solid with six faces and six hooks. Each hook is attached to a face. If you want, -you can also have this function create hooks for edges and vertices as well as for -faces. +The data model contains both the topology and the hooks, and functions handle both topological entities and hooks. Consider the case of a box function, which creates a solid with six faces and six hooks. Each hook is attached to a face. If you want, you can also have this function create hooks for edges and vertices as well as for faces. -Not all functions can define explicit hooks for all topological entities they create, -but all topological entities can be turned into hooks when necessary. This is where -topological naming is necessary. +Not all functions can define explicit hooks for all topological entities they create, but all topological entities can be turned into hooks when necessary. This is where topological naming is necessary. -Consider the following example. A box defines six hooks for the six faces, but a -protrusion created on a face of the box can only define two hooks, one for the top -face, and one for all the lateral faces. As the basic wire defining the protrusion -may change in the future the protrusion function cannot designate the lateral faces -without ambiguity, their number may change. Figure 6 illustrates this example, faces -F1 to F6 of the box each have a hook. Faces F7 to F10, the lateral faces of the protrusion, -share a single hook, and face F11, the top face, has one hook. +Consider the following example. A box defines six hooks for the six faces, but a protrusion created on a face of the box can only define two hooks, one for the top face, and one for all the lateral faces. As the basic wire defining the protrusion may change in the future the protrusion function cannot designate the lateral faces without ambiguity, their number may change. Figure 6 illustrates this example, faces F1 to F6 of the box each have a hook. Faces F7 to F10, the lateral faces of the protrusion, share a single hook, and face F11, the top face, has one hook. @image html /technical_overview/images/technical_overview_occ_0068.png @image latex /technical_overview/images/technical_overview_occ_0068.png @@ -2434,10 +2417,8 @@ attribute you are looking for. For this, find this information using the method ### Function Attributes -A model consists of data and algorithms manipulating with data. OCAF attributes store -data. A Function attribute stores data corresponding to a Function (see the white -paper OCAF Function Mechanism User's Guide). This mechanism manipulates with algorithms -computing the model in the optimal way following the modifications. +A model consists of data and algorithms manipulating with data. OCAF attributes store data. A Function attribute stores data corresponding to a Function (see the white paper Application Framework Function Mechanism ). This mechanism manipulates with algorithms computing the model in the optimal way following the modifications. + @subsection OCCT_TOVW_SECTION_7_3 Persistent Data Storage @@ -2525,9 +2506,7 @@ The standard procedure for an application in writing a container is as follows: @subsubsection OCCT_TOVW_SECTION_7_3_2 Persistent Collections -Persistent collections are classes which handle dynamically sized collections of -data that can be stored in the database. These collections provide three categories -of service: +Persistent collections are classes which handle dynamically sized collections of data that can be stored in the database. These collections provide three categories of service: * persistent strings, * generic arrays of data, @@ -2536,47 +2515,26 @@ of service: Persistent strings are concrete classes that handle sequences of characters based on both ASCII (normal 8-bit) and Unicode (16-bit) character sets. -Arrays are generic classes, that is, they can hold a variety of objects not necessarily -inheriting from a unique root class. These arrays can be instantiated with any kind -of storable or persistent object, and then inserted into the persistent data model -of a user application. +Arrays are generic classes, that is, they can hold a variety of objects not necessarily inheriting from a unique root class. These arrays can be instantiated with any kind of storable or persistent object, and then inserted into the persistent data model of a user application. -The purpose of these data collections is simply to convert transient data into its -persistent equivalent so that it can be stored in the database. To this end, the -collections are used to create the persistent data model and assure the link with -the database. They do not provide editing or query capabilities because it is more -efficient, within the operative data model of the application, to work with transient -data structures (from the TCollection package). +The purpose of these data collections is simply to convert transient data into its persistent equivalent so that it can be stored in the database. To this end, the collections are used to create the persistent data model and assure the link with the database. They do not provide editing or query capabilities because it is more efficient, within the operative data model of the application, to work with transient data structures (from the TCollection package). For this reason: * the persistent strings only provide constructors and functions to convert between transient and persistent strings, and * the persistent data collections are limited to arrays. In other words, PCollection does not include sequences, lists, queues, sets, stacks and so on (unlike TCollection). -Persistent string and array classes are found in the PCollection package. -In addition, PColStd package provides standard, -and frequently used, instantiations of persistent arrays, for very simple objects. +Persistent string and array classes are found in the PCollection package. In addition, PColStd package provides standard, and frequently used, instantiations of persistent arrays, for very simple objects. @subsubsection OCCT_TOVW_SECTION_7_3_3 Persistent Geometry -The Persistent Geometry component describes geometric data structures which can be -stored in the database. These packages provide a way to convert data from the transient -"world" to the persistent "world". +The Persistent Geometry component describes geometric data structures which can be stored in the database. These packages provide a way to convert data from the transient "world" to the persistent "world". -Persistent Geometry consists of a set of atomic data models parallel to the geometric -data structures described in the geometry packages. Geometric data models, independent -of each other, can appear within the data model of any application. The system provides -the means to convert each atomic transient data model into a persistent one, but -it does not provide a way for these data models to share data. +Persistent Geometry consists of a set of atomic data models parallel to the geometric data structures described in the geometry packages. Geometric data models, independent of each other, can appear within the data model of any application. The system provides the means to convert each atomic transient data model into a persistent one, but it does not provide a way for these data models to share data. -Consequently, you can create a data model using these components, store data in, -and retrieve it from a file or a database, using the geometric components provided -in the transient and persistent "worlds". In other words, you customize the system -by declaring your own objects, and the conversion of the geometric components from -persistent to transient and vice versa is automatically managed for you by the system. +Consequently, you can create a data model using these components, store data in, and retrieve it from a file or a database, using the geometric components provided in the transient and persistent "worlds". In other words, you customize the system by declaring your own objects, and the conversion of the geometric components from persistent to transient and vice versa is automatically managed for you by the system. -However, these simple objects cannot be shared within a more complex data model. -To allow data to be shared, you must provide additional tools. +However, these simple objects cannot be shared within a more complex data model. To allow data to be shared, you must provide additional tools. Persistent Geometry is provided by several packages. @@ -2924,7 +2882,6 @@ It is recommended to use option *Inherit access rights from parent*. ### Problems with 3D viewer - If the 3D viewer fails to display the scene properly, or works very slowly, or exhibits another problem, make sure to have the latest version of the graphics card driver installed. If this is not possible or does not help, try to decrease diff --git a/dox/user_guides/brep_wp/brep_wp.md b/dox/user_guides/brep_wp/brep_wp.md index 13404b0aac..fd4b706d94 100644 --- a/dox/user_guides/brep_wp/brep_wp.md +++ b/dox/user_guides/brep_wp/brep_wp.md @@ -1,4 +1,4 @@ - BRep Format Description {#occt_brep_format} + BRep Format {#occt_brep_format} ======================== @tableofcontents @@ -18,6 +18,12 @@ * format file fragment to illustrate the part; * BNF-like definition of the part; * detailed description of the part. + +**Note** that the format is a part of Open CASCADE Technology (OCCT). + +Some data fields of the format have additional values, which are used in OCCT. + +Some data fields of the format are specific for OCCT. @section occt_brep_format_2 Format Common Structure @@ -25,53 +31,35 @@ BREP format uses the following BNF terms: - * <\n>; - * <_\n>; - * <_>; - * ; - * ; - * ; - * <2D point>; - * <3D point>; - * <2D direction>; - * <3D direction>; - * <+>; - - * \n is the operating-system-dependent ASCII character sequence which separates ASCII text strings in the operating system used. - * _\n = " "* \n; - * _ = " "+; - * _ is a not empty sequence of space characters with ASCII code 21h. - * flag = "0" | "1"; - * int is an integer number from @image html brep_wp_image003.gif to @image html brep_wp_image004.gif which is written in denary system. - * real is a real from @image html brep_wp_image005.gif to @image html brep_wp_image006.gif  - which is written in decimal or E form with base 10. - - The point is used as a delimiter of the integer and fractional parts. - - * 2D point = real _ real; - * 3D point = real (_ real) ^ 2; - * 2D direction is a 2D point @image html brep_wp_image007.gif so that @image html brep_wp_image008.gif. - * 3D direction is a 3D point @image html brep_wp_image009.gif so that @image html brep_wp_image010.gif. - + is an arithmetic operation of addition. - + * \<\\n\>: It is the operating-system-dependent ASCII character sequence which separates ASCII text strings in the operating system used; + * \<_\\n\>: = " "*\<\\n\>; + * \<_\>: = " "+; It is a not empty sequence of space characters with ASCII code 21h; + * \: = "0" | "1"; + * \: It is an integer number from -231 to 231-1 which is written in denary system; + * \: It is a real from -1.7976931348623158 @f$\cdot@f$ 10308 to 1.7976931348623158 @f$\cdot@f$ 10308 which is written in decimal or E form with base 10.The point is used as a delimiter of the integer and fractional parts; + * \<2D point\>: = \\<_\>\; + * \<3D point\>: = \(\<_\>\2; + * \<2D direction\>: It is a \<2D point\> *x y* so that *x2 + y2* = 1; + * \<3D direction\>: It is a \<3D point\> *x y z* so that *x2 + y2 + z2* = 1; + * \<+\>: It is an arithmetic operation of addition. + The format consists of the following sections: - * content type; - * version; - * locations; - * geometry; - * shapes. + * \; + * \; + * \; + * \; + * \. - content type = "DBRep_DrawableShape" _\n _\n; - content type have other values [1]. + \ = "DBRep_DrawableShape" \<_\\n\>\<_\\n\>; + \ have other values [1]. - version = ("CASCADE Topology V1, (c) Matra-Datavision" | "CASCADE Topology V2, (c) Matra-Datavision") _\n; - + \ = ("CASCADE Topology V1, (c) Matra-Datavision" | "CASCADE Topology V2, (c) Matra-Datavision")\<_\\n\>; The difference of the versions is described in the document. - Sections , and are described below in separate chapters of the document. + Sections \, \ and \ are described below in separate chapters of the document. -@section occt_brep_format_3 Section locations +@section occt_brep_format_3 Locations **Example** @@ -91,66 +79,83 @@ **BNF-like Definition** @verbatim - locations = location header _\n location records; - location header = "Locations" _ location record count; - location record count = int; - location records = location record ^ location record count; - location record = location record 1 | location record 2; - location record 1 = "1" _\n location data 1; - location record 2 = "2" _ location data 2; - location data 1 = ((_ real) ^ 4 _\n) ^ 3; - location data 2 = (int _ int _)* "0" _\n; + = <_\n> ; + = "Locations" <_> ; + = ; + = ^ ; + = | ; + = "1" <_\n> ; + = "2" <_> ; + = ((<_> ) ^ 4 <_\n>) ^ 3; + = ( <_> <_>)* "0" <_\n>; @endverbatim -****Description**** +**Description** -*location data 1* is interpreted as a 3 x 4 matrix @image html brep_wp_image011.gif  +\ is interpreted as a 3 x 4 matrix +@f$Q = +\begin{pmatrix} +{q}_{1,1} &{q}_{1,2} &{q}_{1,3} &{q}_{1,4}\\ +{q}_{2,1} &{q}_{2,2} &{q}_{2,3} &{q}_{2,4}\\ +{q}_{3,1} &{q}_{3,2} &{q}_{3,3} &{q}_{3,4} +\end{pmatrix}@f$  which describes transformation of 3 dimensional space and satisfies the following constraints: + * @f$ d \neq 0@f$ where @f$d = |Q_{2}|@f$ where + @f$ Q_{2} = \begin{pmatrix} + {q}_{1,1} &{q}_{1,2} &{q}_{1,3} &{q}_{1,4}\\ + {q}_{2,1} &{q}_{2,2} &{q}_{2,3} &{q}_{2,4}\\ + {q}_{3,1} &{q}_{3,2} &{q}_{3,3} &{q}_{3,4} + \end{pmatrix}; @f$ + * @f$ Q_{3}^{T} = Q_{3}^{-1}@f$ where @f$Q_{3} = Q_{2}/d^{1/3}. @f$ - * @image html brep_wp_image012.gif where @image html brep_wp_image013.gif where @image html brep_wp_image014.gif; - * @image html brep_wp_image015.gif where @image html brep_wp_image016.gif. - -The transformation transforms a point +The transformation transforms a point (x, y, z) to another point (u, v, w) by the rule: +@f[ \begin{pmatrix} +u \\ v \\ w +\end{pmatrix} = +Q\cdot(x\;y\;z\;1)^{T} = +\begin{pmatrix} +{q}_{1,1}\cdot x +{q}_{1,2}\cdot y +{q}_{1,3}\cdot z +{q}_{1,4}\\ +{q}_{2,1}\cdot x +{q}_{2,2}\cdot y +{q}_{2,3}\cdot z +{q}_{2,4}\\ +{q}_{3,1}\cdot x +{q}_{3,2}\cdot y +{q}_{3,3}\cdot z +{q}_{3,4} +\end{pmatrix} . @f] -@image html brep_wp_image017.gif  +*Q* may be a composition of matrices for the following elementary transformations: + + * parallel translation – + @f$ \begin{pmatrix} + 1 &0 &0 &{q}_{1,4}\\ + 0 &1 &0 &{q}_{2,4}\\ + 0 &0 &1 &{q}_{3,4} + \end{pmatrix}; @f$ + * rotation around an axis with a direction *D(Dx, Dy, Dz)* by an angle @f$ \varphi @f$ – -to another point - -@image html brep_wp_image018.gif  - -by the rule: + @f[ \begin{pmatrix} + D_{x}^{2} \cdot (1-cos(\varphi)) + cos(\varphi) &D_{x} \cdot D_{y} \cdot (1-cos(\varphi)) - D_{z} \cdot sin(\varphi) &D_{x} \cdot D_{z} \cdot (1-cos(\varphi)) + D_{y} \cdot sin(\varphi) &0\\ + D_{x} \cdot D_{y} \cdot (1-cos(\varphi)) + D_{z} \cdot sin(\varphi) &D_{y}^{2} \cdot (1-cos(\varphi)) + cos(\varphi) &D_{y} \cdot D_{z} \cdot (1-cos(\varphi)) - D_{x} \cdot sin(\varphi) &0\\ + D_{x} \cdot D_{z} \cdot (1-cos(\varphi)) - D_{y} \cdot sin(\varphi) &D_{y} \cdot D_{z} \cdot (1-cos(\varphi)) + D_{x} \cdot sin(\varphi) &D_{z}^{2} \cdot (1-cos(\varphi)) + cos(\varphi) &0 + \end{pmatrix}; @f] -@image html brep_wp_image019.gif. -@image html brep_wp_image020.gif  - -may be a composition of matrices for the following elementary transformations: + * scaling – @f$ \begin{pmatrix} s &0 &0 &0\\ 0 &s &0 &0\\ 0 &0 &s &0 \end{pmatrix} @f$ where @f$ S \in (-\infty,\; \infty)/\left \{ 0 \right \}; @f$ + * central symmetry – @f$ \begin{pmatrix} -1 &0 &0 &0\\ 0 &-1 &0 &0\\ 0 &0 &-1 &0 \end{pmatrix}; @f$ + * axis symmetry – @f$ \begin{pmatrix} -1 &0 &0 &0\\ 0 &-1 &0 &0\\ 0 &0 &1 &0 \end{pmatrix}; @f$ + * plane symmetry – @f$ \begin{pmatrix} 1 &0 &0 &0\\ 0 &1 &0 &0\\ 0 &0 &-1 &0 \end{pmatrix}. @f$ - * parallel translation – @image html brep_wp_image021.gif; - * rotation around an axis with a direction @image html brep_wp_image022.gif by an angle @image html brep_wp_image023.gif – - -@image html brep_wp_image024.gif; - - * scaling – @image html brep_wp_image025.gif where @image html brep_wp_image026.gif; - * central symmetry – @image html brep_wp_image027.gif; - * axis symmetry – @image html brep_wp_image028.gif; - * plane symmetry – @image html brep_wp_image029.gif. - -*location data 2* is interpreted as a composition of locations raised to a power and placed above this location data 2 in the section locations. location data 2 is a sequence @image html brep_wp_image030.gif of @image html brep_wp_image031.gif integer pairs @image html brep_wp_image032.gif (@image html brep_wp_image033.gif). flag 0 is the indicator of the sequence end. The sequence is interpreted as a composition @image html brep_wp_image034.gif where @image html brep_wp_image035.gif is a location from @image html brep_wp_image036.gif-th location record in the section locations. location record numbering starts from1. +\ is interpreted as a composition of locations raised to a power and placed above this \ in the section \. \ is a sequence @f$l_{1}p_{1} ... l_{n}p_{n}@f$ of @f$ n \geq 0 @f$ integer pairs @f$ l_{i}p_{i} \; (1 \leq i \leq n) @f$. \ 0 is the indicator of the sequence end. The sequence is interpreted as a composition @f$ L_{l_{1}}^{p_{1}} \cdot ... \cdot L_{l_{n}}^{p_{n}} @f$ where @f$ L_{l_{i}} @f$ is a location from @f$ l_{i} @f$-th \ in the section locations. \ numbering starts from 1. -@section occt_brep_format_4 Section geometry +@section occt_brep_format_4 Geometry @verbatim - geometry = - 2D curves - 3D curves - 3D polygons - polygons on triangulations - surfaces - triangulations; + = + <2D curves> + <3D curves> + <3D polygons> + + + ; @endverbatim -@subsection occt_brep_format_4_1 Subsection 3D curves +@subsection occt_brep_format_4_1 3D curves **Example** @@ -174,27 +179,27 @@ may be a composition of matrices for the following elementary transformations: **BNF-like Definition** @verbatim - 3D curves = 3D curve header _\n 3D curve records; + <3D curves> = <3D curve header> <_\n> <3D curve records>; - 3D curve header = "Curves" _ 3D curve count; + <3D curve header> = "Curves" <_> <3D curve count>; - 3D curve count = int; + <3D curve count> = ; - 3D curve records = 3D curve record ^ 3D curve count; + <3D curve records> = <3D curve record> ^ <3D curve count>; - 3D curve record = - 3D curve record 1 | - 3D curve record 2 | - 3D curve record 3 | - 3D curve record 4 | - 3D curve record 5 | - 3D curve record 6 | - 3D curve record 7 | - 3D curve record 8 | - 3D curve record 9; + <3D curve record> = + <3D curve record 1> | + <3D curve record 2> | + <3D curve record 3> | + <3D curve record 4> | + <3D curve record 5> | + <3D curve record 6> | + <3D curve record 7> | + <3D curve record 8> | + <3D curve record 9>; + @endverbatim - -@subsubsection occt_brep_format_4_1_1 3D curve record 1 – Line +@subsubsection occt_brep_format_4_1_1 Line - \<3D curve record 1\> **Example** @@ -205,19 +210,19 @@ may be a composition of matrices for the following elementary transformations: **BNF-like Definition** @verbatim - 3D curve record 1 = "1" _ 3D point _ 3D direction _\n; + <3D curve record 1> = "1" <_> <3D point> <_> <3D direction> <_\n>; @endverbatim **Description** -3D curve record 1 describes a line. The line data consist of a 3D point @image html brep_wp_image037.gif and a 3D direction @image html brep_wp_image038.gif. The line passes through the point @image html brep_wp_image037.gif, has the direction @image html brep_wp_image038.gif and is defined by the following parametric equation: +\<3D curve record 1\> describes a line. The line data consist of a 3D point *P* and a 3D direction *D*. The line passes through the point *P*, has the direction *D* and is defined by the following parametric equation: -@image html brep_wp_image039.gif, @image html brep_wp_image040.gif. +@f[ C(u)=P+u \cdot D, \; u \in (-\infty,\; \infty). @f] -The **Example** record is interpreted as a line which passes through a point @image html brep_wp_image041.gif, has a direction @image html brep_wp_image042.gif and is defined by the following parametric equation: @image html brep_wp_image043.gif. +The example record is interpreted as a line which passes through a point *P*=(1, 0, 3), has a direction *D*=(0, 1, 0) and is defined by the following parametric equation: @f$ C(u)=(1,0,3)+u \cdot (0,1,0) @f$. -@subsubsection occt_brep_format_4_1_2 3D curve record 2 – Circle +@subsubsection occt_brep_format_4_1_2 Circle - \<3D curve record 2\> **Example** @@ -228,29 +233,29 @@ The **Example** record is interpreted as a line which passes through a point @i **BNF-like Definition** @verbatim - 3D curve record 2 = "2" _ 3D circle center _ 3D circle N _ 3D circle Dx _ 3D circle Dy _ 3D circle radius _\n; + <3D curve record 2> = "2" <_> <3D circle center> <_> <3D circle N> <_> <3D circle Dx> <_> <3D circle Dy> <_> <3D circle radius> <_\n>; - 3D circle center = 3D point; + <3D circle center> = <3D point>; - 3D circle N = 3D direction; + <3D circle N> = <3D direction>; - 3D circle Dx = 3D direction; + <3D circle Dx> = <3D direction>; - 3D circle Dy = 3D direction; + <3D circle Dy> = <3D direction>; - 3D circle radius = real; + <3D circle radius> = ; @endverbatim **Description** -3D curve record 2 describes a circle. The circle data consist of a 3D point @image html brep_wp_image037.gif, pairwise orthogonal 3D directions @image html brep_wp_image044.gif, @image html brep_wp_image045.gif and @image html brep_wp_image046.gif and a non-negative real @image html brep_wp_image047.gif. The circle has a center @image html brep_wp_image037.gif and is located in a plane with a normal @image html brep_wp_image044.gif. The circle has a radius @image html brep_wp_image047.gif and is defined by the following parametric equation: +\<3D curve record 2\> describes a circle. The circle data consist of a 3D point *P*, pairwise orthogonal 3D directions *N*, *Dx* and *Dy* and a non-negative real *r*. The circle has a center *P* and is located in a plane with a normal *N*. The circle has a radius *r* and is defined by the following parametric equation: -@image html brep_wp_image048.gif, @image html brep_wp_image049.gif. +@f[ C(u)=P+r \cdot (cos(u) \cdot D_{x} + sin(u) \cdot D_{y}), \; u \in [o,\;2 \cdot \pi). @f] -The example record is interpreted as a circle which has its center @image html brep_wp_image050.gif, is located in plane with a normal @image html brep_wp_image051.gif. Directions for the circle are @image html brep_wp_image052.gif and @image html brep_wp_image053.gif. The circle has a radius @image html brep_wp_image054.gif and is defined by the following parametric equation: @image html brep_wp_image055.gif. +The example record is interpreted as a circle which has its center *P*=(1, 2, 3), is located in plane with a normal *N*=(0, 0 ,1). Directions for the circle are *Dx*=(1, 0 ,0) and *Dy*=(0, 1 ,0). The circle has a radius *r*=4 and is defined by the following parametric equation: @f$ C(u) = (1,2,3) + 4 \cdot ( cos(u) \cdot(1,0,0) + sin(u) \cdot (0,1,0) ) @f$. -@subsubsection occt_brep_format_4_1_3 3D curve record 3 – Ellipse +@subsubsection occt_brep_format_4_1_3 Ellipse - \<3D curve record 3\> **Example** @@ -261,31 +266,31 @@ The example record is interpreted as a circle which has its center @image html **BNF-like Definition** @verbatim - 3D curve record 3 = "3" _ 3D ellipse center _ 3D ellipse N _ 3D ellipse Dmaj _ 3D ellipse Dmin _ 3D ellipse Rmaj _ 3D ellipse Rmin _\n; + <3D curve record 3> = "3" <_> <3D ellipse center> <_> <3D ellipse N> <_> <3D ellipse Dmaj> <_> <3D ellipse Dmin> <_> <3D ellipse Rmaj> <_> <3D ellipse Rmin> <_\n>; - 3D ellipse center = 3D point; + <3D ellipse center> = <3D point>; - 3D ellipse N = 3D direction; + <3D ellipse N> = <3D direction>; - 3D ellipse Dmaj = 3D direction; + <3D ellipse Dmaj> = <3D direction>; - 3D ellipse Dmin = 3D direction; + <3D ellipse Dmin> = <3D direction>; - 3D ellipse Rmaj = real; + <3D ellipse Rmaj> = ; - 3D ellipse Rmin = real; + <3D ellipse Rmin> = ; @endverbatim **Description** -3D curve record 3 describes an ellipse. The ellipse data consist of a 3D point @image html brep_wp_image037.gif, pairwise orthogonal 3D directions @image html brep_wp_image044.gif, @image html brep_wp_image056.gif and @image html brep_wp_image057.gif and non-negative reals @image html brep_wp_image058.gif and @image html brep_wp_image059.gif so that @image html brep_wp_image060.gif. The ellipse has its center @image html brep_wp_image037.gif, is located in plane with the normal @image html brep_wp_image044.gif, has major and minor axis directions @image html brep_wp_image056.gif and @image html brep_wp_image057.gif, major and minor radii @image html brep_wp_image058.gif and @image html brep_wp_image059.gif and is defined by the following parametric equation: +\<3D curve record 3\> describes an ellipse. The ellipse data consist of a 3D point *P*, pairwise orthogonal 3D directions *N*, *Dmaj* and *Dmin* and non-negative reals *rmaj* and *rmin* so that *rmin* @f$ \leq @f$ *rmaj*. The ellipse has its center *P*, is located in plane with the normal *N*, has major and minor axis directions *Dmaj* and *Dmin*, major and minor radii *rmaj* and *rmin* and is defined by the following parametric equation: -@image html brep_wp_image061.gif, @image html brep_wp_image049.gif. +@f[ C(u)=P+r_{maj} \cdot cos(u) \cdot D_{maj} + r_{min} \cdot sin(u) \cdot D_{min}, u \in [0, 2 \cdot \pi). @f] -The example record is interpreted as an ellipse which has its center @image html brep_wp_image050.gif, is located in plane with a normal @image html brep_wp_image051.gif, has major and minor axis directions @image html brep_wp_image062.gif and @image html brep_wp_image063.gif, major and minor radii @image html brep_wp_image064.gif and @image html brep_wp_image065.gif and is defined by the following parametric equation: @image html brep_wp_image066.gif. +The example record is interpreted as an ellipse which has its center *P*=(1, 2, 3), is located in plane with a normal *N*=(0, 0, 1), has major and minor axis directions *Dmaj*=(1, 0, 0) and *Dmin*=(0, 1, 0), major and minor radii *rmaj*=5 and *rmin*=4 and is defined by the following parametric equation: @f$ C(u) = (1,2,3) + 5 \cdot cos(u) \cdot(1,0,0) + 4 \cdot sin(u) \cdot (0,1,0) @f$. -@subsubsection occt_brep_format_4_1_4 3D curve record 4 – Parabola +@subsubsection occt_brep_format_4_1_4 Parabola - \<3D curve record 4\> **Example** @@ -296,105 +301,105 @@ The example record is interpreted as an ellipse which has its center @image htm **BNF-like Definition** @verbatim - 3D curve record 4 = "4" _ 3D parabola origin _ 3D parabola N _ 3D parabola Dx _ 3D parabola Dy _ 3D parabola focal length _\n; + <3D curve record 4> = "4" <_> <3D parabola origin> <_> <3D parabola N> <_> <3D parabola Dx> <_> <3D parabola Dy> <_> <3D parabola focal length> <_\n>; - 3D parabola origin = 3D point; + <3D parabola origin> = <3D point>; - 3D parabola N = 3D direction; + <3D parabola N> = <3D direction>; - 3D parabola Dx = 3D direction; + <3D parabola Dx> = <3D direction>; - 3D parabola Dy = 3D direction; + <3D parabola Dy> = <3D direction>; - 3D parabola focal length = real; + <3D parabola focal length> = ; @endverbatim **Description** -3D curve record 4 describes a parabola. The parabola data consist of a 3D point @image html brep_wp_image037.gif, pairwise orthogonal 3D directions @image html brep_wp_image044.gif, @image html brep_wp_image045.gif and @image html brep_wp_image046.gif and a non-negative real @image html brep_wp_image067.gif. The parabola is located in plane which passes through the point @image html brep_wp_image037.gif and has the normal @image html brep_wp_image044.gif. The parabola has a focus length @image html brep_wp_image067.gif and is defined by the following parametric equation: +\<3D curve record 4\> describes a parabola. The parabola data consist of a 3D point *P*, pairwise orthogonal 3D directions *N*, *Dx* and *Dy* and a non-negative real *f*. The parabola is located in plane which passes through the point *P* and has the normal *N*. The parabola has a focus length *f* and is defined by the following parametric equation: -@image html brep_wp_image068.gif, @image html brep_wp_image069.gif Ü @image html brep_wp_image070.gif; +@f[ C(u)=P+\frac{u^{2}}{4 \cdot f} \cdot D_{x} + u \cdot D_{y}, u \in (-\infty,\; \infty) \Leftarrow f \neq 0; @f] +@f[ C(u)=P+u \cdot D_{x}, u \in (-\infty,\; \infty) \Leftarrow f = 0\;(degenerated\;case). @f] -@image html brep_wp_image071.gif, @image html brep_wp_image069.gif Ü @image html brep_wp_image072.gif (degenerated case). - -The example record is interpreted as a parabola in plane which passes through a point @image html brep_wp_image073.gif and has a normal @image html brep_wp_image074.gif. Directions for the parabola are @image html brep_wp_image075.gif and @image html brep_wp_image076.gif. The parabola has a focus length @image html brep_wp_image077.gif and is defined by the following parametric equation: @image html brep_wp_image078.gif. +The example record is interpreted as a parabola in plane which passes through a point *P*=(1, 2, 3) and has a normal *N*=(0, 0, 1). Directions for the parabola are *Dx*=(1, 0, 0) and *Dy*=(0, 1, 0). The parabola has a focus length *f*=16 and is defined by the following parametric equation: @f$ C(u) = (1,2,3) + \frac{u^{2}}{64} \cdot (1,0,0) + u \cdot (0,1,0) @f$. -@subsubsection occt_brep_format_4_1_5 3D curve record 5 – Hyperbola +@subsubsection occt_brep_format_4_1_5 Hyperbola - \<3D curve record 5\> **Example** @verbatim 5 1 2 3 0 0 1 1 0 -0 -0 1 0 5 4 -@verbatim +@endverbatim **BNF-like Definition** @verbatim - 3D curve record 5 = "5" _ 3D hyperbola origin _ 3D hyperbola N _ 3D hyperbola Dx _ 3D hyperbola Dy _ 3D hyperbola Kx _ 3D hyperbola Ky _\n; + <3D curve record 5> = "5" <_> <3D hyperbola origin> <_> <3D hyperbola N> <_> <3D hyperbola Dx> <_> <3D hyperbola Dy> <_> <3D hyperbola Kx> <_> <3D hyperbola Ky> <_\n>; - 3D hyperbola origin = 3D point; + <3D hyperbola origin> = <3D point>; - 3D hyperbola N = 3D direction; + <3D hyperbola N> = <3D direction>; - 3D hyperbola Dx = 3D direction; + <3D hyperbola Dx> = <3D direction>; - 3D hyperbola Dy = 3D direction; + <3D hyperbola Dy> = <3D direction>; - 3D hyperbola Kx = real; + <3D hyperbola Kx> = ; - 3D hyperbola Ky = real; + <3D hyperbola Ky> = ; @endverbatim -Descripton +**Descripton** -3D curve record 5 describes a hyperbola. The hyperbola data consist of a 3D point @image html brep_wp_image037.gif, pairwise orthogonal 3D directions @image html brep_wp_image044.gif, @image html brep_wp_image045.gif and @image html brep_wp_image046.gif and non-negative reals @image html brep_wp_image079.gif and @image html brep_wp_image080.gif. The hyperbola is located in plane which passes through the point @image html brep_wp_image037.gif and has the normal @image html brep_wp_image044.gif. The hyperbola is defined by the following parametric equation: +\<3D curve record 5\> describes a hyperbola. The hyperbola data consist of a 3D point *P*, pairwise orthogonal 3D directions *N*, *Dx* and *Dy* and non-negative reals *kx* and *ky*. The hyperbola is located in plane which passes through the point *P* and has the normal *N*. The hyperbola is defined by the following parametric equation: -@image html brep_wp_image081.gif, @image html brep_wp_image069.gif. +@f[ C(u)=P+k_{x} \cdot cosh(u) \cdot D_{x}+k_{y} \cdot sinh(u) \cdot D_{y} , u \in (-\infty,\; \infty). @f] -The example record is interpreted as a hyperbola in plane which passes through a point @image html brep_wp_image073.gif and has a normal @image html brep_wp_image074.gif. Other hyperbola data are @image html brep_wp_image075.gif, @image html brep_wp_image076.gif, @image html brep_wp_image082.gif and @image html brep_wp_image083.gif. The hyperbola is defined by the following parametric equation: @image html brep_wp_image084.gif. +The example record is interpreted as a hyperbola in plane which passes through a point *P*=(1, 2, 3) and has a normal *N*=(0, 0, 1). Other hyperbola data are *Dx*=(1, 0, 0), *Dy*=(0, 1, 0), *kx*=5 and *ky*=4. The hyperbola is defined by the following parametric equation: @f$ C(u) = (1,2,3) + 5 \cdot cosh(u) \cdot (1,0,0) +4 \cdot sinh(u) \cdot (0,1,0) @f$. -@subsubsection occt_brep_format_4_1_6 3D curve record 6 – Bezier Curve +@subsubsection occt_brep_format_4_1_6 Bezier Curve - \<3D curve record 6\> **Example** @verbatim 6 1 2 0 1 0  4 1 -2 0  5 2 3 0  6 -@verbatim +@endverbatim **BNF-like Definition** @verbatim - 3D curve record 6 = "6" _ 3D Bezier rational flag _ 3D Bezier degree 3D Bezier weight poles _\n; - - 3D Bezier rational flag = flag; - - 3D Bezier degree = int; - - 3D Bezier weight poles = (_ 3D Bezier weight pole) ^ (3D Bezier degree + "1"); - - 3D Bezier weight pole = 3D point [_ real]; -@verbatim + <3D curve record 6> = "6" <_> <3D Bezier rational flag> <_> <3D Bezier degree> + <3D Bezier weight poles> <_\n>; + + <3D Bezier rational flag> = ; + + <3D Bezier degree> = ; + + 3D Bezier weight poles> = (<_> <3D Bezier weight pole>) ^ (<3D Bezier degree> <+> "1"); + + <3D Bezier weight pole> = <3D point> [<_> ]; +@endverbatim **Description** -3D curve record 6 describes a Bezier curve. The curve data consist of a rational flag @image html brep_wp_image047.gif, a degree @image html brep_wp_image085.gif and weight poles. +\<3D curve record 6\> describes a Bezier curve. The curve data consist of a rational *r*, a degree @f$ m \leq 25 @f$ and weight poles. -The weight poles are @image html brep_wp_image086.gif 3D points @image html brep_wp_image087.gif if the flag @image html brep_wp_image047.gif is 0. The weight poles are @image html brep_wp_image086.gif pairs @image html brep_wp_image088.gif if flag @image html brep_wp_image047.gif is 1. Here @image html brep_wp_image089.gif is a 3D point and @image html brep_wp_image090.gif is a positive real (@image html brep_wp_image091.gif). @image html brep_wp_image092.gif (@image html brep_wp_image093.gif) if the flag @image html brep_wp_image047.gif is 0. +The weight poles are *m*+1 3D points *B0 ... Bm* if the flag *r* is 0. The weight poles are *m*+1 pairs *B0h0 ... Bmhm* if flag *r* is 1. Here *Bi* is a 3D point and *hi* is a positive real @f$ (0 \leq i \leq m) @f$. @f$ h_{i}=1\; (0 \leq i \leq m) @f$ if the flag *r* is 0. The Bezier curve is defined by the following parametric equation: -@image html brep_wp_image094.gif, @image html brep_wp_image095.gif +@f[ C(u) = \frac{\sum_{i=0}^{m}B_{i} \cdot h_{i} \cdot C_{m}^{i} \cdot u^{i} \cdot (1-u)^{m-i}}{\sum_{i=0}^{m}h_{i} \cdot C_{m}^{i} \cdot u^{i} \cdot (1-u)^{m-i}},\;u \in [0,\; 1] @f] -where @image html brep_wp_image096.gif. +where @f$ 0^{0} \equiv 1 @f$. -The example record is interpreted as a Bezier curve with a rational flag @image html brep_wp_image097.gif, degree @image html brep_wp_image098.gif and weight poles @image html brep_wp_image099.gif, @image html brep_wp_image100.gif, @image html brep_wp_image101.gif, @image html brep_wp_image102.gif and @image html brep_wp_image103.gif, @image html brep_wp_image104.gif. The Bezier curve is defined by the following parametric equation: +The example record is interpreted as a Bezier curve with a rational flag *r*=1, degree *m*=2 and weight poles *B0*=(0, 1, 0), *h0*=4, *B1*=(1, -2, 0), *h1*=5 and *B2*=(2, 3, 0), *h2*=6. The Bezier curve is defined by the following parametric equation: -@image html brep_wp_image105.gif. +@f[ C(u)=\frac{(0,1,0) \cdot 4 \cdot (1-u)^{2}+(1,-2,0) \cdot 5 \cdot 2 \cdot u \cdot (1-u) + (2,3,0) \cdot 6 \cdot u^{2} )}{4 \cdot (1-u)^{2}+5 \cdot 2 \cdot u \cdot (1-u)+6 \cdot u^{2}}. @f] -@subsubsection occt_brep_format_4_1_7 3D curve record 7 – B-spline Curve +@subsubsection occt_brep_format_4_1_7 B-spline Curve - \<3D curve record 7\> **Example** @@ -406,55 +411,59 @@ The example record is interpreted as a Bezier curve with a rational flag @image **BNF-like Definition** @verbatim - 3D curve record 7 = "7" _ 3D B-spline rational flag _ "0" _ 3D B-spline degree _ 3D B-spline pole count _ 3D B-spline multiplicity knot count 3D B-spline weight poles _\n 3D B-spline multiplicity knots _\n; - - 3D B-spline rational flag = flag; - - 3D B-spline degree = int; - - 3D B-spline pole count = int; - - 3D B-spline multiplicity knot count = int; - - 3D B-spline weight poles = (_ 3D B-spline weight pole) ^ 3D B-spline pole count; - - 3D B-spline weight pole = 3D point [_ real]; - - 3D B-spline multiplicity knots = - (_ 3D B-spline multiplicity knot) ^ 3D B-spline multiplicity knot count; - - 3D B-spline multiplicity knot = real _ int; + <3D curve record 7> = "7" <_> <3D B-spline rational flag> <_> "0" <_> <3D B-spline degree> <_> + <3D B-spline pole count> <_> <3D B-spline multiplicity knot count> <3D B-spline weight poles> + <_\n> <3D B-spline multiplicity knots> <_\n>; + + <3D B-spline rational flag> = ; + + <3D B-spline degree> = ; + + <3D B-spline pole count> = ; + + <3D B-spline multiplicity knot count> = ; + + <3D B-spline weight poles> = (<_> <3D B-spline weight pole>) ^ <3D B-spline pole count>; + + <3D B-spline weight pole> = <3D point> [<_> ]; + + <3D B-spline multiplicity knots> = (<_> <3D B-spline multiplicity knot>) ^ <3D B-spline multiplicity knot count>; + + <3D B-spline multiplicity knot> = <_> ; @endverbatim **Description** -3D curve record 7 describes a B-spline curve. The curve data consist of a rational flag @image html brep_wp_image047.gif, a degree @image html brep_wp_image085.gif, pole count @image html brep_wp_image106.gif, multiplicity knot count @image html brep_wp_image107.gif, weight poles and multiplicity knots. +\<3D curve record 7\> describes a B-spline curve. The curve data consist of a rational flag *r*, a degree @f$ m \leq 25 @f$, pole count @f$ n \geq 2 @f$, multiplicity knot count *k*, weight poles and multiplicity knots. -The weight poles are @image html brep_wp_image108.gif 3D points @image html brep_wp_image109.gif if the flag @image html brep_wp_image047.gif is 0. The weight poles are @image html brep_wp_image108.gif pairs @image html brep_wp_image110.gif if the flag @image html brep_wp_image047.gif is 1. Here @image html brep_wp_image089.gif is a 3D point and @image html brep_wp_image090.gif is a positive real (@image html brep_wp_image111.gif). @image html brep_wp_image092.gif (@image html brep_wp_image111.gif) if the flag @image html brep_wp_image047.gif is 0. +The weight poles are *n* 3D points *B1 ... Bn* if the flag *r* is 0. The weight poles are *n* pairs *B1h1 ... Bnhn* if the flag *r* is 1. Here *Bi* is a 3D point and *hi* is a positive real @f$ (1 \leq i \leq n) @f$. @f$ h_{i}=1\; (1 \leq i \leq n) @f$ if the flag *r* is 0. -The multiplicity knots are @image html brep_wp_image107.gif pairs @image html brep_wp_image112.gif. Here @image html brep_wp_image113.gif is a knot with a multiplicity @image html brep_wp_image114.gif (@image html brep_wp_image115.gif) so that +The multiplicity knots are *k* pairs *u1q1 ... ukqk*. Here *ui* is a knot with a multiplicity @f$ q_{i} \geq 1 \; (1 \leq i \leq k) @f$ so that -@image html brep_wp_image116.gif (@image html brep_wp_image117.gif), -@image html brep_wp_image118.gif, @image html brep_wp_image119.gif, @image html brep_wp_image120.gif (@image html brep_wp_image121.gif), @image html brep_wp_image122.gif. +@f[ u_{i} < u_{i+1} (1 \leq i \leq k-1),@f] +@f[ q_{1} \leq m+1,\; q_{k} \leq m+1,\; q_{i} \leq m\; (2 \leq i \leq k-1), \sum_{i=1}^{k}q_{i}=m+n+1. @f] The B-spline curve is defined by the following parametric equation: -@image html brep_wp_image123.gif, @image html brep_wp_image124.gif +@f[ C(u) = \frac{\sum_{i=1}^{n}B_{i} \cdot h_{i} \cdot N_{i,m+1}(u)}{\sum_{i=1}^{n}h_{i} \cdot N_{i,m+1}(u)},\;u \in [u_{1},\; u_{k}] @f] -where functions @image html brep_wp_image125.gif have the following recursion definition by @image html brep_wp_image126.gif +where functions @f$ N_{i,j} @f$ have the following recursion definition by *j*: + +@f[ N_{i,1}(u)=\left\{\begin{matrix} +1\Leftarrow \bar{u}_{i} \leq u \leq \bar{u}_{i+1}\\ +0\Leftarrow u < \bar{u}_{i} \vee \bar{u}_{i+1} \leq u \end{matrix} \right.,\; +N_{i,j}(u)=\frac{(u-\bar{u}_{i}) \cdot N_{i,j-1}(u) }{\bar{u}_{i+j-1}-\bar{u}_{i}}+ \frac{(\bar{u}_{i+j}-u) \cdot N_{i+1,j-1}(u)}{\bar{u}_{i+j}-\bar{u}_{i+1}},\;(2 \leq j \leq m+1) @f] -@image html brep_wp_image127.gif, @image html brep_wp_image128.gif (@image html brep_wp_image129.gif) +where -where +@f[ \bar{u}_{i} = u_{j},\; (1 \leq j \leq k,\; \sum_{l=1}^{j-1}q_{l}+1 \leq i \leq \sum_{l=1}^{j}q_{l} ). @f] -@image html brep_wp_image130.gif (@image html brep_wp_image131.gif, @image html brep_wp_image132.gif). +The example record is interpreted as a B-spline curve with a rational flag *r*=1, a degree *m*=1, pole count *n*=3, multiplicity knot count *k*=5, weight poles *B1*=(0,1,0), *h1*=4, *B2*=(1,-2,0), *h2*=5 and *B3*=(2,3,0), *h3*=6, multiplicity knots *u1*=0, *q1*=1, *u2*=0.25, *q2*=1, *u3*=0.5, *q3*=1, *u4*=0.75, *q4*=1 and *u5*=1, *q5*=1. The B-spline curve is defined by the following parametric equation: -The example record is interpreted as a B-spline curve with a rational flag @image html brep_wp_image097.gif, a degree @image html brep_wp_image133.gif, pole count @image html brep_wp_image134.gif, multiplicity knot count @image html brep_wp_image135.gif, weight poles @image html brep_wp_image136.gif, @image html brep_wp_image137.gif, @image html brep_wp_image138.gif, @image html brep_wp_image139.gif and @image html brep_wp_image140.gif, @image html brep_wp_image141.gif, multiplicity knots @image html brep_wp_image142.gif, @image html brep_wp_image143.gif, @image html brep_wp_image144.gif, @image html brep_wp_image145.gif, @image html brep_wp_image146.gif, @image html brep_wp_image147.gif, @image html brep_wp_image148.gif, @image html brep_wp_image149.gif and @image html brep_wp_image150.gif, @image html brep_wp_image151.gif. The B-spline curve is defined by the following parametric equation: - -@image html brep_wp_image152.gif. +@f[ C(u)=\frac{(0,1,0) \cdot 4 \cdot N_{1,2}(u) + (1,-2,0) \cdot 5 \cdot N_{2,2}(u)+(2,3,0) \cdot 6 \cdot N_{3,2}(u)}{4 \cdot N_{1,2}(u)+5 \cdot N_{2,2}(u)+6 \cdot N_{3,2}(u)}. @f] -@subsubsection occt_brep_format_4_1_8 3D curve record 8 – Trimmed Curve +@subsubsection occt_brep_format_4_1_8 Trimmed Curve - \<3D curve record 8\> **Example** @@ -466,23 +475,23 @@ The example record is interpreted as a B-spline curve with a rational flag @ima **BNF-like Definition** @verbatim - 3D curve record 8 = "8" _ 3D trimmed curve u min _ 3D trimmed curve u max _\n 3D curve record; - - 3D trimmed curve u min = real; - - 3D trimmed curve u max = real; + <3D curve record 8> = "8" <_> <3D trimmed curve u min> <_> <3D trimmed curve u max> <_\n> <3D curve record>; + + <3D trimmed curve u min> = ; + + <3D trimmed curve u max> = ; @endverbatim **Description** -3D curve record 8 describes a trimmed curve. The trimmed curve data consist of reals @image html brep_wp_image153.gif and @image html brep_wp_image154.gif and 3D curve record so that @image html brep_wp_image155.gif. The trimmed curve is a restriction of the base curve @image html brep_wp_image156.gif described in the record to the segment @image html brep_wp_image157.gif. The trimmed curve is defined by the following parametric equation: +\<3D curve record 8\> describes a trimmed curve. The trimmed curve data consist of reals *umin* and *umax* and \<3D curve record\> so that *umin* < *umax*. The trimmed curve is a restriction of the base curve *B* described in the record to the segment @f$ [u_{min},\;u_{max}]\subseteq domain(B) @f$. The trimmed curve is defined by the following parametric equation: -@image html brep_wp_image158.gif, @image html brep_wp_image159.gif. +@f[ C(u)=B(u),\; u \in [u_{min},\;u_{max}]. @f] -The example record is interpreted as a trimmed curve with @image html brep_wp_image160.gif and @image html brep_wp_image161.gif for the base curve @image html brep_wp_image162.gif. The trimmed curve is defined by the following parametric equation: @image html brep_wp_image163.gif, @image html brep_wp_image164.gif. +The example record is interpreted as a trimmed curve with *umin*=-4 and *umax*=5 for the base curve @f$ B(u)=(1,2,3)+u \cdot (1,0,0) @f$. The trimmed curve is defined by the following parametric equation: @f$ C(u)=(1,2,3)+u \cdot (1,0,0),\; u \in [-4,\; 5] @f$. -@subsubsection occt_brep_format_4_1_9 3D curve record 9 – Offset Curve +@subsubsection occt_brep_format_4_1_9 Offset Curve - \<3D curve record 9\> **Example** @@ -495,24 +504,24 @@ The example record is interpreted as a trimmed curve with @image html brep_wp_i **BNF-like Definition** @verbatim - 3D curve record 9 = "9" _ 3D offset curve distance _\n - 3D offset curve direction _\n - 3D curve record; - - 3D offset curve distance = real; - - 3D offset curve direction = 3D direction; + <3D curve record 9> = "9" <_> <3D offset curve distance> <_\n>; + <3D offset curve direction> <_\n>; + <3D curve record>; + + <3D offset curve distance> = ; + + <3D offset curve direction> = <3D direction>; @endverbatim **Description** -3D curve record 9 describes an offset curve. The offset curve data consist of a distance @image html brep_wp_image165.gif, a 3D direction @image html brep_wp_image038.gif and a 3D curve record. The offset curve is the result of offsetting the base curve @image html brep_wp_image156.gif described in the record to the distance @image html brep_wp_image165.gif along the vector @image html brep_wp_image166.gif. The offset curve is defined by the following parametric equation: +\<3D curve record 9\> describes an offset curve. The offset curve data consist of a distance *d*, a 3D direction *D* and a \<3D curve record\>. The offset curve is the result of offsetting the base curve *B* described in the record to the distance *d* along the vector @f$ [B'(u),\; D] \neq \vec{0} @f$. The offset curve is defined by the following parametric equation: -@image html brep_wp_image167.gif, @image html brep_wp_image168.gif. +@f[ C(u)=B(u)+d \cdot \frac{[B'(u),\; D]}{|[B'(u),\; D]|},\; u \in domain(B) . @f] -The example record is interpreted as an offset curve with a distance @image html brep_wp_image169.gif, direction @image html brep_wp_image170.gif, base curve @image html brep_wp_image162.gif and defined by the following parametric equation: @image html brep_wp_image171.gif. +The example record is interpreted as an offset curve with a distance *d*=2, direction *D*=(0, 1, 0), base curve @f$ B(u)=(1,2,3)+u \cdot (1,0,0) @f$ and defined by the following parametric equation: @f$ C(u)=(1,2,3)+u \cdot (1,0,0)+2 \cdot (0,0,1) @f$. -@subsection occt_brep_format_4_2 Subsection surfaces +@subsection occt_brep_format_4_2 Surfaces **Example** @@ -529,28 +538,28 @@ The example record is interpreted as an offset curve with a distance @image htm **BNF-like Definition** @verbatim - surfaces = surface header _\n surface records; - - surface header = "Surfaces" _ surface count; - - surface records = surface record ^ surface count; - - surface record = - surface record 1 | - surface record 2 | - surface record 3 | - surface record 4 | - surface record 5 | - surface record 6 | - surface record 7 | - surface record 8 | - surface record 9 | - surface record 10 | - surface record 11; -@verbatim - -@subsubsection occt_brep_format_4_2_1 surface record 1 – Plane + = <_\n> ; + + = “Surfaces” <_> ; + + = ^ ; + + = + | + | + | + | + | + | + | + | + | + | + ; +@endverbatim +@subsubsection occt_brep_format_4_2_1 Plane - \< surface record 1 \> + **Example** @verbatim @@ -560,19 +569,19 @@ The example record is interpreted as an offset curve with a distance @image htm **BNF-like Definition** @verbatim -surface record 1 = "1" _ 3D point (_ 3D direction) ^ 3 _\n; + = "1" <_> <3D point> (<_> <3D direction>) ^ 3 <_\n>; @endverbatim **Description** -surface record 1 describes a plane. The plane data consist of a 3D point @image html brep_wp_image037.gif and pairwise orthogonal 3D directions @image html brep_wp_image044.gif, @image html brep_wp_image172.gif and @image html brep_wp_image173.gif. The plane passes through the point @image html brep_wp_image037.gif, has the normal @image html brep_wp_image174.gif and is defined by the following parametric equation: +\ describes a plane. The plane data consist of a 3D point *P* and pairwise orthogonal 3D directions *N*, *Du* and *Dv*. The plane passes through the point *P*, has the normal *N* and is defined by the following parametric equation: -@image html brep_wp_image175.gif, @image html brep_wp_image176.gif. +@f[ S(u,v)=P+u \cdot D_{u}+v \cdot D_{v},\; (u,\;v) \in (-\infty,\; \infty) \times (-\infty,\; \infty). @f] -The example record is interpreted as a plane which passes through a point @image html brep_wp_image177.gif, has a normal @image html brep_wp_image074.gif and is defined by the following parametric equation: @image html brep_wp_image178.gif. +The example record is interpreted as a plane which passes through a point *P*=(0, 0, 3), has a normal *N*=(0, 0, 1) and is defined by the following parametric equation: @f$ S(u,v)=(0,0,3)+u \cdot (1,0,0) + v \cdot (0,1,0) @f$. -@subsubsection occt_brep_format_4_2_2 surface record 2 – Cylinder +@subsubsection occt_brep_format_4_2_2 Cylinder - \< surface record 2 \> **Example** @@ -583,19 +592,19 @@ The example record is interpreted as a plane which passes through a point @imag **BNF-like Definition** @verbatim - surface record 2 = "2" _ 3D point (_ 3D direction) ^ 3 _ real _\n; + = "2" <_> <3D point> (<_> <3D direction>) ^ 3 <_> <_\n>; @endverbatim **Description** -surface record 2 describes a cylinder. The cylinder data consist of a 3D point @image html brep_wp_image037.gif, pairwise orthogonal 3D directions @image html brep_wp_image173.gif, @image html brep_wp_image045.gif and @image html brep_wp_image046.gif and a non-negative real @image html brep_wp_image047.gif. The cylinder axis passes through the point @image html brep_wp_image037.gif and has the direction @image html brep_wp_image173.gif. The cylinder has the radius @image html brep_wp_image179.gif and is defined by the following parametric equation: +\ describes a cylinder. The cylinder data consist of a 3D point *P*, pairwise orthogonal 3D directions *Dv*, *DX* and *DY* and a non-negative real *r*. The cylinder axis passes through the point *P* and has the direction *Dv*. The cylinder has the radius *r* and is defined by the following parametric equation: -@image html brep_wp_image180.gif, @image html brep_wp_image181.gif. +@f[ S(u,v)=P+r \cdot (cos(u) \cdot D_{x}+sin(u) \cdot D_{y} )+v \cdot D_{v},\; (u,v) \in [0,\; 2 \cdot \pi) \times (-\infty,\; \infty) . @f] -The example record is interpreted as a cylinder which axis passes through a point @image html brep_wp_image073.gif and has a direction @image html brep_wp_image182.gif. Directions for the cylinder are @image html brep_wp_image075.gif and @image html brep_wp_image076.gif. The cylinder has a radius @image html brep_wp_image054.gif and is defined by the following parametric equation: @image html brep_wp_image183.gif. +The example record is interpreted as a cylinder which axis passes through a point *P*=(1, 2, 3) and has a direction *Dv*=(0, 0, 1). Directions for the cylinder are *DX*=(1,0,0) and *DY*=(0,1,0). The cylinder has a radius *r*=4 and is defined by the following parametric equation: @f$ S(u,v)=(1,2,3)+4 \cdot ( cos(u) \cdot D_{X} + sin(u) \cdot D_{Y} ) + v \cdot D_{v}. @f$ -@subsubsection occt_brep_format_4_2_3 surface record 3 – Cone +@subsubsection occt_brep_format_4_2_3 Cone - \< surface record 3 \> **Example** @@ -608,20 +617,20 @@ The example record is interpreted as a cylinder which axis passes through a poi **BNF-like Definition** @verbatim - surface record 3 = "3" _ 3D point (_ 3D direction) ^ 3 (_ real) ^ 2 _\n; + = "3" <_> <3D point> (<_> <3D direction>) ^ 3 (<_> ) ^ 2 <_\n>; @endverbatim **Description** -surface record 3 describes a cone. The cone data consist of a 3D point @image html brep_wp_image037.gif, pairwise orthogonal 3D directions @image html brep_wp_image184.gif, @image html brep_wp_image045.gif and @image html brep_wp_image046.gif, a non-negative real @image html brep_wp_image047.gif and a real @image html brep_wp_image185.gif. The cone axis passes through the point @image html brep_wp_image037.gif and has the direction @image html brep_wp_image184.gif. The plane which passes through the point @image html brep_wp_image037.gif and is parallel to directions @image html brep_wp_image045.gif and @image html brep_wp_image046.gif is the cone referenced plane. The cone section by the plane is a circle with the radius @image html brep_wp_image047.gif. The direction from the point @image html brep_wp_image037.gif to the cone apex is @image html brep_wp_image186.gif. The cone has a half-angle @image html brep_wp_image187.gif and is defined by the following parametric equation: +\ describes a cone. The cone data consist of a 3D point *P*, pairwise orthogonal 3D directions *DZ*, *DX* and *DY*, a non-negative real *r* and a real @f$ \varphi \in (-\pi /2,\; \pi/2)/\left \{ 0 \right \} @f$. The cone axis passes through the point *P* and has the direction *DZ*. The plane which passes through the point *P* and is parallel to directions *DX* and *DY* is the cone referenced plane. The cone section by the plane is a circle with the radius *r*. The direction from the point *P* to the cone apex is @f$ -sgn(\varphi) \cdot D_{Z} @f$. The cone has a half-angle @f$| \varphi | @f$ and is defined by the following parametric equation: -@image html brep_wp_image188.gif, @image html brep_wp_image189.gif. +@f[ S(u,v)=P+(r+v \cdot sin(\varphi)) \cdot (cos(u) \cdot D_{X}+sin(u) \cdot D_{Y})+v \cdot cos(\varphi) \cdot D_{Z}, (u,v) \in [0,\; 2 \cdot \pi) \times (-\infty,\; \infty) . @f] -The example record is interpreted as a cone with an axis which passes through a point @image html brep_wp_image073.gif and has a direction @image html brep_wp_image190.gif. Other cone data are @image html brep_wp_image075.gif, @image html brep_wp_image076.gif, @image html brep_wp_image191.gif and @image html brep_wp_image192.gif. The cone is defined by the following parametric equation: -@image html brep_wp_image193.gif. +The example record is interpreted as a cone with an axis which passes through a point *P*=(1, 2, 3) and has a direction *DZ*=(0, 0, 1). Other cone data are *DX*=(1, 0, 0), *DY*=(0, 1, 0), *r*=4 and @f$ \varphi = 0.75 @f$. The cone is defined by the following parametric equation: +@f[ S(u,v)=(1,2,3)+( 4 + v \cdot sin(0.75)) \cdot ( cos(u) \cdot (1,0,0) + sin(u) \cdot (0,1,0) ) + v \cdot cos(0.75) \cdot (0,0,1) . @f] -@subsubsection occt_brep_format_4_2_4 surface record 4 – Sphere +@subsubsection occt_brep_format_4_2_4 Sphere - \< surface record 4 \> **Example** @@ -632,20 +641,20 @@ The example record is interpreted as a cone with an axis which passes through a **BNF-like Definition** @verbatim - surface record 4 = "4" _ 3D point (_ 3D direction) ^ 3 _ real _\n; + = "4" <_> <3D point> (<_> <3D direction>) ^ 3 <_> <_\n>; @endverbatim **Description** -surface record 4 describes a sphere. The sphere data consist of a 3D point @image html brep_wp_image037.gif, pairwise orthogonal 3D directions @image html brep_wp_image184.gif, @image html brep_wp_image045.gif and @image html brep_wp_image046.gif and a non-negative real @image html brep_wp_image047.gif. The sphere has the center @image html brep_wp_image194.gif, radius @image html brep_wp_image179.gif and is defined by the following parametric equation: +\ describes a sphere. The sphere data consist of a 3D point *P*, pairwise orthogonal 3D directions *DZ*, *DX* and *DY* and a non-negative real *r*. The sphere has the center *P*, radius *r* and is defined by the following parametric equation: -@image html brep_wp_image195.gif, @image html brep_wp_image196.gif. +@f[ S(u,v)=P+r \cdot cos(v) \cdot (cos(u) \cdot D_{x}+sin(u) \cdot D_{y} ) +r \cdot sin(v) \cdot D_{Z},\; (u,v) \in [0,\;2 \cdot \pi) \times [-\pi /2,\; \pi /2] . @f] -The example record is interpreted as a sphere with its center @image html brep_wp_image073.gif. Directions for the sphere are @image html brep_wp_image190.gif, @image html brep_wp_image075.gif and @image html brep_wp_image076.gif. The sphere has a radius @image html brep_wp_image191.gif and is defined by the following parametric equation: -@image html brep_wp_image197.gif. +The example record is interpreted as a sphere with its center *P*=(1, 2, 3). Directions for the sphere are *DZ*=(0, 0, 1), *DX*=(1, 0, 0) and *DY*=(0, 1, 0). The sphere has a radius *r*=4 and is defined by the following parametric equation: +@f[ S(u,v)=(1,2,3)+ 4 \cdot cos(v) \cdot ( cos(u) \cdot (1,0,0) + sin(u) \cdot (0,1,0) ) + 4 \cdot sin(v) \cdot (0,0,1) . @f] -@subsubsection occt_brep_format_4_2_5 surface record 5 – Torus +@subsubsection occt_brep_format_4_2_5 Torus - \< surface record 5 \> **Example** @@ -656,20 +665,20 @@ The example record is interpreted as a sphere with its center @image html brep_ **BNF-like Definition** @verbatim - surface record 5 = "5" _ 3D point (_ 3D direction) ^ 3 (_ real) ^ 2 _\n; + = "5" <_> <3D point> (<_> <3D direction>) ^ 3 (<_> ) ^ 2 <_\n>; @endverbatim **Description** -surface record 5 describes a torus. The torus data consist of a 3D point @image html brep_wp_image037.gif, pairwise orthogonal 3D directions @image html brep_wp_image184.gif, @image html brep_wp_image045.gif and @image html brep_wp_image046.gif and non-negative reals @image html brep_wp_image198.gif and @image html brep_wp_image199.gif. The torus axis passes through the point @image html brep_wp_image037.gif and has the direction @image html brep_wp_image184.gif. @image html brep_wp_image198.gif is the distance from the torus circle center to the axis. The torus circle has the radius @image html brep_wp_image199.gif. The torus is defined by the following parametric equation: +\ describes a torus. The torus data consist of a 3D point *P*, pairwise orthogonal 3D directions *DZ*, *DX* and *DY* and non-negative reals *r1* and *r2*. The torus axis passes through the point *P* and has the direction *DZ*. *r1* is the distance from the torus circle center to the axis. The torus circle has the radius *r2*. The torus is defined by the following parametric equation: -@image html brep_wp_image200.gif, @image html brep_wp_image201.gif. +@f[ S(u,v)=P+(r_{1}+r_{2} \cdot cos(v)) \cdot (cos(u) \cdot D_{x}+sin(u) \cdot D_{y} ) +r_{2} \cdot sin(v) \cdot D_{Z},\; (u,v) \in [0,\;2 \cdot \pi) \times [0,\; 2 \cdot \pi) . @f] -The example record is interpreted as a torus with an axis which passes through a point @image html brep_wp_image073.gif and has a direction @image html brep_wp_image190.gif. @image html brep_wp_image075.gif, @image html brep_wp_image076.gif, @image html brep_wp_image202.gif and @image html brep_wp_image203.gif for the torus. The torus is defined by the following parametric equation: -@image html brep_wp_image204.gif. +The example record is interpreted as a torus with an axis which passes through a point *P*=(1, 2, 3) and has a direction *DZ*=(0, 0, 1). *DX*=(1, 0, 0), *DY*=(0, 1, 0), *r1*=8 and *r2*=4 for the torus. The torus is defined by the following parametric equation: +@f[ S(u,v)=(1,2,3)+ (8+4 \cdot cos(v)) \cdot ( cos(u) \cdot (1,0,0) + sin(u) \cdot (0,1,0) ) + 4 \cdot sin(v) \cdot (0,0,1) . @f] -@subsubsection occt_brep_format_4_2_6 surface record 6 – Linear Extrusion +@subsubsection occt_brep_format_4_2_6 Linear Extrusion - \< surface record 6 \> **Example** @@ -681,19 +690,21 @@ The example record is interpreted as a torus with an axis which passes through **BNF-like Definition** @verbatim - surface record 6 = "6" _ 3D direction _\n 3D curve record; + = "6" <_> <3D direction> <_\n> <3D curve record>; @endverbatim **Description** -surface record 6 describes a linear extrusion surface. The surface data consist of a 3D direction @image html brep_wp_image173.gif and a 3D curve record. The linear extrusion surface has the direction @image html brep_wp_image173.gif, the base curve @image html brep_wp_image205.gif described in the record and is defined by the following parametric equation: +\ describes a linear extrusion surface. The surface data consist of a 3D direction *Dv* and a \<3D curve record\>. The linear extrusion surface has the direction *Dv*, the base curve *C* described in the record and is defined by the following parametric equation: -@image html brep_wp_image206.gif, @image html brep_wp_image207.gif. +@f[ S(u,v)=C(u)+v \cdot D_{v},\; (u,v) \in domain(C) \times (-\infty,\; \infty) . @f] -The example record is interpreted as a linear extrusion surface with a direction @image html brep_wp_image208.gif. The base curve is a circle for the surface. The surface is defined by the following parametric equation: @image html brep_wp_image209.gif, @image html brep_wp_image189.gif. +The example record is interpreted as a linear extrusion surface with a direction *Dv*=(0, 0.6, 0.8). The base curve is a circle for the surface. The surface is defined by the following parametric equation: + +@f[ S(u,v)=(1,2,3)+4 \cdot (cos(u) \cdot (1,0,0)+sin(u) \cdot (0,1,0))+v \cdot (0, 0.6, 0.8),\; (u,v) \in [0,\; 2 \cdot \pi) \times (-\infty,\; \infty). @f] -@subsubsection occt_brep_format_4_2_7 surface record 7 – Revolution Surface +@subsubsection occt_brep_format_4_2_7 Revolution Surface - \< surface record 7 \> **Example** @@ -705,22 +716,25 @@ The example record is interpreted as a linear extrusion surface with a directio **BNF-like Definition** @verbatim - surface record 7 = "7" _ 3D point _ 3D direction _\n 3D curve record; + = "7" <_> <3D point> <_> <3D direction> <_\n> <3D curve record>; @endverbatim **Description** -surface record 7 describes a revolution surface. The surface data consist of a 3D point @image html brep_wp_image037.gif, a 3D direction @image html brep_wp_image038.gif and a 3D curve record. The surface axis passes through the point @image html brep_wp_image037.gif and has the direction @image html brep_wp_image038.gif. The base curve @image html brep_wp_image210.gif described by the record and the axis are coplanar. The surface is defined by the following parametric equation: +\ describes a revolution surface. The surface data consist of a 3D point *P*, a 3D direction *D* and a \<3D curve record\>. The surface axis passes through the point *P* and has the direction *D*. The base curve *C* described by the record and the axis are coplanar. The surface is defined by the following parametric equation: -@image html brep_wp_image211.gif, @image html brep_wp_image212.gif +@f[ S(u,v)= P+V_{D}(v)+cos(u) \cdot (V(v)-V_{D}(v))+sin(u) \cdot [D,V(v)],\;(u,v) \in [0,\; 2 \cdot \pi)\times domain(C) @f] -where @image html brep_wp_image213.gif, @image html brep_wp_image214.gif. +where @f$ V(v)=C(v)-P, V_{D}(v)=(D,V(v)) \cdot D @f$. -The example record is interpreted as a revolution surface with an axis which passes through a point @image html brep_wp_image215.gif and has a direction @image html brep_wp_image170.gif. The base curve is a circle for the surface. The surface is defined by the following parametric equation: -@image html brep_wp_image216.gif, @image html brep_wp_image201.gif where @image html brep_wp_image217.gif, @image html brep_wp_image218.gif. +The example record is interpreted as a revolution surface with an axis which passes through a point *P*=(-4, 0, 3) and has a direction *D*=(0, 1, 0). The base curve is a circle for the surface. The surface is defined by the following parametric equation: + +@f[ S(u,v)= (-4,0,3)+V_{D}(v)+cos(u) \cdot (V(v)-V_{D}(v))+sin(u) \cdot [(0,1,0),V(v)],\;(u,v) \in [0,\; 2 \cdot \pi)\times [0,\; 2 \cdot \pi) @f] + +where @f$ V(v)=(5,2,0)+4 \cdot (cos(v) \cdot (1,0,0)+sin(v) \cdot (0,1,0)), V_{D}(v)=((0,1,0),V(v)) \cdot (0,1,0) @f$. -@subsubsection occt_brep_format_4_2_8 surface record 8 – Bezier Surface +@subsubsection occt_brep_format_4_2_8 Bezier Surface - \< surface record 8 \> **Example** @@ -733,43 +747,49 @@ The example record is interpreted as a revolution surface with an axis which pa **BNF-like Definition** @verbatim - surface record 8 = "8" _ Bezier surface u rational flag _ Bezier surface v rational flag _ Bezier surface u degree _ Bezier surface v degree _ Bezier surface weight poles; - - Bezier surface u rational flag = flag; - - Bezier surface v rational flag = flag; - - Bezier surface u degree = int; - - Bezier surface v degree = int; - - Bezier surface weight poles = - (Bezier surface weight pole group _\n) ^ (Bezier surface u degree + "1"); - - Bezier surface weight pole group = Bezier surface weight pole - (_ Bezier surface weight pole) ^ Bezier surface v degree; - - Bezier surface weight pole = 3D point [_ real]; - @endverbatim + = "8" <_> <_> <_> <_> <_> + ; + + = ; + + = ; + + = ; + + = ; + + = + ( <_\n>) ^ ( <+> "1"); + + = + (<_> ) ^ ; + + = <3D point> [<_> ]; +@endverbatim **Description** -surface record 8 describes a Bezier surface. The surface data consist of a u rational flag @image html brep_wp_image219.gif, v rational flag @image html brep_wp_image220.gif, u degree @image html brep_wp_image221.gif, v degree @image html brep_wp_image222.gif and weight poles. +\ describes a Bezier surface. The surface data consist of a u rational flag *ru*, v rational flag *rv*, u degree @f$ m_{u} \leq 25 @f$, v degree @f$ m_{v} \leq 25 @f$ and weight poles. -The weight poles are @image html brep_wp_image223.gif 3D points @image html brep_wp_image224.gif (@image html brep_wp_image225.gif) if @image html brep_wp_image226.gif. The weight poles are @image html brep_wp_image223.gif pairs @image html brep_wp_image227.gif (@image html brep_wp_image225.gif) if @image html brep_wp_image228.gif. Here @image html brep_wp_image224.gif is a 3D point and @image html brep_wp_image229.gif is a positive real (@image html brep_wp_image225.gif). @image html brep_wp_image230.gif (@image html brep_wp_image225.gif) if @image html brep_wp_image226.gif. +The weight poles are @f$ (m_{u}+1) \cdot (m_{v}+1) @f$ 3D points @f$ B_{i,j}\; ((i,j) \in \left \{ 0,...,m_{u} \right \} \times \left \{ 0,...,m_{v} \right \}) @f$ if @f$ r_{u}+r_{v}=0 @f$. The weight poles are @f$ (m_{u}+1) \cdot (m_{v}+1) @f$ pairs @f$ B_{i,j}h_{i,j}\; ((i,j) \in \left \{ 0,...,m_{u} \right \} \times \left \{ 0,...,m_{v} \right \}) @f$ if @f$ r_{u}+r_{v} \neq 0 @f$. Here @f$ B_{i,j} @f$ is a 3D point and @f$ h_{i,j} @f$ is a positive real @f$ ((i,j) \in \left \{ 0,...,m_{u} \right \} \times \left \{ 0,...,m_{v} \right \}) @f$. @f$ h_{i,j}=1\; ((i,j) \in \left \{ 0,...,m_{u} \right \} \times \left \{ 0,...,m_{v} \right \}) @f$ if @f$ r_{u}+r_{v} = 0 @f$. The Bezier surface is defined by the following parametric equation: -@image html brep_wp_image231.gif, @image html brep_wp_image232.gif +@f[ S(u,v)=\frac{\sum_{i=0}^{m_{u}} \sum_{j=0}^{m_{v}} B_{i,j} \cdot h_{i,j} \cdot C_{m_{u}}^{i} \cdot u^{i} \cdot (1-u)^{m_{u}-i} \cdot C_{m_{v}}^{j} \cdot v^{j} \cdot (1-v)^{m_{v}-j}}{\sum_{i=0}^{m_{u}} \sum_{j=0}^{m_{v}} h_{i,j} \cdot C_{m_{u}}^{i} \cdot u^{i} \cdot (1-u)^{m_{u}-i} \cdot C_{m_{v}}^{j} \cdot v^{j} \cdot (1-v)^{m_{v}-j}}, (u,v) \in [0,1] \times [0,1] @f] -where @image html brep_wp_image096.gif. +where @f$ 0^{0} \equiv 1 @f$. -The example record is interpreted as a Bezier surface with a u rational flag @image html brep_wp_image233.gif, v rational flag @image html brep_wp_image234.gif, u degree @image html brep_wp_image235.gif, v degree @image html brep_wp_image236.gif, weight poles @image html brep_wp_image237.gif, @image html brep_wp_image238.gif, @image html brep_wp_image239.gif, @image html brep_wp_image240.gif, @image html brep_wp_image241.gif, @image html brep_wp_image242.gif, @image html brep_wp_image243.gif, @image html brep_wp_image244.gif, @image html brep_wp_image245.gif, @image html brep_wp_image246.gif and @image html brep_wp_image247.gif, @image html brep_wp_image248.gif. The surface is defined by the following parametric equation: +The example record is interpreted as a Bezier surface with a u rational flag *ru*=1, v rational flag *rv*=1, u degree *mu*=2, v degree *mv*=1, weight poles *B0,0*=(0, 0, 1), *h0,0*=7, *B0,1*=(1, 0, -4), *h0,1*=10, *B1,0*=(0, 1, -2), *h1,0*=8, *B1,1*=(1, 1, 5), *h1,1*=11, *B2,0*=(0, 2, 3), *h2,0*=9 and *B2,1*=(1, 2, 6), *h2,1*=12. The surface is defined by the following parametric equation: -@image html brep_wp_image249.gif +@f[ S(u,v)= [ (0,0,1) \cdot 7 \cdot (1-u)^{2} \cdot (1-v)+(1,0,-4) \cdot 10 \cdot (1-u)^{2} \cdot v+ \\ +(0,1,-2) \cdot 8 \cdot 2 \cdot u \cdot (1-u) \cdot (1-v)+(1,1,5) \cdot 11 \cdot 2 \cdot u \cdot (1-u) \cdot v+ \\ +(0,2,3) \cdot 9 \cdot u^{2} \cdot (1-v)+(1,2,6) \cdot 12 \cdot u^{2} \cdot v] \div \\ +[7 \cdot (1-u)^{2} \cdot (1-v)+10 \cdot (1-u)^{2} \cdot v+ \\ +8 \cdot 2 \cdot u \cdot (1-u) \cdot (1-v)+11 \cdot 2 \cdot u \cdot (1-u) \cdot v+ \\ +9 \cdot u^{2} \cdot (1-v)+12 \cdot u^{2} \cdot v ] @f] -@subsubsection occt_brep_format_4_2_9 surface record 9 – B-spline Surface +@subsubsection occt_brep_format_4_2_9 B-spline Surface - \< surface record 9 \> **Example** @@ -793,80 +813,94 @@ The example record is interpreted as a Bezier surface with a u rational flag @i **BNF-like Definition** @verbatim - surface record 9 = "9" _ B-spline surface u rational flag _ B-spline surface v rational flag _ "0" _ "0" _ B-spline surface u degree _ B-spline surface v degree _ B-spline surface u pole count _ B-spline surface v pole count _ B-spline surface u multiplicity knot count _ B-spline surface v multiplicity knot count _ B-spline surface weight poles _\n B-spline surface u multiplicity knots _\n B-spline surface v multiplicity knots; - - B-spline surface u rational flag = flag; - - B-spline surface v rational flag = flag; - - B-spline surface u degree = int; - - B-spline surface v degree = int; - - B-spline surface u pole count = int; - - B-spline surface v pole count = int; - - B-spline surface u multiplicity knot count = int; - - B-spline surface v multiplicity knot count = int; - - B-spline surface weight poles = - (B-spline surface weight pole group _\n) ^ B-spline surface u pole count; - - B-spline surface weight pole group = - (B-spline surface weight pole _) ^ B-spline surface v pole count; - - B-spline surface weight pole = 3D point [_ real]; - - B-spline surface u multiplicity knots = - (B-spline surface u multiplicity knot _\n) ^ B-spline surface u multiplicity knot count; - - B-spline surface u multiplicity knot = real _ int; - - B-spline surface v multiplicity knots = - (B-spline surface v multiplicity knot _\n) ^ B-spline surface v multiplicity knot count; - - B-spline surface v multiplicity knot = real _ int; + = "9" <_> <_> + <_> "0" <_> "0" <_> <_> + <_> <_> + <_> <_> + <_> <_\n> + <_\n> ; + + = ; + + = ; + + = ; + + = ; + + = ; + + = ; + + = ; + + = ; + + = + ( <_\n>) ^ ; + + = + ( <_>) ^ ; + + = <3D point> [<_> ]; + + = + ( <_\n>) ^ ; + + = <_> ; + + = + ( <_\n>) ^ ; + + = <_> ; @endverbatim **Description** -surface record 9 describes a B-spline surface. The surface data consist of a u rational flag @image html brep_wp_image219.gif, v rational flag @image html brep_wp_image220.gif, u degree @image html brep_wp_image221.gif, v degree @image html brep_wp_image222.gif, u pole count @image html brep_wp_image250.gif, v pole count @image html brep_wp_image251.gif, u multiplicity knot count @image html brep_wp_image252.gif, v multiplicity knot count @image html brep_wp_image253.gif, weight poles, u multiplicity knots, v multiplicity knots. +\ describes a B-spline surface. The surface data consist of a u rational flag *ru*, v rational flag *rv*, u degree @f$ m_{u} \leq 25 @f$, v degree @f$ m_{v} \leq 25 @f$, u pole count @f$ n_{u} \geq 2 @f$, v pole count @f$ n_{v} \geq 2 @f$, u multiplicity knot count *ku*, v multiplicity knot count *kv*, weight poles, u multiplicity knots, v multiplicity knots. -The weight poles are @image html brep_wp_image254.gif 3D points @image html brep_wp_image224.gif (@image html brep_wp_image255.gif) if @image html brep_wp_image226.gif. The weight poles are @image html brep_wp_image254.gif pairs @image html brep_wp_image227.gif (@image html brep_wp_image255.gif) if @image html brep_wp_image228.gif. Here @image html brep_wp_image224.gif is a 3D point and @image html brep_wp_image229.gif is a positive real (@image html brep_wp_image255.gif). @image html brep_wp_image230.gif (@image html brep_wp_image255.gif) if @image html brep_wp_image226.gif. +The weight poles are @f$ n_{u} \cdot n_{v} @f$ 3D points @f$ B_{i,j}\; ((i,j) \in \left \{ 1,...,n_{u} \right \} \times \left \{ 1,...,n_{v} \right \}) @f$ if @f$ r_{u}+r_{v}=0 @f$. The weight poles are @f$ n_{u} \cdot n_{v} @f$ pairs @f$ B_{i,j}h_{i,j}\; ((i,j) \in \left \{ 1,...,n_{u} \right \} \times \left \{ 1,...,n_{v} \right \}) @f$ if @f$ r_{u}+r_{v} \neq 0 @f$. Here @f$ B_{i,j} @f$ is a 3D point and @f$ h_{i,j} @f$ is a positive real @f$ ((i,j) \in \left \{ 1,...,n_{u} \right \} \times \left \{ 1,...,n_{v} \right \}) @f$. @f$ h_{i,j}=1\; ((i,j) \in \left \{ 1,...,n_{u} \right \} \times \left \{ 1,...,n_{v} \right \}) @f$ if @f$ r_{u}+r_{v} = 0 @f$. -The u multiplicity knots are @image html brep_wp_image252.gif pairs @image html brep_wp_image256.gif. Here @image html brep_wp_image113.gif is a knot with multiplicity @image html brep_wp_image114.gif (@image html brep_wp_image257.gif) so that +The u multiplicity knots are *ku* pairs @f$ u_{1}q_{1} ... u_{k_{u}}q_{k_{u}} @f$. Here @f$ u_{i} @f$ is a knot with multiplicity @f$ q_{i} \geq 1 \;(1\leq i\leq k_{u}) @f$ so that -@image html brep_wp_image116.gif (@image html brep_wp_image258.gif), -@image html brep_wp_image259.gif, @image html brep_wp_image260.gif, @image html brep_wp_image261.gif (@image html brep_wp_image262.gif), @image html brep_wp_image263.gif. +@f[ u_{i} < u_{i+1} \; (1\leq i\leq k_{u}-1), \\ +q_{1} \leq m_{u}+1,\; q_{k_{u}} \leq m_{u}+1,\; q_{i} \leq m_{u}\; (2\leq i\leq k_{u}-1),\; \sum_{i=1}^{k_{u}}q_{i}=m_{u}+n_{u}+1. @f] -The v multiplicity knots are @image html brep_wp_image253.gif pairs @image html brep_wp_image264.gif. Here @image html brep_wp_image265.gif is a knot with multiplicity @image html brep_wp_image266.gif (@image html brep_wp_image267.gif) so that +The v multiplicity knots are *kv* pairs @f$ v_{1}t_{1} ... v_{k_{v}}t_{k_{v}} @f$. Here @f$ v_{j} @f$ is a knot with multiplicity @f$ t_{i} \geq 1\;(1\leq i\leq k_{v}) @f$ so that -@image html brep_wp_image268.gif (@image html brep_wp_image269.gif), -@image html brep_wp_image270.gif, @image html brep_wp_image271.gif, @image html brep_wp_image272.gif (@image html brep_wp_image273.gif), @image html brep_wp_image274.gif. +@f[ v_{j} < v_{j+1} \; (1\leq j\leq k_{v}-1), \\ +t_{1} \leq m_{v}+1,\; t_{k_{v}} \leq m_{v}+1,\; t_{j} \leq m_{v}\; (2\leq j\leq k_{v}-1),\; \sum_{j=1}^{k_{v}}t_{j}=m_{v}+n_{v}+1. @f] The B-spline surface is defined by the following parametric equation: -@image html brep_wp_image275.gif, @image html brep_wp_image276.gif +@f[ S(u,v)=\frac{\sum_{i=1}^{n_{u}} \sum_{j=1}^{n_{v}} B_{i,j} \cdot h_{i,j} \cdot N_{i,m_{u}+1}(u) \cdot M_{j,m_{v}+1}(v)}{\sum_{i=1}^{n_{u}} \sum_{j=1}^{n_{v}} h_{i,j} \cdot N_{i,m_{u}+1}(u) \cdot M_{j,m_{v}+1}(v)}, (u,v) \in [u_{1},u_{k_{u}}] \times [v_{1},v_{k_{v}}] @f] -where functions @image html brep_wp_image277.gif and @image html brep_wp_image278.gif have the following recursion definition by @image html brep_wp_image279.gif +where functions *Ni,j* and *Mi,j* have the following recursion definition by *j*: -@image html brep_wp_image127.gif, @image html brep_wp_image128.gif (@image html brep_wp_image280.gif); +@f[ N_{i,1}(u)=\left\{\begin{matrix} +1\Leftarrow \bar{u}_{i} \leq u \leq \bar{u}_{i+1}\\ +0\Leftarrow u < \bar{u}_{i} \vee \bar{u}_{i+1} \leq u \end{matrix} \right.,\; +N_{i,j}(u)=\frac{(u-\bar{u}_{i}) \cdot N_{i,j-1}(u) }{\bar{u}_{i+j-1}-\bar{u}_{i}}+ \frac{(\bar{u}_{i+j}-u) \cdot N_{i+1,j-1}(u)}{\bar{u}_{i+j}-\bar{u}_{i+1}},\;(2 \leq j \leq m_{u}+1); \\ +M_{i,1}(v)=\left\{\begin{matrix} +1\Leftarrow \bar{v}_{i} \leq v \leq \bar{v}_{i+1}\\ +0\Leftarrow v < \bar{v}_{i} \vee \bar{v}_{i+1} \leq v \end{matrix} \right.,\; +M_{i,j}(v)=\frac{(v-\bar{v}_{i}) \cdot M_{i,j-1}(v) }{\bar{v}_{i+j-1}-\bar{v}_{i}}+ \frac{(\bar{v}_{i+j}-v) \cdot M_{i+1,j-1}(v)}{\bar{v}_{i+j}-\bar{v}_{i+1}},\;(2 \leq j \leq m_{v}+1); @f] + +where +@f[ \bar{u}_{i}=u_{j}\; (1 \leq j \leq k_{u},\; \sum_{l=1}^{j-1}q_{l} \leq i \leq \sum_{l=1}^{j}q_{l}), \\ + \bar{v}_{i}=v_{j}\; (1 \leq j \leq k_{v},\; \sum_{l=1}^{j-1}t_{l} \leq i \leq \sum_{l=1}^{j}t_{l}); @f] -@image html brep_wp_image281.gif, @image html brep_wp_image282.gif (@image html brep_wp_image283.gif); -where +The example record is interpreted as a B-spline surface with a u rational flag *ru*=1, v rational flag *rv*=1, u degree *mu*=1, v degree *mv*=1, u pole count *nu*=3, v pole count *nv*=2, u multiplicity knot count *ku*=5, v multiplicity knot count *kv*=4, weight poles *B1,1*=(0, 0, 1), *h1,1*=7, *B1,2*=(1, 0, -4), *h1,2*=10, *B2,1*=(0, 1, -2), *h2,1*=8, *B2,2*=(1, 1, 5), *h2,2*=11, *B3,1*=(0, 2, 3), *h3,1*=9 and *B3,2*=(1, 2, 6), *h3,2*=12, u multiplicity knots *u1*=0, *q1*=1, *u2*=0.25, *q2*=1, *u3*=0.5, *q3*=1, *u4*=0.75, *q4*=1 and *u5*=1, *q5*=1, v multiplicity knots *v1*=0, *r1*=1, *v2*=0.3, *r2*=1, *v3*=0.7, *r3*=1 and *v4*=1, *r4*=1. The B-spline surface is defined by the following parametric equation: -@image html brep_wp_image284.gif (@image html brep_wp_image285.gif,@image html brep_wp_image286.gif), - -@image html brep_wp_image287.gif (@image html brep_wp_image288.gif,@image html brep_wp_image289.gif). - -The example record is interpreted as a B-spline surface with a u rational flag @image html brep_wp_image233.gif, v rational flag @image html brep_wp_image234.gif, u degree @image html brep_wp_image290.gif, v degree @image html brep_wp_image236.gif, u pole count @image html brep_wp_image291.gif, v pole count @image html brep_wp_image292.gif, u multiplicity knot count @image html brep_wp_image293.gif, v multiplicity knot count @image html brep_wp_image294.gif, weight poles @image html brep_wp_image295.gif, @image html brep_wp_image296.gif, @image html brep_wp_image297.gif, @image html brep_wp_image298.gif, @image html brep_wp_image299.gif, @image html brep_wp_image300.gif, @image html brep_wp_image301.gif, @image html brep_wp_image302.gif, @image html brep_wp_image303.gif, @image html brep_wp_image304.gif and @image html brep_wp_image305.gif, @image html brep_wp_image306.gif, u multiplicity knots @image html brep_wp_image142.gif, @image html brep_wp_image143.gif, @image html brep_wp_image144.gif, @image html brep_wp_image145.gif, @image html brep_wp_image146.gif, @image html brep_wp_image147.gif, @image html brep_wp_image148.gif, @image html brep_wp_image149.gif and @image html brep_wp_image150.gif, @image html brep_wp_image151.gif, v multiplicity knots @image html brep_wp_image307.gif, @image html brep_wp_image308.gif, @image html brep_wp_image309.gif, @image html brep_wp_image310.gif, @image html brep_wp_image311.gif, @image html brep_wp_image312.gif and @image html brep_wp_image313.gif, @image html brep_wp_image314.gif. The B-spline surface is defined by the following parametric equation: - -@image html brep_wp_image315.gif +@f[ S(u,v)= [ (0,0,1) \cdot 7 \cdot N_{1,2}(u) \cdot M_{1,2}(v)+(1,0,-4) \cdot 10 \cdot N_{1,2}(u) \cdot M_{2,2}(v)+ \\ +(0,1,-2) \cdot 8 \cdot N_{2,2}(u) \cdot M_{1,2}(v)+(1,1,5) \cdot 11 \cdot N_{2,2}(u) \cdot M_{2,2}(v)+ \\ +(0,2,3) \cdot 9 \cdot N_{3,2}(u) \cdot M_{1,2}(v)+(1,2,6) \cdot 12 \cdot N_{3,2}(u) \cdot M_{2,2}(v)] \div \\ +[7 \cdot N_{1,2}(u) \cdot M_{1,2}(v)+10 \cdot N_{1,2}(u) \cdot M_{2,2}(v)+ \\ +8 \cdot N_{2,2}(u) \cdot M_{1,2}(v)+11 \cdot N_{2,2}(u) \cdot M_{2,2}(v)+ \\ +9 \cdot N_{3,2}(u) \cdot M_{1,2}(v)+12 \cdot N_{3,2}(u) \cdot M_{2,2}(v) ] @f] -@subsubsection occt_brep_format_4_2_10 surface record 10 – Rectangular Trim Surface +@subsubsection occt_brep_format_4_2_10 Rectangular Trim Surface - \< surface record 10 \> **Example** @@ -878,57 +912,57 @@ The example record is interpreted as a B-spline surface with a u rational flag **BNF-like Definition** @verbatim - surface record 10 = "10" _ trim surface u min _ trim surface u max _ trim surface v min _ trim surface v max _\n surface record; - - trim surface u min = real; - - trim surface u max = real; - - trim surface v min = real; - - trim surface v max = real; -@verbatim + = "10" <_> <_> <_> + <_> <_\n> ; + + = ; + + = ; + + = ; + + = ; +@endverbatim **Description** -surface record 10 describes a rectangular trim surface. The surface data consist of reals @image html brep_wp_image153.gif, @image html brep_wp_image154.gif, @image html brep_wp_image316.gif and @image html brep_wp_image317.gif and a surface record so that @image html brep_wp_image155.gif and @image html brep_wp_image318.gif. The rectangular trim surface is a restriction of the base surface @image html brep_wp_image156.gif described in the record to the set @image html brep_wp_image319.gif. The rectangular trim surface is defined by the following parametric equation: +\ describes a rectangular trim surface. The surface data consist of reals *umin*, *umax*, *vmin* and *vmax* and a \ so that *umin* < *umax* and *vmin* < *vmax*. The rectangular trim surface is a restriction of the base surface *B* described in the record to the set @f$ [u_{min},u_{max}] \times [v_{min},v_{max}] \subseteq domain(B) @f$. The rectangular trim surface is defined by the following parametric equation: -@image html brep_wp_image320.gif, @image html brep_wp_image321.gif. +@f[ S(u,v)=B(u,v),\; (u,v) \in [u_{min},u_{max}] \times [v_{min},v_{max}] . @f] -The example record is interpreted as a rectangular trim surface to the set @image html brep_wp_image322.gif for the base surface @image html brep_wp_image323.gif. The rectangular trim surface is defined by the following parametric equation: @image html brep_wp_image324.gif, @image html brep_wp_image325.gif. +The example record is interpreted as a rectangular trim surface to the set [-1, 2]x[-3, 4] for the base surface @f$ B(u,v)=(1,2,3)+u \cdot (1,0,0)+v \cdot (0,1,0) @f$. The rectangular trim surface is defined by the following parametric equation: @f$ B(u,v)=(1,2,3)+u \cdot (1,0,0)+ v \cdot (0,1,0),\; (u,v) \in [-1,2] \times [-3,4] @f$. -@subsubsection occt_brep_format_4_2_11 surface record 11 – Offset Surface +@subsubsection occt_brep_format_4_2_11 Offset Surface - \< surface record 11 \> **Example** @verbatim 11 -2 1 1 2 3 0 0 1 1 0 -0 -0 1 0 -@verbatim +@endverbatim **BNF-like Definition** @verbatim - surface record 11 = "11" _ surface record distance _\n surface record; - - surface record distance = real; -@verbatim + = "11" <_> <_\n> ; + + = ; +@endverbatim **Description** -surface record 11 describes an offset surface. -The offset surface data consist of a distance @image html brep_wp_image165.gif and a surface record. The offset surface is the result of offsetting the base surface @image html brep_wp_image326.gif described in the record to the distance @image html brep_wp_image165.gif along the normal @image html brep_wp_image044.gif of surface @image html brep_wp_image156.gif. The offset surface is defined by the following parametric equation: +\ describes an offset surface. +The offset surface data consist of a distance *d* and a \. The offset surface is the result of offsetting the base surface *B* described in the record to the distance *d* along the normal *N* of surface *B*. The offset surface is defined by the following parametric equation: -@image html brep_wp_image327.gif, @image html brep_wp_image328.gif. +@f[ S(u,v)=B(u,v)+d \cdot N(u,v),\; (u,v) \in domain(B) . \\ +N(u,v) = [S'_{u}(u,v),S'_{v}(u,v)] @f] +if @f$ [S'_{u}(u,v),S'_{v}(u,v)] \neq \vec{0} @f$. -@image html brep_wp_image329.gif -if @image html brep_wp_image330.gif. - -The example record is interpreted as an offset surface with a distance @image html brep_wp_image331.gif and base surface @image html brep_wp_image323.gif. The offset surface is defined by the following parametric equation: @image html brep_wp_image332.gif. +The example record is interpreted as an offset surface with a distance *d*=-2 and base surface @f$ B(u,v)=(1,2,3)+u \cdot (1,0,0)+v \cdot (0,1,0) @f$. The offset surface is defined by the following parametric equation: @f$ S(u,v)=(1,2,3)+u \cdot (1,0,0)+v \cdot (0,1,0)-2 \cdot (0,0,1) @f$. -@subsection occt_brep_format_4_3 Subsection 2D curves +@subsection occt_brep_format_4_3 2D curves **Example** @@ -963,27 +997,27 @@ The example record is interpreted as an offset surface with a distance @image h **BNF-like Definition** @verbatim - 2D curves = 2D curve header _\n 2D curve records; - - 2D curve header = "Curve2ds" _ 2D curve count; - - 2D curve count = int; - - 2D curve records = 2D curve record ^ 2D curve count; - - 2D curve record = - 2D curve record 1 | - 2D curve record 2 | - 2D curve record 3 | - 2D curve record 4 | - 2D curve record 5 | - 2D curve record 6 | - 2D curve record 7 | - 2D curve record 8 | - 2D curve record 9; + <2D curves> = <2D curve header> <_\n> <2D curve records>; + + <2D curve header> = "Curve2ds" <_> <2D curve count>; + + <2D curve count> = ; + + <2D curve records> = <2D curve record> ^ <2D curve count>; + + <2D curve record> = + <2D curve record 1> | + <2D curve record 2> | + <2D curve record 3> | + <2D curve record 4> | + <2D curve record 5> | + <2D curve record 6> | + <2D curve record 7> | + <2D curve record 8> | + <2D curve record 9>; @endverbatim -@subsubsection occt_brep_format_4_3_1 2D curve record 1 – Line +@subsubsection occt_brep_format_4_3_1 Line - \<2D curve record 1\> **Example** @@ -994,19 +1028,19 @@ The example record is interpreted as an offset surface with a distance @image h **BNF-like Definition** @verbatim - 2D curve record 1 = "1" _ 2D point _ 2D direction _\n; + <2D curve record 1> = "1" <_> <2D point> <_> <2D direction> <_\n>; @endverbatim **Description** -2D curve record 1 describes a line. The line data consist of a 2D point @image html brep_wp_image037.gif and a 2D direction @image html brep_wp_image038.gif. The line passes through the point @image html brep_wp_image037.gif, has the direction @image html brep_wp_image038.gif and is defined by the following parametric equation: +\<2D curve record 1\> describes a line. The line data consist of a 2D point *P* and a 2D direction *D*. The line passes through the point *P*, has the direction *D* and is defined by the following parametric equation: -@image html brep_wp_image333.gif, @image html brep_wp_image069.gif. +@f[ C(u)=P+u \cdot D, \; u \in (-\infty,\; \infty). @f] -The example record is interpreted as a line which passes through a point@image html brep_wp_image334.gif, has a direction @image html brep_wp_image335.gif and is defined by the following parametric equation: @image html brep_wp_image336.gif. +The example record is interpreted as a line which passes through a point *P*=(3,0), has a direction *D*=(0,-1) and is defined by the following parametric equation: @f$ C(u)=(3,0)+ u \cdot (0,-1) @f$. -@subsubsection occt_brep_format_4_3_2 2D curve record 2 – Circle +@subsubsection occt_brep_format_4_3_2 Circle - \<2D curve record 2\> **Example** @@ -1017,27 +1051,27 @@ The example record is interpreted as a line which passes through a point@image **BNF-like Definition** @verbatim - 2D curve record 2 = "2" _ 2D circle center _ 2D circle Dx _ 2D circle Dy _ 2D circle radius _\n; - - 2D circle center = 2D point; - - 2D circle Dx = 2D direction; - - 2D circle Dy = 2D direction; - - 2D circle radius = real; + <2D curve record 2> = "2" <_> <2D circle center> <_> <2D circle Dx> <_> <2D circle Dy> <_> <2D circle radius> <_\n>; + + <2D circle center> = <2D point>; + + <2D circle Dx> = <2D direction>; + + <2D circle Dy> = <2D direction>; + + <2D circle radius> = ; @endverbatim **Description** -2D curve record 2 describes a circle. The circle data consist of a 2D point @image html brep_wp_image037.gif, orthogonal 2D directions @image html brep_wp_image045.gif and @image html brep_wp_image046.gif and a non-negative real @image html brep_wp_image047.gif. The circle has a center @image html brep_wp_image037.gif. The circle plane is parallel to directions @image html brep_wp_image045.gif and @image html brep_wp_image046.gif. The circle has a radius @image html brep_wp_image047.gif and is defined by the following parametric equation: +\<2D curve record 2\> describes a circle. The circle data consist of a 2D point *P*, orthogonal 2D directions *Dx* and *Dy* and a non-negative real *r*. The circle has a center *P*. The circle plane is parallel to directions *Dx* and *Dy*. The circle has a radius *r* and is defined by the following parametric equation: -@image html brep_wp_image337.gif, @image html brep_wp_image338.gif. +@f[ C(u)=P+r \cdot (cos(u) \cdot D_{x} + sin(u) \cdot D_{y}),\; u \in [0,\; 2 \cdot \pi) . @f] -The example record is interpreted as a circle which has a center @image html brep_wp_image339.gif. The circle plane is parallel to directions @image html brep_wp_image340.gif and @image html brep_wp_image341.gif. The circle has a radius @image html brep_wp_image342.gif and is defined by the following parametric equation: @image html brep_wp_image343.gif. +The example record is interpreted as a circle which has a center *P*=(1,2). The circle plane is parallel to directions *Dx*=(1,0) and *Dy*=(0,1). The circle has a radius *r*=3 and is defined by the following parametric equation: @f$ C(u)=(1,2)+3 \cdot (cos(u) \cdot (1,0) + sin(u) \cdot (0,1)) @f$. -@subsubsection occt_brep_format_4_3_3 2D curve record 3 – Ellipse +@subsubsection occt_brep_format_4_3_3 Ellipse - \<2D curve record 3\> **Example** @@ -1048,29 +1082,30 @@ The example record is interpreted as a circle which has a center @image html br **BNF-like Definition** @verbatim - 2D curve record 3 = "3" _ 2D ellipse center _ 2D ellipse Dmaj _ 2D ellipse Dmin _ 2D ellipse Rmaj _ 2D ellipse Rmin _\n; - - 2D ellipse center = 2D point; - - 2D ellipse Dmaj = 2D direction; - - 2D ellipse Dmin = 2D direction; - - 2D ellipse Rmaj = real; - - 2D ellipse Rmin = real; + <2D curve record 3> = "3" <_> <2D ellipse center> <_> <2D ellipse Dmaj> <_> + <2D ellipse Dmin> <_> <2D ellipse Rmaj> <_> <2D ellipse Rmin> <_\n>; + + <2D ellipse center> = <2D point>; + + <2D ellipse Dmaj> = <2D direction>; + + <2D ellipse Dmin> = <2D direction>; + + <2D ellipse Rmaj> = ; + + <2D ellipse Rmin> = ; @endverbatim **Description** -2D curve record 3 describes an ellipse. The ellipse data are 2D point @image html brep_wp_image037.gif, orthogonal 2D directions @image html brep_wp_image056.gif and @image html brep_wp_image057.gif and non-negative reals @image html brep_wp_image344.gif and @image html brep_wp_image345.gif that @image html brep_wp_image346.gif. The ellipse has a center @image html brep_wp_image037.gif, major and minor axis directions @image html brep_wp_image056.gif and @image html brep_wp_image057.gif, major and minor radii @image html brep_wp_image058.gif and @image html brep_wp_image345.gif and is defined by the following parametric equation: +\<2D curve record 3\> describes an ellipse. The ellipse data are 2D point *P*, orthogonal 2D directions *Dmaj* and *Dmin* and non-negative reals *rmaj* and *rmin* that *rmaj* @f$ \leq @f$ *rmin*. The ellipse has a center *P*, major and minor axis directions *Dmaj* and *Dmin*, major and minor radii *rmaj* and *rmin* and is defined by the following parametric equation: -@image html brep_wp_image347.gif, @image html brep_wp_image338.gif. +@f[ C(u)=P+r_{maj} \cdot cos(u) \cdot D_{maj}+r_{min} \cdot sin(u) \cdot D_{min},\; u \in [0,\; 2 \cdot \pi) . @f] -The example record is interpreted as an ellipse which has a center @image html brep_wp_image339.gif, major and minor axis directions @image html brep_wp_image348.gif and @image html brep_wp_image349.gif, major and minor radii @image html brep_wp_image350.gif and @image html brep_wp_image351.gif and is defined by the following parametric equation: @image html brep_wp_image352.gif. +The example record is interpreted as an ellipse which has a center *P*=(1,2), major and minor axis directions *Dmaj*=(1,0) and *Dmin*=(0,1), major and minor radii *rmaj*=4 and *rmin*=3 and is defined by the following parametric equation: @f$ C(u)=(1,2)+4 \cdot cos(u) \cdot (1,0)+3 \cdot sin(u) \cdot (0,1) @f$. -@subsubsection occt_brep_format_4_3_4 2D curve record 4 – Parabola +@subsubsection occt_brep_format_4_3_4 Parabola - \<2D curve record 4\> **Example** @@ -1081,30 +1116,29 @@ The example record is interpreted as an ellipse which has a center @image html **BNF-like Definition** @verbatim - 2D curve record 4 = "4" _ 2D parabola origin _ 2D parabola Dx _ 2D parabola Dy _ 2D parabola focal length _\n; - - 2D parabola origin = 2D point; - - 2D parabola Dx = 2D direction; - - 2D parabola Dy = 2D direction; - - 2D parabola focal length = real; + <2D curve record 4> = "4" <_> <2D parabola origin> <_> <2D parabola Dx> <_> + <2D parabola Dy> <_> <2D parabola focal length> <_\n>; + + <2D parabola origin> = <2D point>; + + <2D parabola Dx> = <2D direction>; + + <2D parabola Dy> = <2D direction>; + + <2D parabola focal length> = ; @endverbatim **Description** -2D curve record 4 describes a parabola. The parabola data consist of a 2D point @image html brep_wp_image037.gif, orthogonal 2D directions @image html brep_wp_image045.gif and @image html brep_wp_image046.gif and a non-negative real @image html brep_wp_image067.gif. The parabola coordinate system has its origin @image html brep_wp_image037.gif and axis directions @image html brep_wp_image045.gif and @image html brep_wp_image046.gif. The parabola has a focus length @image html brep_wp_image067.gif and is defined by the following parametric equation: +\<2D curve record 4\> describes a parabola. The parabola data consist of a 2D point *P*, orthogonal 2D directions *Dx* and *Dy* and a non-negative real *f*. The parabola coordinate system has its origin *P* and axis directions *Dx* and *Dy*. The parabola has a focus length *f* and is defined by the following parametric equation: -@image html brep_wp_image068.gif, @image html brep_wp_image069.gif Ü @image html brep_wp_image070.gif; +@f[ C(u)=P+\frac{u^{2}}{4 \cdot f} \cdot D_{x}+u \cdot D_{y},\; u \in (-\infty,\; \infty) \Leftarrow f \neq 0;\\ +C(u)=P+u \cdot D_{x},\; u \in (-\infty,\; \infty) \Leftarrow f = 0\; (degenerated\;case). @f] -@image html brep_wp_image071.gif, @image html brep_wp_image069.gif Ü @image html brep_wp_image072.gif (degenerated case). - -The example record is interpreted as a parabola in plane which passes through a point @image html brep_wp_image339.gif and is parallel to directions @image html brep_wp_image340.gif and @image html brep_wp_image341.gif. The parabola has a focus length @image html brep_wp_image077.gif and is defined by the following parametric equation: @image html brep_wp_image353.gif. +The example record is interpreted as a parabola in plane which passes through a point *P*=(1,2) and is parallel to directions *Dx*=(1,0) and *Dy*=(0,1). The parabola has a focus length *f*=16 and is defined by the following parametric equation: @f$ C(u)=(1,2)+ \frac{u^{2}}{64} \cdot (1,0)+u \cdot (0,1) @f$. -@subsubsection occt_brep_format_4_3_5 2D curve record 5 – Hyperbola - +@subsubsection occt_brep_format_4_3_5 Hyperbola - \<2D curve record 5\> **Example** 5 1 2 1 0 -0 1 3 4 @@ -1113,29 +1147,30 @@ The example record is interpreted as a parabola in plane which passes through a **BNF-like Definition** @verbatim - 2D curve record 5 = "5" _ 2D hyperbola origin _ 2D hyperbola Dx _ 2D hyperbola Dy _ 2D hyperbola Kx _ 2D hyperbola Ky _\n; - - 2D hyperbola origin = 2D point; - - 2D hyperbola Dx = 2D direction; - - 2D hyperbola Dy = 2D direction; - - 2D hyperbola Kx = real; - - 2D hyperbola Ky = real; + <2D curve record 5> = "5" <_> <2D hyperbola origin> <_> <2D hyperbola Dx> <_> + <2D hyperbola Dy> <_> <2D hyperbola Kx> <_> <2D hyperbola Ky> <_\n>; + + <2D hyperbola origin> = <2D point>; + + <2D hyperbola Dx> = <2D direction>; + + <2D hyperbola Dy> = <2D direction>; + + <2D hyperbola Kx> = ; + + <2D hyperbola Ky> = ; @endverbatim **Description** -2D curve record 5 describes a hyperbola. The hyperbola data consist of a 2D point @image html brep_wp_image037.gif, orthogonal 2D directions @image html brep_wp_image045.gif and @image html brep_wp_image046.gif and non-negative reals @image html brep_wp_image079.gif and @image html brep_wp_image080.gif. The hyperbola coordinate system has origin @image html brep_wp_image037.gif and axis directions @image html brep_wp_image045.gif and @image html brep_wp_image046.gif. The hyperbola is defined by the following parametric equation: +\<2D curve record 5\> describes a hyperbola. The hyperbola data consist of a 2D point *P*, orthogonal 2D directions *Dx* and *Dy* and non-negative reals *kx* and *ky*. The hyperbola coordinate system has origin *P* and axis directions *Dx* and *Dy*. The hyperbola is defined by the following parametric equation: -@image html brep_wp_image081.gif, @image html brep_wp_image069.gif. +@f[ C(u)=P+k_{x} \cdot cosh(u) D_{x}+k_{y} \cdot sinh(u) \cdot D_{y},\; u \in (-\infty,\; \infty). @f] -The example record is interpreted as a hyperbola with coordinate system which has origin @image html brep_wp_image339.gif and axis directions @image html brep_wp_image354.gif and @image html brep_wp_image341.gif. Other data for the hyperbola are @image html brep_wp_image082.gif and @image html brep_wp_image083.gif. The hyperbola is defined by the following parametric equation: @image html brep_wp_image355.gif. +The example record is interpreted as a hyperbola with coordinate system which has origin *P*=(1,2) and axis directions *Dx*=(1,0) and *Dy*=(0,1). Other data for the hyperbola are *kx*=5 and *ky*=4. The hyperbola is defined by the following parametric equation: @f$ C(u)=(1,2)+3 \cdot cosh(u) \cdot (1,0)+4 \cdot sinh(u) \cdot (0,1) @f$. -@subsubsection occt_brep_format_4_3_6 2D curve record 6 – Bezier Curve +@subsubsection occt_brep_format_4_3_6 Bezier Curve - \<2D curve record 6\> **Example** @@ -1146,35 +1181,36 @@ The example record is interpreted as a hyperbola with coordinate system which h **BNF-like Definition** @verbatim -2D curve record 6 = "6" _ 2D Bezier rational flag _ 2D Bezier degree 2D Bezier weight poles _\n; - -2D Bezier rational flag = flag; - -2D Bezier degree = int; - -2D Bezier weight poles = (_ 2D Bezier weight pole) ^ (2D Bezier degree + "1"); - -2D Bezier weight pole = 2D point [_ real]; + <2D curve record 6> = "6" <_> <2D Bezier rational flag> <_> <2D Bezier degree> + <2D Bezier weight poles> <_\n>; + + <2D Bezier rational flag> = ; + + <2D Bezier degree> = ; + + <2D Bezier weight poles> = (<_> <2D Bezier weight pole>) ^ (<2D Bezier degree> <+> “1”); + + <2D Bezier weight pole> = <2D point> [<_> ]; @endverbatim **Description** -2D curve record 6 describes a Bezier curve. The curve data consist of a rational flag @image html brep_wp_image047.gif, a degree @image html brep_wp_image085.gif and weight poles. +\<2D curve record 6\> describes a Bezier curve. The curve data consist of a rational flag *r*, a degree @f$ m \leq 25 @f$ and weight poles. -The weight poles are @image html brep_wp_image086.gif 2D points @image html brep_wp_image087.gif if the flag @image html brep_wp_image047.gif is 0. The weight poles are @image html brep_wp_image086.gif pairs @image html brep_wp_image088.gif if the flag @image html brep_wp_image047.gif is 1. Here @image html brep_wp_image089.gif is a 2D point and @image html brep_wp_image090.gif is a positive real (@image html brep_wp_image093.gif). @image html brep_wp_image092.gif (@image html brep_wp_image093.gif) if the flag @image html brep_wp_image047.gif is 0. +The weight poles are *m*+1 2D points *B0 ... Bm* if the flag *r* is 0. The weight poles are *m*+1 pairs *B0h0 ... Bmhm* if the flag *r* is 1. Here *Bi* is a 2D point and *hi* is a positive real @f$ (0\leq i\leq m) @f$. *hi*=1 @f$(0\leq i\leq m) @f$ if the flag *r* is 0. The Bezier curve is defined by the following parametric equation: -@image html brep_wp_image094.gif, @image html brep_wp_image095.gif +@f[ C(u)= \frac{\sum_{i=0}^{m} B_{i} \cdot h_{i} \cdot C_{m}^{i} \cdot u^{i} \cdot (1-u)^{m-i}}{\sum_{i=0}^{m} h_{i} \cdot C_{m}^{i} \cdot u^{i} \cdot (1-u)^{m-i}},\; u \in [0,1] @f] -where @image html brep_wp_image096.gif. +where @f$ 0^{0} \equiv 1 @f$. -The example record is interpreted as a Bezier curve with a rational flag @image html brep_wp_image097.gif, a degree @image html brep_wp_image098.gif and weight poles @image html brep_wp_image356.gif, @image html brep_wp_image100.gif, @image html brep_wp_image357.gif, @image html brep_wp_image102.gif and @image html brep_wp_image358.gif, @image html brep_wp_image104.gif. The Bezier curve is defined by the following parametric equation: +The example record is interpreted as a Bezier curve with a rational flag *r*=1, a degree *m*=2 and weight poles *B0*=(0,1), *h0*=4, *B1*=(1,-2), *h1*=5 and *B2*=(2,3), *h2*=6. The Bezier curve is defined by the following parametric equation: -@image html brep_wp_image359.gif. +@f[ C(u)= \frac{(0,1) \cdot 4 \cdot (1-u)^{2}+(1,-2) \cdot 5 \cdot 2 \cdot u \cdot (1-u)+(2,3) \cdot 6 \cdot u^{2}}{ 4 \cdot (1-u)^{2}+5 \cdot 2 \cdot u \cdot (1-u)+6 \cdot u^{2}} . @f] -@subsubsection occt_brep_format_4_3_7 2D curve record 7 – B-spline Curve +@subsubsection occt_brep_format_4_3_7 B-spline Curve - \<2D curve record 7\> **Example** @@ -1186,55 +1222,58 @@ The example record is interpreted as a Bezier curve with a rational flag @image **BNF-like Definition** @verbatim - 2D curve record 7 = "7" _ 2D B-spline rational flag _ "0" _ 2D B-spline degree _ 2D B-spline pole count _ 2D B-spline multiplicity knot count 2D B-spline weight poles _\n 2D B-spline multiplicity knots _\n; - - 2D B-spline rational flag = flag; - - 2D B-spline degree = int; - - 2D B-spline pole count = int; - - 2D B-spline multiplicity knot count = int; - - 2D B-spline weight poles = 2D B-spline weight pole ^ 2D B-spline pole count; - - 2D B-spline weight pole = _ 2D point [_ real]; - - 2D B-spline multiplicity knots = - 2D B-spline multiplicity knot ^ 2D B-spline multiplicity knot count; - - 2D B-spline multiplicity knot = _ real _ int; + <2D curve record 7> = "7" <_> <2D B-spline rational flag> <_> "0" <_> <2D B-spline degree> <_> <2D B-spline pole count> <_> <2D B-spline multiplicity knot count> <2D B-spline weight poles> <_\n> <2D B-spline multiplicity knots> <_\n>; + + <2D B-spline rational flag> = ; + + <2D B-spline degree> = ; + + <2D B-spline pole count> = ; + + <2D B-spline multiplicity knot count> = ; + + <2D B-spline weight poles> = <2D B-spline weight pole> ^ <2D B-spline pole count>; + + <2D B-spline weight pole> = <_> <2D point> [<_> ]; + + <2D B-spline multiplicity knots> = + <2D B-spline multiplicity knot> ^ <2D B-spline multiplicity knot count>; + + <2D B-spline multiplicity knot> = <_> <_> ; @endverbatim **Description** -2D curve record 7 describes a B-spline curve. The curve data consist of a rational flag @image html brep_wp_image047.gif, a degree @image html brep_wp_image085.gif, a pole count @image html brep_wp_image106.gif, a multiplicity knot count @image html brep_wp_image107.gif, weight poles and multiplicity knots. +\<2D curve record 7\> describes a B-spline curve. The curve data consist of a rational flag *r*, a degree @f$ m \leq 25 @f$, a pole count @f$ n \geq 2 @f$, a multiplicity knot count *k*, weight poles and multiplicity knots. -The weight poles are @image html brep_wp_image108.gif 2D points @image html brep_wp_image109.gif if the flag @image html brep_wp_image047.gif is 0. The weight poles are @image html brep_wp_image108.gif pairs @image html brep_wp_image110.gif if the flag @image html brep_wp_image047.gif is 1. Here @image html brep_wp_image089.gif is a 2D point and @image html brep_wp_image090.gif is a positive real (@image html brep_wp_image111.gif). @image html brep_wp_image092.gif (@image html brep_wp_image111.gif) if the flag @image html brep_wp_image047.gif is 0. +The weight poles are *n* 2D points *B1 ... Bn* if the flag *r* is 0. The weight poles are *n* pairs *B1h1 ... Bnhn* if the flag *r* is 1. Here *Bi* is a 2D point and *hi* is a positive real @f$ (1\leq i\leq n) @f$. *hi*=1 @f$(1\leq i\leq n) @f$ if the flag *r* is 0. -The multiplicity knots are @image html brep_wp_image107.gif pairs @image html brep_wp_image112.gif. Here @image html brep_wp_image113.gif is a knot with multiplicity @image html brep_wp_image114.gif (@image html brep_wp_image115.gif) so that +The multiplicity knots are *k* pairs *u1q1 ... ukqk*. Here *ui* is a knot with multiplicity @f$ q_{i} \geq 1\; (1 \leq i \leq k) @f$ so that -@image html brep_wp_image360.gif (@image html brep_wp_image361.gif), -@image html brep_wp_image362.gif, @image html brep_wp_image363.gif, @image html brep_wp_image364.gif (@image html brep_wp_image365.gif), @image html brep_wp_image366.gif. +@f[ u_{i} < u_{i+1}\; (1 \leq i \leq k-1), \\ +q_{1} \leq m+1,\; q_{k} \leq m+1,\; q_{i} \leq m\; (2 \leq i \leq k-1),\; \sum_{i=1}^{k}q_{i}=m+n+1 . @f] The B-spline curve is defined by the following parametric equation: -@image html brep_wp_image367.gif, @image html brep_wp_image368.gif +@f[ C(u)= \frac{\sum_{i=1}^{n} B_{i} \cdot h_{i} \cdot N_{i,m+1}(u) }{\sum_{i=1}^{n} h_{i} \cdot N_{i,m+1}(u)},\; u \in [u_{1},\; u_{k}] @f] -where functions @image html brep_wp_image125.gif have the following recursion definition by @image html brep_wp_image126.gif +where functions *Ni,j* have the following recursion definition by *j* -@image html brep_wp_image127.gif, @image html brep_wp_image128.gif (@image html brep_wp_image129.gif) +@f[ N_{i,1}(u)=\left\{\begin{matrix} +1\Leftarrow \bar{u}_{i} \leq u \leq \bar{u}_{i+1}\\ +0\Leftarrow u < \bar{u}_{i} \vee \bar{u}_{i+1} \leq u \end{matrix} \right.,\; +N_{i,j}(u)=\frac{(u-\bar{u}_{i}) \cdot N_{i,j-1}(u) }{\bar{u}_{i+j-1}-\bar{u}_{i}}+ \frac{(\bar{u}_{i+j}-u) \cdot N_{i+1,j-1}(u)}{\bar{u}_{i+j}-\bar{u}_{i+1}},\;(2 \leq j \leq m+1) @f] where -@image html brep_wp_image284.gif (@image html brep_wp_image369.gif,@image html brep_wp_image286.gif). +@f[ \bar{u}_{i}=u_{j}\; (1\leq j\leq k,\; \sum_{l=1}^{j-1}q_{l}+1 \leq i \leq \sum_{l=1}^{j}q_{l}) . @f] -The example record is interpreted as a B-spline curve with a rational flag @image html brep_wp_image097.gif, a degree @image html brep_wp_image133.gif, a pole count @image html brep_wp_image134.gif, a multiplicity knot count @image html brep_wp_image135.gif, weight poles @image html brep_wp_image370.gif, @image html brep_wp_image137.gif, @image html brep_wp_image371.gif, @image html brep_wp_image139.gif and @image html brep_wp_image372.gif, @image html brep_wp_image141.gif and multiplicity knots @image html brep_wp_image373.gif, @image html brep_wp_image374.gif, @image html brep_wp_image375.gif, @image html brep_wp_image376.gif, @image html brep_wp_image377.gif, @image html brep_wp_image378.gif, @image html brep_wp_image379.gif, @image html brep_wp_image380.gif and @image html brep_wp_image381.gif, @image html brep_wp_image382.gif. The B-spline curve is defined by the following parametric equation: +The example record is interpreted as a B-spline curve with a rational flag *r*=1, a degree *m*=1, a pole count *n*=3, a multiplicity knot count *k*=5, weight poles *B1*=(0,1), *h1*=4, *B2*=(1,-2), *h2*=5 and *B3*=(2,3), *h3*=6 and multiplicity knots *u1*=0, *q1*=1, *u2*=0.25, *q2*=1, *u3*=0.5, *q3*=1, *u4*=0.75, *q4*=1 and *u5*=1, *q5*=1. The B-spline curve is defined by the following parametric equation: -@image html brep_wp_image383.gif. +@f[ C(u)= \frac{(0,1) \cdot 4 \cdot N_{1,2}(u)+(1,-2) \cdot 5 \cdot N_{2,2}(u)+(2,3) \cdot 6 \cdot N_{3,2}(u)}{ 4 \cdot N_{1,2}(u)+5 \cdot N_{2,2}(u)+6 \cdot N_{3,2}(u)} . @f] -@subsubsection occt_brep_format_4_3_8 2D curve record 8 – Trimmed Curve +@subsubsection occt_brep_format_4_3_8 Trimmed Curve - \<2D curve record 8\> **Example** @@ -1246,23 +1285,24 @@ The example record is interpreted as a B-spline curve with a rational flag @ima **BNF-like Definition** @verbatim - 2D curve record 8 = "8" _ 2D trimmed curve u min _ 2D trimmed curve u max _\n 2D curve record; - - 2D trimmed curve u min = real; - - 2D trimmed curve u max = real; + <2D curve record 8> = "8" <_> <2D trimmed curve u min> <_> <2D trimmed curve u max> <_\n> + <2D curve record>; + + <2D trimmed curve u min> = ; + + <2D trimmed curve u max> = ; @endverbatim -**Description** +**Description** -2D curve record 8 describes a trimmed curve. The trimmed curve data consist of reals @image html brep_wp_image153.gif and @image html brep_wp_image154.gif and a 2D curve record so that @image html brep_wp_image155.gif. The trimmed curve is a restriction of the base curve @image html brep_wp_image326.gif described in the record to the segment @image html brep_wp_image157.gif. The trimmed curve is defined by the following parametric equation: +\<2D curve record 8\> describes a trimmed curve. The trimmed curve data consist of reals *umin* and *umax* and a \<2D curve record\> so that *umin* < *umax*. The trimmed curve is a restriction of the base curve *B* described in the record to the segment @f$ [u_{min},\;u_{max}]\subseteq domain(B) @f$. The trimmed curve is defined by the following parametric equation: -@image html brep_wp_image158.gif, @image html brep_wp_image159.gif. +@f[ C(u)=B(u),\; u \in [u_{min},\;u_{max}] . @f] -The example record is interpreted as a trimmed curve with @image html brep_wp_image160.gif, @image html brep_wp_image161.gif and base curve @image html brep_wp_image384.gif. The trimmed curve is defined by the following parametric equation: @image html brep_wp_image385.gif, @image html brep_wp_image164.gif. +The example record is interpreted as a trimmed curve with *umin*=-4, *umax*=5 and base curve @f$ B(u)=(1,2)+u \cdot (1,0) @f$. The trimmed curve is defined by the following parametric equation: @f$ C(u)=(1,2)+u \cdot (1,0),\; u \in [-4,5] @f$. -@subsubsection occt_brep_format_4_3_9 2D curve record 9 – Offset Curve +@subsubsection occt_brep_format_4_3_9 Offset Curve - \<2D curve record 9\> **Example** @@ -1274,22 +1314,20 @@ The example record is interpreted as a trimmed curve with @image html brep_wp_im **BNF-like Definition** @verbatim -2D curve record 9 = "9" _ 2D offset curve distance _\n 2D curve record; - -2D offset curve distance = real; +<2D curve record 9> = "9" <_> <2D offset curve distance> <_\n> <2D curve record>; + +<2D offset curve distance> = ; @endverbatim **Description** -2D curve record 9 describes an offset curve. The offset curve data consist of a distance @image html brep_wp_image165.gif and a 2D curve record. The offset curve is the result of offsetting the base curve @image html brep_wp_image156.gif described in the record to the distance @image html brep_wp_image165.gif along the vector @image html brep_wp_image386.gif where @image html brep_wp_image387.gif. The offset curve is defined by the following parametric equation: +\<2D curve record 9\> describes an offset curve. The offset curve data consist of a distance *d* and a \<2D curve record\>. The offset curve is the result of offsetting the base curve *B* described in the record to the distance *d* along the vector @f$ (B'_{Y}(u),\; -B'_{X}(u)) \neq \vec{0} @f$ where @f$ B(u)=(B'_{X}(u),\; B'_{Y}(u)) @f$. The offset curve is defined by the following parametric equation: -@image html brep_wp_image388.gif, @image html brep_wp_image168.gif. +@f[ C(u)=B(u)+d \cdot (B'_{Y}(u),\; -B'_{X}(u)),\; u \in domain(B) . @f] -The example record is interpreted as an offset curve with a distance -@image html brep_wp_image169.gif and base curve @image html brep_wp_image384.gif and -is defined by the following parametric equation: @image html brep_wp_image389.gif. +The example record is interpreted as an offset curve with a distance *d*=2 and base curve @f$ B(u)=(1,2)+u \cdot (1,0) @f$ and is defined by the following parametric equation: @f$ C(u)=(1,2)+u \cdot (1,0)+2 \cdot (0,-1) @f$. -@subsection occt_brep_format_4_4 Subsection 3D polygons +@subsection occt_brep_format_4_4 3D polygons **Example** @@ -1304,47 +1342,47 @@ is defined by the following parametric equation: @image html brep_wp_image389.gi **BNF-like Definition** @verbatim - 3D polygons = 3D polygon header _\n 3D polygon records; - - 3D polygon header = "Polygon3D" _ 3D polygon record count; - - 3D polygon records = 3D polygon record ^ 3D polygon record count; - - 3D polygon record = - 3D polygon node count _ 3D polygon flag of parameter presence _\n - 3D polygon deflection _\n - 3D polygon nodes _\n - [3D polygon parameters _\n]; - - 3D polygon node count = int; - - 3D polygon flag of parameter presence = flag; - - 3D polygon deflection = real; - - 3D polygon nodes = (3D polygon node _) ^ 3D polygon node count; - - 3D polygon node = 3D point; - - 3D polygon u parameters = (3D polygon u parameter _) ^ 3D polygon node count; - - 3D polygon u parameter = real; + <3D polygons> = <3D polygon header> <_\n> <3D polygon records>; + + <3D polygon header> = "Polygon3D" <_> <3D polygon record count>; + + <3D polygon records> = <3D polygon record> ^ <3D polygon record count>; + + <3D polygon record> = + <3D polygon node count> <_> <3D polygon flag of parameter presence> <_\n> + <3D polygon deflection> <_\n> + <3D polygon nodes> <_\n> + [<3D polygon parameters> <_\n>]; + + <3D polygon node count> = ; + + <3D polygon flag of parameter presence> = ; + + <3D polygon deflection> = ; + + <3D polygon nodes> = (<3D polygon node> <_>) ^ <3D polygon node count>; + + <3D polygon node> = <3D point>; + + <3D polygon u parameters> = (<3D polygon u parameter> <_>) ^ <3D polygon node count>; + + <3D polygon u parameter> = ; @endverbatim **Description** -3D polygon record describes a 3D polyline @image html brep_wp_image390.gif which approximates a 3D curve @image html brep_wp_image205.gif. The polyline data consist of a node count @image html brep_wp_image391.gif, a parameter presence flag @image html brep_wp_image392.gif, a deflection @image html brep_wp_image393.gif, nodes @image html brep_wp_image394.gif (@image html brep_wp_image395.gif) and parameters @image html brep_wp_image113.gif (@image html brep_wp_image395.gif). The parameters are present only if@image html brep_wp_image396.gif. The polyline @image html brep_wp_image390.gif passes through the nodes. The deflection @image html brep_wp_image165.gif describes the deflection of polyline @image html brep_wp_image390.gif from the curve @image html brep_wp_image205.gif: +\<3D polygons\> record describes a 3D polyline *L* which approximates a 3D curve *C*. The polyline data consist of a node count @f$ m \geq 2 @f$, a parameter presence flag *p*, a deflection @f$ d \geq 0 @f$, nodes @f$ N_{i}\; (1\leq i \leq m) @f$ and parameters @f$ u_{i}\; (1\leq i \leq m) @f$. The parameters are present only if *p*=1. The polyline *L* passes through the nodes. The deflection *d* describes the deflection of polyline *L* from the curve *C*: -@image html brep_wp_image397.gif. +@f[ \underset{P \in C}{max}\; \underset{Q \in L}{min}|Q-P| \leq d . @f] -The parameter @image html brep_wp_image113.gif (@image html brep_wp_image395.gif) is the parameter of the node @image html brep_wp_image394.gif on the curve @image html brep_wp_image205.gif: +The parameter @f$ u_{i}\; (1\leq i \leq m) @f$ is the parameter of the node *Ni* on the curve *C*: -@image html brep_wp_image398.gif. +@f[ C(u_{i})=N_{i} . @f] -The example record describes a polyline from @image html brep_wp_image098.gif nodes with a parameter presence flag @image html brep_wp_image396.gif, a deflection @image html brep_wp_image399.gif, nodes @image html brep_wp_image400.gif and @image html brep_wp_image401.gif and parameters @image html brep_wp_image142.gif and @image html brep_wp_image402.gif. +The example record describes a polyline from *m*=2 nodes with a parameter presence flag *p*=1, a deflection *d*=0.1, nodes *N1*=(1,0,0) and *N2*=(2,0,0) and parameters *u1*=0 and *u2*=1. -@subsection occt_brep_format_4_5 Subsection triangulations +@subsection occt_brep_format_4_5 Triangulations **Example** @@ -1367,46 +1405,48 @@ The example record describes a polyline from @image html brep_wp_image098.gif n **BNF-like Definition** @verbatim - triangulations = triangulation header _\n triangulation records; - - triangulation header = "Triangulations" _ triangulation count; - - triangulation records = triangulation record ^ triangulation count; - - triangulation record = triangulation node count _ triangulation triangle count _ triangulation parameter presence flag _ triangulation deflection _\n triangulation nodes [_ triangulation u v parameters] _ triangulation triangles _\n; - - triangulation node count = int; - - triangulation triangle count = int; - - triangulation parameter presence flag = flag; - - triangulation deflection = real; - - triangulation nodes = (triangulation node _) ^ triangulation node count; - - triangulation node = 3D point; - - triangulation u v parameters =           (triangulation u v parameter pair _) ^ triangulation node count; - - triangulation u v parameter pair = real _ real; - - triangulation triangles = (triangulation triangle _) ^ triangulation triangle count; - - triangulation triangle = int _ int _ int. + = <_\n> ; + + = "Triangulations" <_> ; + + = ^ ; + + = <_> <_> <_> <_\n> + [<_> ] <_> <_\n>; + + = ; + + = ; + + = ; + + = ; + + = ( <_>) ^ ; + + = <3D point>; + + = + ( <_>) ^ ; + + = <_> ; + + = ( <_>) ^ ; + + = <_> <_> . @endverbatim **Description** -triangulation record describes a triangulation @image html brep_wp_image403.gif which approximates a surface @image html brep_wp_image404.gif. The triangulation data consist of a node count @image html brep_wp_image405.gif, a triangle count @image html brep_wp_image406.gif, a parameter presence flag @image html brep_wp_image392.gif, a deflection @image html brep_wp_image393.gif, nodes @image html brep_wp_image394.gif (@image html brep_wp_image395.gif), parameter pairs @image html brep_wp_image407.gif (@image html brep_wp_image395.gif), triangles @image html brep_wp_image408.gif (@image html brep_wp_image131.gif, @image html brep_wp_image409.gif (@image html brep_wp_image410.gif)). The parameters are present only if @image html brep_wp_image396.gif. The deflection describes the triangulation deflection from the surface: +\ describes a triangulation *T* which approximates a surface *S*. The triangulation data consist of a node count @f$ m \geq 3 @f$, a triangle count @f$ k \geq 1 @f$, a parameter presence flag *p*, a deflection @f$ d \geq 0 @f$, nodes @f$ N_{i}\; (1\leq i \leq m) @f$, parameter pairs @f$ u_{i}\; v_{i}\; (1\leq i \leq m) @f$, triangles @f$ n_{j,1}\; n_{j,2}\; n_{j,3}\; (1\leq j \leq k,\; n_{j,l} \in \left \{1,...,m \right \}\; (1\leq l\leq 3)) @f$. The parameters are present only if *p*=1. The deflection describes the triangulation deflection from the surface: -@image html brep_wp_image411.gif. +@f[ \underset{P \in S}{max}\; \underset{Q \in T}{min}|Q-P| \leq d . @f] -The parameter pair @image html brep_wp_image407.gif (@image html brep_wp_image395.gif) describes the parameters of node @image html brep_wp_image394.gif on the surface: +The parameter pair @f$ u_{i}\; v_{i}\; (1\leq i \leq m) @f$ describes the parameters of node *Ni* on the surface: -@image html brep_wp_image412.gif. +@f[ S(u_{i},v_{i})=N_{i} . @f] -The triangle @image html brep_wp_image408.gif (@image html brep_wp_image131.gif) is interpreted as a triangle of nodes @image html brep_wp_image413.gif, @image html brep_wp_image414.gif and @image html brep_wp_image415.gif with circular traversal of the nodes in the order @image html brep_wp_image413.gif, @image html brep_wp_image414.gif and @image html brep_wp_image415.gif. From any side of the triangulation @image html brep_wp_image403.gif all its triangles have the same direction of the node circular traversal: either clockwise or counterclockwise. +The triangle @f$ n_{j,1}\; n_{j,2}\; n_{j,3}\; (1\leq j \leq k) @f$ is interpreted as a triangle of nodes @f$ N_{n_{j},1}\; N_{n_{j},2}@f$ and @f$ N_{n_{j},3} @f$ with circular traversal of the nodes in the order @f$ N_{n_{j},1}\; N_{n_{j},2}@f$ and @f$ N_{n_{j},3} @f$. From any side of the triangulation *T* all its triangles have the same direction of the node circular traversal: either clockwise or counterclockwise. Triangulation record @@ -1415,10 +1455,10 @@ Triangulation record 0 0 0 0 0 3 0 2 3 0 2 0 0 0 3 0 3 -2 0 -2 2 4 3 2 1 4 @endverbatim -describes a triangulation with @image html brep_wp_image416.gif nodes, @image html brep_wp_image417.gif triangles, parameter presence flag @image html brep_wp_image396.gif, deflection @image html brep_wp_image418.gif, nodes @image html brep_wp_image419.gif, @image html brep_wp_image420.gif, @image html brep_wp_image421.gif and @image html brep_wp_image422.gif, parameters @image html brep_wp_image423.gif, @image html brep_wp_image424.gif, @image html brep_wp_image425.gif and @image html brep_wp_image426.gif, and triangles @image html brep_wp_image427.gif and @image html brep_wp_image428.gif. From the point @image html brep_wp_image429.gif (@image html brep_wp_image430.gif) the triangles have clockwise (counterclockwise) direction of the node circular traversal. +describes a triangulation with *m*=4 nodes, *k*=2 triangles, parameter presence flag *p*=1, deflection *d*=0, nodes *N1*=(0,0,0), *N2*=(0,0,3), *N3*=(0,2,3) and *N4*=(0,2,0), parameters (*u1*, *v1*)=(0,0), (*u2*, *v2*)=(3,0), (*u3*, *v3*)=(3,-2) and (*u4*, *v4*)=(0,-2), and triangles (*n1,1*, *n1,2*, *n1,3*)=(2,4,3) and (*n2,1*, *n2,2*, *n2,3*)=(2,1,4). From the point (1,0,0) ((-1,0,0)) the triangles have clockwise (counterclockwise) direction of the node circular traversal. -@subsection occt_brep_format_4_6 Subsection polygons on triangulations +@subsection occt_brep_format_4_6 Polygons on triangulations **Example** @@ -1477,55 +1517,55 @@ describes a triangulation with @image html brep_wp_image416.gif nodes, @image h **BNF-like Definition** @verbatim - polygons on triangulations = polygons on triangulations header _\n - polygons on triangulations records; - - polygons on triangulations header = - "PolygonOnTriangulations" _ polygons on triangulations record count; - - polygons on triangulations record count = int; - - polygons on triangulations records = - polygons on triangulations record ^ polygons on triangulations record count; - - polygons on triangulations record = - polygons on triangulations node count _ polygons on triangulations node numbers _\n - "p" _ polygons on triangulations deflection _ - polygons on triangulations parameter presence flag - [_ polygons on triangulations u parameters] _\n; - - polygons on triangulations node count = int; - - polygons on triangulations node numbers = - polygons on triangulations node number ^ polygons on triangulations node count; - - polygons on triangulations node number = int; - - polygons on triangulations deflection = real; - - polygons on triangulations parameter presence flag = flag; - - polygons on triangulations u parameters = - (polygons on triangulations u parameter _) ^ polygons on triangulations node count; - - polygons on triangulations u parameter = real; + = <_\n> + ; + + = + "PolygonOnTriangulations" <_> ; + + = ; + + = + ^ ; + + = + <_> <_\n> + "p" <_> <_> + + [<_> ] <_\n>; + + = ; + + = + ^ ; + + = ; + + = ; + + = ; + + = + ( <_>) ^ ; + + = ; @endverbatim **Description** -polygons on triangulations describes a polyline @image html brep_wp_image390.gif on a triangulation which approximates a curve @image html brep_wp_image205.gif. The polyline data consist of a node count @image html brep_wp_image391.gif, node numbers @image html brep_wp_image431.gif, deflection @image html brep_wp_image393.gif, a parameter presence flag @image html brep_wp_image392.gif and parameters @image html brep_wp_image432.gif (@image html brep_wp_image395.gif). The parameters are present only if @image html brep_wp_image396.gif. The deflection @image html brep_wp_image165.gif describes the deflection of polyline @image html brep_wp_image390.gif from the curve @image html brep_wp_image205.gif: +\ describes a polyline *L* on a triangulation which approximates a curve *C*. The polyline data consist of a node count @f$ m \geq 2 @f$, node numbers @f$ n_{i} \geq 1 @f$, deflection @f$ d \geq 0 @f$, a parameter presence flag *p* and parameters @f$ u_{i}\; (1\leq i\leq m) @f$. The parameters are present only if *p*=1. The deflection *d* describes the deflection of polyline *L* from the curve *C*: -@image html brep_wp_image397.gif. +@f[ \underset{P \in C}{max}\; \underset{Q \in L}{min}|Q-P| \leq d . @f] -Parameter @image html brep_wp_image113.gif (@image html brep_wp_image395.gif) is @image html brep_wp_image433.gif-th node @image html brep_wp_image434.gif parameter on curve @image html brep_wp_image205.gif. +Parameter @f$ u_{i}\; (1\leq i\leq m) @f$ is *ni*-th node *C(ui)* parameter on curve *C*. @subsection occt_brep_format_4_7 Geometric Sense of a Curve -Geometric sense of curve @image html brep_wp_image210.gif described above is determined by the direction of parameter @image html brep_wp_image435.gif increasing. +Geometric sense of curve *C* described above is determined by the direction of parameter *u* increasing. -@section occt_brep_format_5 Section shapes +@section occt_brep_format_5 Shapes An example of section shapes and a whole *.brep file are given in chapter 7 "Appendix". @@ -1533,69 +1573,68 @@ An example of section shapes and a whole *.brep file are given in chapter 7 "Ap **BNF-like Definition** @verbatim - shapes = shape header _\n shape records _\n shape final record; - - shape header = "TShapes" _ shape count; - - shape count = int; - - shape records = shape record ^ shape count; - - shape record = shape subrecord _\n shape flag word _\n shape subshapes _\n; - - shape flag word = flag ^ 7; - - shape subshapes = (shape subshape _)* "*"; - - shape subshape = - shape subshape orientation shape subshape number _ shape location number; - - shape subshape orientation = "+" | "-" | "i" | "e"; - - shape subshape number = int; - - shape location number = int; - - shape final record = shape subshape; - - shape subrecord = - ("Ve" _\n vertex data _\n) | - ("Ed" _\n edge data _\n) | - ("Wi" _\n _\n) | - ("Fa" _\n face data) | - ("Sh" _\n _\n) | - ("So" _\n _\n) | - ("CS" _\n _\n) | - ("Co" _\n _\n); + = <_\n> <_\n> ; + + = "TShapes" <_> ; + + = ; + + = ^ ; + + = <_\n> <_\n> <_\n>; + + = ^ 7; + + = ( <_>)* "*"; + + = + <_> ; + + = "+" | "-" | "i" | "e"; + + = ; + + = ; + + = ; + + = + ("Ve" <_\n> <_\n>) | + ("Ed" <_\n> <_\n>) | + ("Wi" <_\n> <_\n>) | + ("Fa" <_\n> ) | + ("Sh" <_\n> <_\n>) | + ("So" <_\n> <_\n>) | + ("CS" <_\n> <_\n>) | + ("Co" <_\n> <_\n>); @endverbatim **Description** -shape flag word @image html brep_wp_image436.gif flags @image html brep_wp_image437.gif  -(@image html brep_wp_image438.gif) are interpreted as shape flags in the following way: +\ @f$ f_{1}\; f_{2}\; f_{3}\; f_{4}\; f_{5}\; f_{6}\; f_{7} @f$ \s @f$ f_{i}\;(1\leq i \leq 7) @f$ are interpreted as shape flags in the following way: - * @image html brep_wp_image439.gif – free; - * @image html brep_wp_image440.gif – modified; - * @image html brep_wp_image441.gif – IGNORED (version * / checked (version * ; - * @image html brep_wp_image442.gif – orientable; - * @image html brep_wp_image443.gif – closed; - * @image html brep_wp_image444.gif – infinite; - * @image html brep_wp_image445.gif – convex. + * @f$ f_{1} @f$ – free; + * @f$ f_{2} @f$ – modified; + * @f$ f_{3} @f$ – IGNORED(version 1) \\ checked (version 2); + * @f$ f_{4} @f$ – orientable; + * @f$ f_{5} @f$ – closed; + * @f$ f_{6} @f$ – infinite; + * @f$ f_{7} @f$ – convex. The flags are used in a special way [1]. -shape subshape orientation is interpreted in the following way: +\ is interpreted in the following way: * + – forward; * - – reversed; * i – internal; * e – external. -shape subshape orientation is used in a special way [1]. +\ is used in a special way [1]. -shape subshape number is the number of a shape record which is located in this section above the shape subshape number. shape record numbering is backward and starts from 1. +\ is the number of a \ which is located in this section above the \. \ numbering is backward and starts from 1. -shape subrecord types are interpreted in the following way: +\ types are interpreted in the following way: * "Ve" – vertex; * "Ed" – edge; @@ -1606,203 +1645,201 @@ shape subrecord types are interpreted in the following way: * "CS" – compsolid; * "Co" – compound. -shape final record determines the orientation and location for the whole model. +\ determines the orientation and location for the whole model. @subsection occt_brep_format_5_1 Common Terms -The terms below are used by vertex data, edge data and face data. +The terms below are used by \, \ and \. **BNF-like Definition** @verbatim - location number = int; - - 3D curve number = int; - - surface number = int; - - 2D curve number = int; - - 3D polygon number = int; - - triangulation number = int; - - polygon on triangulation number = int; - - curve parameter minimal and maximal values = real _ real; - - curve values for parameter minimal and maximal values = - real _ real _ real _ real; + = ; + + <3D curve number> = ; + + = ; + + <2D curve number> = ; + + <3D polygon number> = ; + + = ; + + = ; + + = <_> ; + + = + real> <_> <_> <_> ; @endverbatim **Description** -location number is the number of location record from section locations. location record numbering starts from 1. location number 0 is interpreted as the identity location. +\ is the number of \ from section locations. \ numbering starts from 1. \ 0 is interpreted as the identity location. -3D curve number is the number of a 3D curve record from subsection 3D curves of section geometry. 3D curve record numbering starts from 1. +\<3D curve number\> is the number of a \<3D curve record\> from subsection \<3D curves\> of section \. \<3D curve record\> numbering starts from 1. -surface number is the number of a surface record from subsection surfaces of section geometry. surface record numbering starts from 1. +\ is the number of a \ from subsection \ of section \. \ numbering starts from 1. -2D curve number is the number of a 2D curve record from subsection 2D curves of section geometry. 2D curve record numbering starts from 1. +\<2D curve number\> is the number of a \<2D curve record\> from subsection \<2D curves\> of section \. \<2D curve record\> numbering starts from 1. -3D polygon number is the number of a 3D polygon record from subsection 3D polygons of section geometry. 3D polygon record numbering starts from 1. +\<3D polygon number\> is the number of a \<3D polygon record\> from subsection \<3D polygons\> of section \. \<3D polygon record\> numbering starts from 1. -triangulation number is the number of a triangulation record from subsection triangulations of section geometry. triangulation record numbering starts from 1. +\ is the number of a \ from subsection \ of section \. \ numbering starts from 1. -polygon on triangulation number is the number of a polygons on triangulations record from subsection polygons on triangulations of section geometry. -polygons on triangulations record numbering starts from 1. +\ number is the number of a \ from subsection \ of section \. +\ numbering starts from 1. -curve parameter minimal and maximal values @image html brep_wp_image446.gif and @image html brep_wp_image447.gif are the curve parameter@image html brep_wp_image448.gif bounds: @image html brep_wp_image449.gif. +\ *umin* and *umax* are the curve parameter *u* bounds: *umin* @f$ \leq @f$ *u* @f$ \leq @f$ *umax*. -curve values for parameter minimal and maximal values @image html brep_wp_image446.gif and @image html brep_wp_image447.gif are real pairs @image html brep_wp_image450.gif and @image html brep_wp_image451.gif that @image html brep_wp_image452.gif and @image html brep_wp_image453.gif where @image html brep_wp_image205.gif is a parametric equation of the curve. +\ *umin* and *umax* are real pairs *xmin ymin* and *xmax ymax* that (*xmin*, *ymin*)= *C* (*umin*) and (*xmax*, *ymax*)= *C* (*umax*) where *C* is a parametric equation of the curve. -@subsection occt_brep_format_5_2 vertex data +@subsection occt_brep_format_5_2 Vertex data **BNF-like Definition** @verbatim - vertex data = vertex data tolerance _\n vertex data 3D representation _\n vertex data representations; - - vertex data tolerance = real; - - vertex data 3D representation = 3D point; - - vertex data representations = (vertex data representation _\n)* "0 0"; - - vertex data representation = vertex data representation u parameter _ - vertex data representation data _ location number; - - vertex data representation u parameter = real; - - vertex data representation data = - ("1" _ vertex data representation data * | - ("2" _ vertex data representation data * | - ("3" _ vertex data representation data * ; - - vertex data representation data 1 = 3D curve number; - - vertex data representation data 2 = 2D curve number _ surface number; - - vertex data representation data 3 = - vertex data representation v parameter _ surface number; - - vertex data representation v parameter = real; + = <_\n> <_\n> + ; + + = ; + + = <3D point>; + + = ( <_\n>)* "0 0"; + + = <_> + <_> ; + + = ; + + = + ("1" <_> ) | + ("2" <_> ) | + ("3" <_> ); + + = <3D curve number>; + + = <2D curve number> <_> ; + + = + <_> ; + + = ; @endverbatim **Description** -The usage of vertex data representation u parameter @image html brep_wp_image448.gif is described below. +The usage of \ *U* is described below. -vertex data representation data 1 and parameter @image html brep_wp_image448.gif describe the position of the vertex @image html brep_wp_image454.gif on a 3D curve @image html brep_wp_image205.gif. Parameter @image html brep_wp_image448.gif is a parameter of the vertex @image html brep_wp_image454.gif on the curve @image html brep_wp_image205.gif: @image html brep_wp_image455.gif. +\ and parameter *U* describe the position of the vertex *V* on a 3D curve *C*. Parameter *U* is a parameter of the vertex *V* on the curve *C*: *C(u)=V*. -vertex data representation data 2 and parameter @image html brep_wp_image448.gif describe the position of the vertex @image html brep_wp_image454.gif on a 2D curve @image html brep_wp_image205.gif which is located on a surface. Parameter @image html brep_wp_image448.gif is a parameter of the vertex @image html brep_wp_image454.gif on the curve @image html brep_wp_image205.gif: @image html brep_wp_image455.gif. +\ and parameter *U* describe the position of the vertex *V* on a 2D curve *C* which is located on a surface. Parameter *U* is a parameter of the vertex *V* on the curve *C*: *C(u)=V*. -vertex data representation data 3 and parameter @image html brep_wp_image435.gif describe the position of the vertex @image html brep_wp_image454.gif on a surface @image html brep_wp_image404.gif through vertex data representation v parameter @image html brep_wp_image456.gif: @image html brep_wp_image457.gif. +\ and parameter *u* describe the position of the vertex *V* on a surface *S* through \ *v*: *S(u,v)=V*. -vertex data tolerance @image html brep_wp_image458.gif describes the maximum distance from the vertex @image html brep_wp_image454.gif to the set @image html brep_wp_image459.gif* *of vertex @image html brep_wp_image454.gif representations: +\ *t* describes the maximum distance from the vertex *V* to the set *R* of vertex *V* representations: -@image html brep_wp_image460.gif. +@f[ \underset{P \in R }{max} |P-V| \leq t . @f] -@subsection occt_brep_format_5_3 edge data +@subsection occt_brep_format_5_3 Edge data **BNF-like Definition** @verbatim - edge data = _ edge data tolerance _ edge data same parameter flag _ edge data same range flag _ edge data degenerated flag _\n edge data representations; - - edge data tolerance = real; - - edge data same parameter flag = flag; - - edge data same range flag = flag; - - edge data degenerated flag = flag; - - edge data representations = (edge data representation _\n)* "0"; - - edge data representation = - "1" _ edge data representation data 1 - "2" _ edge data representation data 2 - "3" _ edge data representation data 3 - "4" _ edge data representation data 4 - "5" _ edge data representation data 5 - "6" _ edge data representation data 6 - "7" _ edge data representation data 7; - - edge data representation data 1 = 3D curve number _ location number _ - curve parameter minimal and maximal values; - - edge data representation data 2 = 2D curve number _ surface number _ location number _ curve parameter minimal and maximal values - [_\n curve values for parameter minimal and maximal values]; - - edge data representation data 3 = (2D curve number _) ^ 2 continuity order _ surface number _ location number _ curve parameter minimal and maximal values \n curve values for parameter minimal and maximal values]; - - continuity order = "C0" | "C1" | "C2" | "C3" | "CN" | "G1" | "G2". - - edge data representation data 4 = - continuity order (_ surface number _ location number) ^ 2; - - edge data representation data 5 = 3D polygon number _ location number; - - edge data representation data 6 = - polygon on triangulation number _ triangulation number _ location number; - - edge data representation data 7 = (polygon on triangulation number _) ^ 2 triangulation number _ location number; + = <_> <_> <_> edge data same range flag> <_> <_\n> ; + + = ; + + = ; + + = ; + + = ; + + = ( <_\n>)* "0"; + + = + "1" <_> + "2" <_> + "3" <_> + "4" <_> + "5" <_> + "6" <_> + "7" <_> ; + + = <3D curve number> <_> <_> + ; + + = <2D curve number> <_> <_> + <_> + [<_\n> ]; + + = (<2D curve number> <_>) ^ 2 <_> <_> <_> <\n> ]; + + = "C0" | "C1" | "C2" | "C3" | "CN" | "G1" | "G2". + + = + (<_> <_> ) ^ 2; + + = <3D polygon number> <_> ; + + = + <_> <_> ; + + = ( <_>) ^ 2 + <_> ; @endverbatim **Description** -Flags edge data same parameter flag, edge data same range flag and edge data degenerated flag are used in a special way [1]. +Flags \, \ and \ are used in a special way [1]. -edge data representation data 1 describes a 3D curve. +\ describes a 3D curve. -edge data representation data 2 describes a 2D curve on a surface. -curve values for parameter minimal and maximal values are used only in version 2. +\ describes a 2D curve on a surface. +\ are used only in version 2. -edge data representation data 3 describes a 2D curve on a closed surface. -curve values for parameter minimal and maximal values are used only in version 2. +\ describes a 2D curve on a closed surface. +\ are used only in version 2. -edge data representation data 5 describes a 3D polyline. +\ describes a 3D polyline. -edge data representation data 6 describes a polyline on a triangulation. +\ describes a polyline on a triangulation. -edge data tolerance @image html brep_wp_image458.gif describes the maximum distance from the edge @image html brep_wp_image461.gif to the set @image html brep_wp_image459.gif of edge @image html brep_wp_image461.gif representations: +\ *t* describes the maximum distance from the edge *E* to the set *R* of edge *E* representations: -@image html brep_wp_image462.gif. +@f[ \underset{C \in R}{max}\;\underset{P \in E}{max}\;\underset{Q \in C}{min}|Q-P| \leq t @f] -@subsection occt_brep_format_5_4 face data +@subsection occt_brep_format_5_4 Face data **BNF-like Definition** @verbatim - face data = face data natural restriction flag _ face data tolerance _ surface number _ location number \n ["2" _ triangulation number]; - - face data natural restriction flag = flag; - - face data tolerance = real; + = <_> <_> <_> <\n> ["2" <_> ]; + + = ; + + = ; @endverbatim **Description** -face data describes a surface @image html brep_wp_image404.gif of face @image html brep_wp_image463.gif and a triangulation @image html brep_wp_image403.gif of face @image html brep_wp_image463.gif. The surface @image html brep_wp_image404.gif may be empty: surface number = 0. +\ describes a surface *S* of face *F* and a triangulation *T* of face *F*. The surface *S* may be empty: \ = 0. -face data tolerance @image html brep_wp_image458.gif describes the maximum distance from the face @image html brep_wp_image463.gif to the surface @image html brep_wp_image404.gif: +\ *t* describes the maximum distance from the face *F* to the surface *S*: -@image html brep_wp_image464.gif. +@f[ \underset{P \in F}{max}\;\underset{Q \in S}{min}|Q-P| \leq t @f] - Flag face data natural restriction flag is used in a special way [1]. +Flag \ is used in a special way [1]. -@section occt_brep_format_6 References - - The format is part of Open CASCADE Technology (OCCT). - Some data fields of the format have additional values which are used in OCCT. - Some data fields of the format are specific for OCCT. - -@section occt_brep_format_7 Appendix + +@section occt_brep_format_6 Appendix This chapter contains a *.brep file example. diff --git a/dox/user_guides/brep_wp/images/brep_wp_image003.gif b/dox/user_guides/brep_wp/images/brep_wp_image003.gif deleted file 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b/dox/user_guides/voxels_wp/images/voxels_wp_image010.jpg deleted file mode 100644 index f1219fc2b1..0000000000 Binary files a/dox/user_guides/voxels_wp/images/voxels_wp_image010.jpg and /dev/null differ diff --git a/dox/user_guides/voxels_wp/images/voxels_wp_image010.png b/dox/user_guides/voxels_wp/images/voxels_wp_image010.png new file mode 100644 index 0000000000..b024013e00 Binary files /dev/null and b/dox/user_guides/voxels_wp/images/voxels_wp_image010.png differ diff --git a/dox/user_guides/voxels_wp/voxels_wp.md b/dox/user_guides/voxels_wp/voxels_wp.md index e577e983ac..b8d44134fa 100644 --- a/dox/user_guides/voxels_wp/voxels_wp.md +++ b/dox/user_guides/voxels_wp/voxels_wp.md @@ -16,8 +16,8 @@ Voxel Package {#occt_voxels_wp} To produce a voxel representation the 3D space is split by equal intervals along the main orthogonal coordinate axes to obtain nx x ny x nz voxels (small cubes): -@image html voxels_wp_image003.jpg "A cube of 3 X 3 X 3  = 9 voxels." -@image latex voxels_wp_image003.jpg "A cube of 3 X 3 X 3  = 9 voxels." +@image html voxels_wp_image003.png "A cube of 3 X 3 X 3  = 9 voxels." +@image latex voxels_wp_image003.png "A cube of 3 X 3 X 3  = 9 voxels." The data are attached to each voxel and remain the same within the voxel. It means that we obtain the 3D space with discrete data distribution. @@ -34,10 +34,11 @@ Voxel Package {#occt_voxels_wp} as colored or black/white points and cubes, displaying only the voxels visible from the user's point of view. -@image html voxels_wp_image004.jpg "A shape and its voxel representation" -@image html voxels_wp_image005.jpg "A shape and its voxel representation" -@image latex voxels_wp_image004.jpg "A shape and its voxel representation" -@image latex voxels_wp_image005.jpg "A shape and its voxel representation" +@image html voxels_wp_image004.png "A shape and its voxel representation" +@image html voxels_wp_image005.png "A shape and its voxel representation" + +@image latex voxels_wp_image004.png "A shape and its voxel representation" +@image latex voxels_wp_image005.png "A shape and its voxel representation" In these images a boundary representation is displayed to the left. In the center and to the right there are 3D discrete representations (or 3D discrete topology). Any solid shape can be translated into a voxel representation. @@ -100,8 +101,8 @@ The algorithm can run in multi-threaded mode (the number of threads is unlimite It is possible to display the voxels from a particular range of values (iso-volume): -@image html voxels_wp_image006.jpg "Iso-volume of a shape" -@image latex voxels_wp_image006.jpg "Iso-volume of a shape" +@image html voxels_wp_image006.png "Iso-volume of a shape" +@image latex voxels_wp_image006.png "Iso-volume of a shape" The voxels are displayed by means of "direct drawing in Open GL" technology or "user draw" technology. Therefore, some visualization files are compiled within Open CASCADE Technology, but the files of "direct drawing" are compiled  by the end-user application. @@ -115,15 +116,15 @@ It is necessary to include the files *Voxel_VisData.h*, *VoxelClient_VisDrawer. The *File* menu allows creation of canonical shapes (box, cylinder, sphere, torus) or loading of shapes in BREP format: -@image html voxels_wp_image007.jpg "Demo-application. Creation or loading of a shape" -@image latex voxels_wp_image007.jpg "Demo-application. Creation or loading of a shape" +@image html voxels_wp_image007.png "Demo-application. Creation or loading of a shape" +@image latex voxels_wp_image007.png "Demo-application. Creation or loading of a shape" The menu *Converter* voxelizes the shape. Two types of voxels can be obtained: 1-bit or 4-bit voxels. * 1-bit voxels are displayed in white color on black background. * 4-bit voxels use 16 colors filling the model in a special way for demonstrative purposes: -@image html voxels_wp_image008.jpg "Demo-application. Voxelization" -@image latex voxels_wp_image008.jpg "Demo-application. Voxelization" +@image html voxels_wp_image008.png "Demo-application. Voxelization" +@image latex voxels_wp_image008.png "Demo-application. Voxelization" The converter uses two threads (two processors, if available) to perform voxelization. @@ -131,13 +132,13 @@ The menu *Converter* voxelizes the shape. Two types of voxels can be obtained allows defining the size of points and boxes (quadrangles), the minimum and the maximum displayed color, and the boundaries of the bounding box for displayed voxels: -@image html voxels_wp_image009.jpg "Demo-application. Visualization" -@image latex voxels_wp_image009.jpg "Demo-application. Visualization" +@image html voxels_wp_image009.png "Demo-application. Visualization" +@image latex voxels_wp_image009.png "Demo-application. Visualization" The last menu, *Demo* contains a demo-command for running waves of 4-bit voxels: -@image html voxels_wp_image010.jpg "Demo-application. Running waves" -@image latex voxels_wp_image010.jpg "Demo-application. Running waves" +@image html voxels_wp_image010.png "Demo-application. Running waves" +@image latex voxels_wp_image010.png "Demo-application. Running waves" @section occt_voxels_wp_6 Future development