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Added {.cpp} highlighting markers to the C++ code blocks in documentation;
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2294 lines
88 KiB
Markdown
2294 lines
88 KiB
Markdown
BRep Format {#specification__brep_format}
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========================
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@tableofcontents
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@section specification__brep_format_1 Introduction
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BREP format is used to store 3D models and allows to store a model which consists
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of vertices, edges, wires, faces, shells, solids, compsolids, compounds, edge triangulations,
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face triangulations, polylines on triangulations, space location and orientation.
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Any set of such models may be stored as a single model which is a compound of the models.
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The format is described in an order which is convenient for understanding
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rather than in the order the format parts follow each other.
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BNF-like definitions are used in this document.
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Most of the chapters contain BREP format descriptions in the following order:
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* format file fragment to illustrate the part;
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* BNF-like definition of the part;
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* detailed description of the part.
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**Note** that the format is a part of Open CASCADE Technology (OCCT).
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Some data fields of the format have additional values, which are used in OCCT.
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Some data fields of the format are specific for OCCT.
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@section specification__brep_format_2 Storage of shapes
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*BRepTools* and *BinTools* packages contain methods *Read* and *Write* allowing to read and write a Shape to/from a stream or a file.
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The methods provided by *BRepTools* package use ASCII storage format; *BinTools* package uses binary format.
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Each of these methods has two arguments:
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- a *TopoDS_Shape* object to be read/written;
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- a stream object or a file name to read from/write to.
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The following sample code reads a shape from ASCII file and writes it to a binary one:
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~~~~{.cpp}
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TopoDS_Shape aShape;
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if (BRepTools::Read (aShape, "source_file.txt")) {
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BinTools::Write (aShape, "result_file.bin");
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}
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~~~~
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@section specification__brep_format_3 Format Common Structure
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ASCII encoding is used to read/write BREP format from/to file. The format data are stored in a file as text data.
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BREP format uses the following BNF terms:
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* \<\\n\>: It is the operating-system-dependent ASCII character sequence which separates ASCII text strings in the operating system used;
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* \<_\\n\>: = " "*\<\\n\>;
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* \<_\>: = " "+; It is a not empty sequence of space characters with ASCII code 21h;
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* \<flag\>: = "0" | "1";
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* \<int\>: It is an integer number from -2<sup>31</sup> to 2<sup>31</sup>-1 which is written in denary system;
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* \<real\>: It is a real from -1.7976931348623158 @f$\cdot@f$ 10<sup>308</sup> to 1.7976931348623158 @f$\cdot@f$ 10<sup>308</sup> which is written in decimal or E form with base 10.The point is used as a delimiter of the integer and fractional parts;
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* \<short real\>: It is a real from -3.402823 @f$\cdot@f$ 10<sup>38</sup> to 3.402823 @f$\cdot@f$ 10<sup>38</sup> which is written in decimal or E form with base 10.The point is used as a delimiter of the integer and fractional parts;
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* \<2D point\>: = \<real\>\<_\>\<real\>;
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* \<3D point\>: = \<real\>(\<_\>\<real)\><sup>2</sup>;
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* \<2D direction\>: It is a \<2D point\> *x y* so that *x<sup>2</sup> + y<sup>2</sup>* = 1;
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* \<3D direction\>: It is a \<3D point\> *x y z* so that *x<sup>2</sup> + y<sup>2</sup> + z<sup>2</sup>* = 1;
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* \<+\>: It is an arithmetic operation of addition.
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The format consists of the following sections:
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* \<content type\>;
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* \<version\>;
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* \<locations\>;
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* \<geometry\>;
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* \<shapes\>.
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\<content type\> = "DBRep_DrawableShape" \<_\\n\>\<_\\n\>;
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\<content type\> have other values [1].
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\<version\> = ("CASCADE Topology V1, (c) Matra-Datavision" | "CASCADE Topology V2, (c) Matra-Datavision" | "CASCADE Topology V3, (c) Open Cascade")\<_\\n\>;
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The difference of the versions is described in the document.
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Sections \<locations\>, \<geometry\> and \<shapes\> are described below in separate chapters of the document.
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@section specification__brep_format_4 Locations
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**Example**
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@verbatim
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Locations 3
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1
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0 0 1 0
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1 0 0 0
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0 1 0 0
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1
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1 0 0 4
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0 1 0 5
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0 0 1 6
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2 1 1 2 1 0
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@endverbatim
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**BNF-like Definition**
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@verbatim
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<locations> = <location header> <_\n> <location records>;
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<location header> = "Locations" <_> <location record count>;
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<location record count> = <int>;
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<location records> = <location record> ^ <location record count>;
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<location record> = <location record 1> | <location record 2>;
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<location record 1> = "1" <_\n> <location data 1>;
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<location record 2> = "2" <_> <location data 2>;
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<location data 1> = ((<_> <real>) ^ 4 <_\n>) ^ 3;
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<location data 2> = (<int> <_> <int> <_>)* "0" <_\n>;
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@endverbatim
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**Description**
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\<location data 1\> is interpreted as a 3 x 4 matrix
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@f$Q =
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\begin{pmatrix}
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{q}_{1,1} &{q}_{1,2} &{q}_{1,3} &{q}_{1,4}\\
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{q}_{2,1} &{q}_{2,2} &{q}_{2,3} &{q}_{2,4}\\
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{q}_{3,1} &{q}_{3,2} &{q}_{3,3} &{q}_{3,4}
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\end{pmatrix}@f$
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which describes transformation of 3 dimensional space and satisfies the following constraints:
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* @f$ d \neq 0@f$ where @f$d = |Q_{2}|@f$ where
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@f$ Q_{2} = \begin{pmatrix}
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{q}_{1,1} &{q}_{1,2} &{q}_{1,3} &{q}_{1,4}\\
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{q}_{2,1} &{q}_{2,2} &{q}_{2,3} &{q}_{2,4}\\
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{q}_{3,1} &{q}_{3,2} &{q}_{3,3} &{q}_{3,4}
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\end{pmatrix}; @f$
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* @f$ Q_{3}^{T} = Q_{3}^{-1}@f$ where @f$Q_{3} = Q_{2}/d^{1/3}. @f$
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The transformation transforms a point (x, y, z) to another point (u, v, w) by the rule:
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@f[ \begin{pmatrix}
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u \\ v \\ w
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\end{pmatrix} =
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Q\cdot(x\;y\;z\;1)^{T} =
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\begin{pmatrix}
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{q}_{1,1}\cdot x +{q}_{1,2}\cdot y +{q}_{1,3}\cdot z +{q}_{1,4}\\
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{q}_{2,1}\cdot x +{q}_{2,2}\cdot y +{q}_{2,3}\cdot z +{q}_{2,4}\\
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{q}_{3,1}\cdot x +{q}_{3,2}\cdot y +{q}_{3,3}\cdot z +{q}_{3,4}
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\end{pmatrix} . @f]
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*Q* may be a composition of matrices for the following elementary transformations:
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* parallel translation --
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@f$ \begin{pmatrix}
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1 &0 &0 &{q}_{1,4}\\
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0 &1 &0 &{q}_{2,4}\\
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0 &0 &1 &{q}_{3,4}
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\end{pmatrix}; @f$
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* rotation around an axis with a direction *D(D<sub>x</sub>, D<sub>y</sub>, D<sub>z</sub>)* by an angle @f$ \varphi @f$ --
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@f[ \begin{pmatrix}
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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\\
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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\\
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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
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\end{pmatrix}; @f]
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* 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$
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* central symmetry -- @f$ \begin{pmatrix} -1 &0 &0 &0\\ 0 &-1 &0 &0\\ 0 &0 &-1 &0 \end{pmatrix}; @f$
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* axis symmetry -- @f$ \begin{pmatrix} -1 &0 &0 &0\\ 0 &-1 &0 &0\\ 0 &0 &1 &0 \end{pmatrix}; @f$
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* plane symmetry -- @f$ \begin{pmatrix} 1 &0 &0 &0\\ 0 &1 &0 &0\\ 0 &0 &-1 &0 \end{pmatrix}. @f$
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\<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 @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$. \<flag\> 0 is the indicator of the sequence end. The sequence is interpreted as a composition @f$ 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 \<location record\> in the section locations. \<location record\> numbering starts from 1.
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@section specification__brep_format_5 Geometry
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@verbatim
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<geometry> =
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<2D curves>
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<3D curves>
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<3D polygons>
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<polygons on triangulations>
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<surfaces>
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<triangulations>;
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@endverbatim
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@subsection specification__brep_format_5_1 3D curves
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**Example**
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@verbatim
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Curves 13
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1 0 0 0 0 0 1
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1 0 0 3 -0 1 0
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1 0 2 0 0 0 1
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1 0 0 0 -0 1 0
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1 1 0 0 0 0 1
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1 1 0 3 0 1 0
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1 1 2 0 0 0 1
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1 1 0 0 -0 1 0
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1 0 0 0 1 0 -0
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1 0 0 3 1 0 -0
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1 0 2 0 1 0 -0
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1 0 2 3 1 0 -0
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1 1 0 0 1 0 0
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@endverbatim
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**BNF-like Definition**
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@verbatim
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<3D curves> = <3D curve header> <_\n> <3D curve records>;
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<3D curve header> = "Curves" <_> <3D curve count>;
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<3D curve count> = <int>;
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<3D curve records> = <3D curve record> ^ <3D curve count>;
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<3D curve record> =
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<3D curve record 1> |
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<3D curve record 2> |
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<3D curve record 3> |
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<3D curve record 4> |
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<3D curve record 5> |
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<3D curve record 6> |
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<3D curve record 7> |
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<3D curve record 8> |
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<3D curve record 9>;
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@endverbatim
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@subsubsection specification__brep_format_5_1_1 Line - \<3D curve record 1\>
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**Example**
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@verbatim
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1 1 0 3 0 1 0
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@endverbatim
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**BNF-like Definition**
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@verbatim
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<3D curve record 1> = "1" <_> <3D point> <_> <3D direction> <_\n>;
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@endverbatim
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**Description**
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\<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:
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@f[ C(u)=P+u \cdot D, \; u \in (-\infty,\; \infty). @f]
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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$.
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@subsubsection specification__brep_format_5_1_2 Circle - \<3D curve record 2\>
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**Example**
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@verbatim
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2 1 2 3 0 0 1 1 0 -0 -0 1 0 4
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@endverbatim
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**BNF-like Definition**
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~~~~{.cpp}
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<3D curve record 2> = "2" <_> <3D circle center> <_> <3D circle N> <_> <3D circle Dx> <_> <3D circle Dy> <_> <3D circle radius> <_\n>;
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<3D circle center> = <3D point>;
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<3D circle N> = <3D direction>;
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<3D circle Dx> = <3D direction>;
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<3D circle Dy> = <3D direction>;
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<3D circle radius> = <real>;
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~~~~
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**Description**
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\<3D curve record 2\> describes a circle. The circle data consist of a 3D point *P*, pairwise orthogonal 3D directions *N*, *D<sub>x</sub>* and *D<sub>y</sub>* 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:
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@f[ C(u)=P+r \cdot (cos(u) \cdot D_{x} + sin(u) \cdot D_{y}), \; u \in [o,\;2 \cdot \pi). @f]
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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 *D<sub>x</sub>*=(1, 0 ,0) and *D<sub>y</sub>*=(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$.
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@subsubsection specification__brep_format_5_1_3 Ellipse - \<3D curve record 3\>
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**Example**
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@verbatim
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3 1 2 3 0 0 1 1 0 -0 -0 1 0 5 4
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@endverbatim
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**BNF-like Definition**
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~~~~{.cpp}
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<3D curve record 3> = "3" <_> <3D ellipse center> <_> <3D ellipse N> <_> <3D ellipse Dmaj> <_> <3D ellipse Dmin> <_> <3D ellipse Rmaj> <_> <3D ellipse Rmin> <_\n>;
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<3D ellipse center> = <3D point>;
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<3D ellipse N> = <3D direction>;
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<3D ellipse Dmaj> = <3D direction>;
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<3D ellipse Dmin> = <3D direction>;
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<3D ellipse Rmaj> = <real>;
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<3D ellipse Rmin> = <real>;
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~~~~
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**Description**
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\<3D curve record 3\> describes an ellipse. The ellipse data consist of a 3D point *P*, pairwise orthogonal 3D directions *N*, *D<sub>maj</sub>* and *D<sub>min</sub>* and non-negative reals *r<sub>maj</sub>* and *r<sub>min</sub>* so that *r<sub>min</sub>* @f$ \leq @f$ *r<sub>maj</sub>*. The ellipse has its center *P*, is located in plane with the normal *N*, has major and minor axis directions *D<sub>maj</sub>* and *D<sub>min</sub>*, major and minor radii *r<sub>maj</sub>* and *r<sub>min</sub>* and is defined by the following parametric equation:
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@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]
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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 *D<sub>maj</sub>*=(1, 0, 0) and *D<sub>min</sub>*=(0, 1, 0), major and minor radii *r<sub>maj</sub>*=5 and *r<sub>min</sub>*=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$.
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@subsubsection specification__brep_format_5_1_4 Parabola - \<3D curve record 4\>
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**Example**
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@verbatim
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4 1 2 3 0 0 1 1 0 -0 -0 1 0 16
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@endverbatim
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**BNF-like Definition**
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~~~~{.cpp}
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<3D curve record 4> = "4" <_> <3D parabola origin> <_> <3D parabola N> <_> <3D parabola Dx> <_> <3D parabola Dy> <_> <3D parabola focal length> <_\n>;
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<3D parabola origin> = <3D point>;
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<3D parabola N> = <3D direction>;
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<3D parabola Dx> = <3D direction>;
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<3D parabola Dy> = <3D direction>;
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<3D parabola focal length> = <real>;
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~~~~
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**Description**
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\<3D curve record 4\> describes a parabola. The parabola data consist of a 3D point *P*, pairwise orthogonal 3D directions *N*, *D<sub>x</sub>* and *D<sub>y</sub>* 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:
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@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]
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@f[ C(u)=P+u \cdot D_{x}, u \in (-\infty,\; \infty) \Leftarrow f = 0\;(degenerated\;case). @f]
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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 *D<sub>x</sub>*=(1, 0, 0) and *D<sub>y</sub>*=(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$.
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@subsubsection specification__brep_format_5_1_5 Hyperbola - \<3D curve record 5\>
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**Example**
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@verbatim
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5 1 2 3 0 0 1 1 0 -0 -0 1 0 5 4
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@endverbatim
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**BNF-like Definition**
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~~~~{.cpp}
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<3D curve record 5> = "5" <_> <3D hyperbola origin> <_> <3D hyperbola N> <_> <3D hyperbola Dx> <_> <3D hyperbola Dy> <_> <3D hyperbola Kx> <_> <3D hyperbola Ky> <_\n>;
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<3D hyperbola origin> = <3D point>;
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<3D hyperbola N> = <3D direction>;
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<3D hyperbola Dx> = <3D direction>;
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<3D hyperbola Dy> = <3D direction>;
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<3D hyperbola Kx> = <real>;
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<3D hyperbola Ky> = <real>;
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~~~~
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**Description**
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\<3D curve record 5\> describes a hyperbola. The hyperbola data consist of a 3D point *P*, pairwise orthogonal 3D directions *N*, *D<sub>x</sub>* and *D<sub>y</sub>* and non-negative reals *k<sub>x</sub>* and *k<sub>y</sub>*. 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:
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|
||
@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 *P*=(1, 2, 3) and has a normal *N*=(0, 0, 1). Other hyperbola data are *D<sub>x</sub>*=(1, 0, 0), *D<sub>y</sub>*=(0, 1, 0), *k<sub>x</sub>*=5 and *k<sub>y</sub>*=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$.
|
||
|
||
|
||
@subsubsection specification__brep_format_5_1_6 Bezier Curve - \<3D curve record 6\>
|
||
|
||
**Example**
|
||
|
||
@verbatim
|
||
6 1 2 0 1 0 4 1 -2 0 5 2 3 0 6
|
||
@endverbatim
|
||
|
||
**BNF-like Definition**
|
||
|
||
@verbatim
|
||
<3D curve record 6> = "6" <_> <3D Bezier rational flag> <_> <3D Bezier degree>
|
||
<3D Bezier weight poles> <_\n>;
|
||
|
||
<3D Bezier rational flag> = <flag>;
|
||
|
||
<3D Bezier degree> = <int>;
|
||
|
||
3D Bezier weight poles> = (<_> <3D Bezier weight pole>) ^ (<3D Bezier degree> <+> "1");
|
||
|
||
<3D Bezier weight pole> = <3D point> [<_> <real>];
|
||
@endverbatim
|
||
|
||
**Description**
|
||
|
||
\<3D curve record 6\> describes a Bezier curve. The curve data consist of a rational *r*, a degree @f$ m \leq 25 @f$ and weight poles.
|
||
|
||
The weight poles are *m*+1 3D points *B<sub>0</sub> ... B<sub>m</sub>* if the flag *r* is 0. The weight poles are *m*+1 pairs *B<sub>0</sub>h<sub>0</sub> ... B<sub>m</sub>h<sub>m</sub>* if flag *r* is 1. Here *B<sub>i</sub>* is a 3D point and *h<sub>i</sub>* 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:
|
||
|
||
@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 @f$ 0^{0} \equiv 1 @f$.
|
||
|
||
The example record is interpreted as a Bezier curve with a rational flag *r*=1, degree *m*=2 and weight poles *B<sub>0</sub>*=(0, 1, 0), *h<sub>0</sub>*=4, *B<sub>1</sub>*=(1, -2, 0), *h<sub>1</sub>*=5 and *B<sub>2</sub>*=(2, 3, 0), *h<sub>2</sub>*=6. The Bezier curve is defined by the following parametric equation:
|
||
|
||
@f[ C(u)=\frac{(0,1,0) \cdot 4 \cdot (1-u)^{2}+(1,-2,0) \cdot 5 \cdot 2 \cdot u \cdot (1-u) + (2,3,0) \cdot 6 \cdot u^{2} )}{4 \cdot (1-u)^{2}+5 \cdot 2 \cdot u \cdot (1-u)+6 \cdot u^{2}}. @f]
|
||
|
||
|
||
@subsubsection specification__brep_format_5_1_7 B-Spline Curve - \<3D curve record 7\>
|
||
|
||
**Example**
|
||
|
||
@verbatim
|
||
7 1 0 1 3 5 0 1 0 4 1 -2 0 5 2 3 0 6
|
||
0 1 0.25 1 0.5 1 0.75 1 1 1
|
||
@endverbatim
|
||
|
||
**BNF-like Definition**
|
||
|
||
~~~~{.cpp}
|
||
<3D curve record 7> = "7" <_> <3D B-spline rational flag> <_> "0" <_> <3D B-spline degree> <_>
|
||
<3D B-spline pole count> <_> <3D B-spline multiplicity knot count> <3D B-spline weight poles>
|
||
<_\n> <3D B-spline multiplicity knots> <_\n>;
|
||
|
||
<3D B-spline rational flag> = <flag>;
|
||
|
||
<3D B-spline degree> = <int>;
|
||
|
||
<3D B-spline pole count> = <int>;
|
||
|
||
<3D B-spline multiplicity knot count> = <int>;
|
||
|
||
<3D B-spline weight poles> = (<_> <3D B-spline weight pole>) ^ <3D B-spline pole count>;
|
||
|
||
<3D B-spline weight pole> = <3D point> [<_> <real>];
|
||
|
||
<3D B-spline multiplicity knots> = (<_> <3D B-spline multiplicity knot>) ^ <3D B-spline multiplicity knot count>;
|
||
|
||
<3D B-spline multiplicity knot> = <real> <_> <int>;
|
||
~~~~
|
||
|
||
**Description**
|
||
|
||
\<3D curve record 7\> describes a B-spline curve. The curve data consist of a rational flag *r*, a degree @f$ m \leq 25 @f$, pole count @f$ n \geq 2 @f$, multiplicity knot count *k*, weight poles and multiplicity knots.
|
||
|
||
The weight poles are *n* 3D points *B<sub>1</sub> ... B<sub>n</sub>* if the flag *r* is 0. The weight poles are *n* pairs *B<sub>1</sub>h<sub>1</sub> ... B<sub>n</sub>h<sub>n</sub>* if the flag *r* is 1. Here *B<sub>i</sub>* is a 3D point and *h<sub>i</sub>* is a positive real @f$ (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 *k* pairs *u<sub>1</sub>q<sub>1</sub> ... u<sub>k</sub>q<sub>k</sub>*. Here *u<sub>i</sub>* is a knot with a multiplicity @f$ q_{i} \geq 1 \; (1 \leq i \leq k) @f$ so that
|
||
|
||
@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:
|
||
|
||
@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 @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]
|
||
|
||
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]
|
||
|
||
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 *B<sub>1</sub>*=(0,1,0), *h<sub>1</sub>*=4, *B<sub>2</sub>*=(1,-2,0), *h<sub>2</sub>*=5 and *B<sub>3</sub>*=(2,3,0), *h<sub>3</sub>*=6, multiplicity knots *u<sub>1</sub>*=0, *q<sub>1</sub>*=1, *u<sub>2</sub>*=0.25, *q<sub>2</sub>*=1, *u<sub>3</sub>*=0.5, *q<sub>3</sub>*=1, *u<sub>4</sub>*=0.75, *q<sub>4</sub>*=1 and *u<sub>5</sub>*=1, *q<sub>5</sub>*=1. The B-spline curve is defined by the following parametric equation:
|
||
|
||
@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 specification__brep_format_5_1_8 Trimmed Curve - \<3D curve record 8\>
|
||
|
||
**Example**
|
||
|
||
@verbatim
|
||
8 -4 5
|
||
1 1 2 3 1 0 0
|
||
@endverbatim
|
||
|
||
**BNF-like Definition**
|
||
|
||
~~~~{.cpp}
|
||
<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>;
|
||
~~~~
|
||
|
||
**Description**
|
||
|
||
\<3D curve record 8\> describes a trimmed curve. The trimmed curve data consist of reals *u<sub>min</sub>* and *u<sub>max</sub>* and \<3D curve record\> so that *u<sub>min</sub>* < *u<sub>max</sub>*. 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$. The trimmed curve is defined by the following parametric equation:
|
||
|
||
@f[ C(u)=B(u),\; u \in [u_{min},\;u_{max}]. @f]
|
||
|
||
The example record is interpreted as a trimmed curve with *u<sub>min</sub>*=-4 and *u<sub>max</sub>*=5 for the base curve @f$ 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$.
|
||
|
||
|
||
@subsubsection specification__brep_format_5_1_9 Offset Curve - \<3D curve record 9\>
|
||
|
||
**Example**
|
||
|
||
@verbatim
|
||
9 2
|
||
0 1 0
|
||
1 1 2 3 1 0 0
|
||
@endverbatim
|
||
|
||
**BNF-like Definition**
|
||
|
||
@verbatim
|
||
<3D curve record 9> = "9" <_> <3D offset curve distance> <_\n>;
|
||
<3D offset curve direction> <_\n>;
|
||
<3D curve record>;
|
||
|
||
<3D offset curve distance> = <real>;
|
||
|
||
<3D offset curve direction> = <3D direction>;
|
||
@endverbatim
|
||
|
||
**Description**
|
||
|
||
\<3D curve record 9\> describes an offset curve. The offset curve data consist of a distance *d*, a 3D direction *D* and a \<3D curve record\>. The offset curve is the result of offsetting the base curve *B* described in the record to the distance *d* along the vector @f$ [B'(u),\; D] \neq \vec{0} @f$. The offset curve is defined by the following parametric equation:
|
||
|
||
@f[ C(u)=B(u)+d \cdot \frac{[B'(u),\; D]}{|[B'(u),\; D]|},\; u \in domain(B) . @f]
|
||
|
||
The example record is interpreted as an offset curve with a distance *d*=2, direction *D*=(0, 1, 0), base curve @f$ 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$.
|
||
|
||
@subsection specification__brep_format_5_2 Surfaces
|
||
|
||
**Example**
|
||
|
||
@verbatim
|
||
Surfaces 6
|
||
1 0 0 0 1 0 -0 0 0 1 0 -1 0
|
||
1 0 0 0 -0 1 0 0 0 1 1 0 -0
|
||
1 0 0 3 0 0 1 1 0 -0 -0 1 0
|
||
1 0 2 0 -0 1 0 0 0 1 1 0 -0
|
||
1 0 0 0 0 0 1 1 0 -0 -0 1 0
|
||
1 1 0 0 1 0 -0 0 0 1 0 -1 0
|
||
@endverbatim
|
||
|
||
**BNF-like Definition**
|
||
|
||
@verbatim
|
||
<surfaces> = <surface header> <_\n> <surface records>;
|
||
|
||
<surface header> = “Surfaces” <_> <surface count>;
|
||
|
||
<surface records> = <surface record> ^ <surface count>;
|
||
|
||
<surface record> =
|
||
<surface record 1> |
|
||
<surface record 2> |
|
||
<surface record 3> |
|
||
<surface record 4> |
|
||
<surface record 5> |
|
||
<surface record 6> |
|
||
<surface record 7> |
|
||
<surface record 8> |
|
||
<surface record 9> |
|
||
<surface record 10> |
|
||
<surface record 11>;
|
||
@endverbatim
|
||
|
||
@subsubsection specification__brep_format_5_2_1 Plane - \< surface record 1 \>
|
||
|
||
**Example**
|
||
|
||
@verbatim
|
||
1 0 0 3 0 0 1 1 0 -0 -0 1 0
|
||
@endverbatim
|
||
|
||
**BNF-like Definition**
|
||
|
||
@verbatim
|
||
<surface record 1> = "1" <_> <3D point> (<_> <3D direction>) ^ 3 <_\n>;
|
||
@endverbatim
|
||
|
||
**Description**
|
||
|
||
\<surface record 1\> describes a plane. The plane data consist of a 3D point *P* and pairwise orthogonal 3D directions *N*, *D<sub>u</sub>* and *D<sub>v</sub>*. The plane passes through the point *P*, has the normal *N* and is defined by the following parametric equation:
|
||
|
||
@f[ S(u,v)=P+u \cdot D_{u}+v \cdot D_{v},\; (u,\;v) \in (-\infty,\; \infty) \times (-\infty,\; \infty). @f]
|
||
|
||
The example record is interpreted as a plane which passes through a point *P*=(0, 0, 3), has a normal *N*=(0, 0, 1) and is defined by the following parametric equation: @f$ S(u,v)=(0,0,3)+u \cdot (1,0,0) + v \cdot (0,1,0) @f$.
|
||
|
||
|
||
@subsubsection specification__brep_format_5_2_2 Cylinder - \< surface record 2 \>
|
||
|
||
**Example**
|
||
|
||
@verbatim
|
||
2 1 2 3 0 0 1 1 0 -0 -0 1 0 4
|
||
@endverbatim
|
||
|
||
**BNF-like Definition**
|
||
|
||
@verbatim
|
||
<surface record 2> = "2" <_> <3D point> (<_> <3D direction>) ^ 3 <_> <real> <_\n>;
|
||
@endverbatim
|
||
|
||
**Description**
|
||
|
||
\<surface record 2\> describes a cylinder. The cylinder data consist of a 3D point *P*, pairwise orthogonal 3D directions *D<sub>v</sub>*, *D<sub>X</sub>* and *D<sub>Y</sub>* and a non-negative real *r*. The cylinder axis passes through the point *P* and has the direction *D<sub>v</sub>*. The cylinder has the radius *r* and is defined by the following parametric equation:
|
||
|
||
@f[ S(u,v)=P+r \cdot (cos(u) \cdot D_{x}+sin(u) \cdot D_{y} )+v \cdot D_{v},\; (u,v) \in [0,\; 2 \cdot \pi) \times (-\infty,\; \infty) . @f]
|
||
|
||
The example record is interpreted as a cylinder which axis passes through a point *P*=(1, 2, 3) and has a direction *D<sub>v</sub>*=(0, 0, 1). Directions for the cylinder are *D<sub>X</sub>*=(1,0,0) and *D<sub>Y</sub>*=(0,1,0). The cylinder has a radius *r*=4 and is defined by the following parametric equation: @f$ S(u,v)=(1,2,3)+4 \cdot ( cos(u) \cdot D_{X} + sin(u) \cdot D_{Y} ) + v \cdot D_{v}. @f$
|
||
|
||
|
||
@subsubsection specification__brep_format_5_2_3 Cone - \< surface record 3 \>
|
||
|
||
**Example**
|
||
|
||
@verbatim
|
||
3 1 2 3 0 0 1 1 0 -0 -0 1 0 4
|
||
0.75
|
||
@endverbatim
|
||
|
||
|
||
**BNF-like Definition**
|
||
|
||
@verbatim
|
||
<surface record 3> = "3" <_> <3D point> (<_> <3D direction>) ^ 3 (<_> <real>) ^ 2 <_\n>;
|
||
@endverbatim
|
||
|
||
**Description**
|
||
|
||
\<surface record 3\> describes a cone. The cone data consist of a 3D point *P*, pairwise orthogonal 3D directions *D<sub>Z</sub>*, *D<sub>X</sub>* and *D<sub>Y</sub>*, a non-negative real *r* and a real @f$ \varphi \in (-\pi /2,\; \pi/2)/\left \{ 0 \right \} @f$. The cone axis passes through the point *P* and has the direction *D<sub>Z</sub>*. The plane which passes through the point *P* and is parallel to directions *D<sub>X</sub>* and *D<sub>Y</sub>* is the cone referenced plane. The cone section by the plane is a circle with the radius *r*. The direction from the point *P* to the cone apex is @f$ -sgn(\varphi) \cdot D_{Z} @f$. The cone has a half-angle @f$| \varphi | @f$ and is defined by the following parametric equation:
|
||
|
||
@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 *P*=(1, 2, 3) and has a direction *D<sub>Z</sub>*=(0, 0, 1). Other cone data are *D<sub>X</sub>*=(1, 0, 0), *D<sub>Y</sub>*=(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 specification__brep_format_5_2_4 Sphere - \< surface record 4 \>
|
||
|
||
**Example**
|
||
|
||
@verbatim
|
||
4 1 2 3 0 0 1 1 0 -0 -0 1 0 4
|
||
@endverbatim
|
||
|
||
**BNF-like Definition**
|
||
|
||
@verbatim
|
||
<surface record 4> = "4" <_> <3D point> (<_> <3D direction>) ^ 3 <_> <real> <_\n>;
|
||
@endverbatim
|
||
|
||
**Description**
|
||
|
||
\<surface record 4\> describes a sphere. The sphere data consist of a 3D point *P*, pairwise orthogonal 3D directions *D<sub>Z</sub>*, *D<sub>X</sub>* and *D<sub>Y</sub>* and a non-negative real *r*. The sphere has the center *P*, radius *r* and is defined by the following parametric equation:
|
||
|
||
@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 *P*=(1, 2, 3). Directions for the sphere are *D<sub>Z</sub>*=(0, 0, 1), *D<sub>X</sub>*=(1, 0, 0) and *D<sub>Y</sub>*=(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 specification__brep_format_5_2_5 Torus - \< surface record 5 \>
|
||
|
||
**Example**
|
||
|
||
@verbatim
|
||
5 1 2 3 0 0 1 1 0 -0 -0 1 0 8 4
|
||
@endverbatim
|
||
|
||
**BNF-like Definition**
|
||
|
||
@verbatim
|
||
<surface record 5> = "5" <_> <3D point> (<_> <3D direction>) ^ 3 (<_> <real>) ^ 2 <_\n>;
|
||
@endverbatim
|
||
|
||
**Description**
|
||
|
||
\<surface record 5\> describes a torus. The torus data consist of a 3D point *P*, pairwise orthogonal 3D directions *D<sub>Z</sub>*, *D<sub>X</sub>* and *D<sub>Y</sub>* and non-negative reals *r<sub>1</sub>* and *r<sub>2</sub>*. The torus axis passes through the point *P* and has the direction *D<sub>Z</sub>*. *r<sub>1</sub>* is the distance from the torus circle center to the axis. The torus circle has the radius *r<sub>2</sub>*. The torus is defined by the following parametric equation:
|
||
|
||
@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 *P*=(1, 2, 3) and has a direction *D<sub>Z</sub>*=(0, 0, 1). *D<sub>X</sub>*=(1, 0, 0), *D<sub>Y</sub>*=(0, 1, 0), *r<sub>1</sub>*=8 and *r<sub>2</sub>*=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 specification__brep_format_5_2_6 Linear Extrusion - \< surface record 6 \>
|
||
|
||
**Example**
|
||
|
||
@verbatim
|
||
6 0 0.6 0.8
|
||
2 1 2 3 0 0 1 1 0 -0 -0 1 0 4
|
||
@endverbatim
|
||
|
||
**BNF-like Definition**
|
||
|
||
@verbatim
|
||
<surface record 6> = "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 *D<sub>v</sub>* and a \<3D curve record\>. The linear extrusion surface has the direction *D<sub>v</sub>*, the base curve *C* described in the record and is defined by the following parametric equation:
|
||
|
||
@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 *D<sub>v</sub>*=(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 specification__brep_format_5_2_7 Revolution Surface - \< surface record 7 \>
|
||
|
||
**Example**
|
||
|
||
@verbatim
|
||
7 -4 0 3 0 1 0
|
||
2 1 2 3 0 0 1 1 0 -0 -0 1 0 4
|
||
@endverbatim
|
||
|
||
**BNF-like Definition**
|
||
|
||
@verbatim
|
||
<surface record 7> = "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 *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:
|
||
|
||
@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 @f$ V(v)=C(v)-P, V_{D}(v)=(D,V(v)) \cdot D @f$.
|
||
|
||
The example record is interpreted as a revolution surface with an axis which passes through a point *P*=(-4, 0, 3) and has a direction *D*=(0, 1, 0). The base curve is a circle for the surface. The surface is defined by the following parametric equation:
|
||
|
||
@f[ S(u,v)= (-4,0,3)+V_{D}(v)+cos(u) \cdot (V(v)-V_{D}(v))+sin(u) \cdot [(0,1,0),V(v)],\;(u,v) \in [0,\; 2 \cdot \pi)\times [0,\; 2 \cdot \pi) @f]
|
||
|
||
where @f$ 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$.
|
||
|
||
|
||
@subsubsection specification__brep_format_5_2_8 Bezier Surface - \< surface record 8 \>
|
||
|
||
**Example**
|
||
|
||
@verbatim
|
||
8 1 1 2 1 0 0 1 7 1 0 -4 10
|
||
0 1 -2 8 1 1 5 11
|
||
0 2 3 9 1 2 6 12
|
||
@endverbatim
|
||
|
||
**BNF-like Definition**
|
||
|
||
~~~~{.cpp}
|
||
<surface record 8> = "8" <_> <Bezier surface u rational flag> <_> <Bezier surface v rational flag> <_> <Bezier surface u degree> <_> <Bezier surface v degree> <_>
|
||
<Bezier surface weight poles>;
|
||
|
||
<Bezier surface u rational flag> = <flag>;
|
||
|
||
<Bezier surface v rational flag> = <flag>;
|
||
|
||
<Bezier surface u degree> = <int>;
|
||
|
||
<Bezier surface v degree> = <int>;
|
||
|
||
<Bezier surface weight poles> =
|
||
(<Bezier surface weight pole group> <_\n>) ^ (<Bezier surface u degree> <+> "1");
|
||
|
||
<Bezier surface weight pole group> = <Bezier surface weight pole>
|
||
(<_> <Bezier surface weight pole>) ^ <Bezier surface v degree>;
|
||
|
||
<Bezier surface weight pole> = <3D point> [<_> <real>];
|
||
~~~~
|
||
|
||
**Description**
|
||
|
||
\<surface record 8\> describes a Bezier surface. The surface data consist of a u rational flag *r<sub>u</sub>*, v rational flag *r<sub>v</sub>*, u degree @f$ m_{u} \leq 25 @f$, v degree @f$ m_{v} \leq 25 @f$ and weight poles.
|
||
|
||
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$.
|
||
|
||
The Bezier surface is defined by the following parametric equation:
|
||
|
||
@f[ S(u,v)=\frac{\sum_{i=0}^{m_{u}} \sum_{j=0}^{m_{v}} B_{i,j} \cdot h_{i,j} \cdot C_{m_{u}}^{i} \cdot u^{i} \cdot (1-u)^{m_{u}-i} \cdot C_{m_{v}}^{j} \cdot v^{j} \cdot (1-v)^{m_{v}-j}}{\sum_{i=0}^{m_{u}} \sum_{j=0}^{m_{v}} h_{i,j} \cdot C_{m_{u}}^{i} \cdot u^{i} \cdot (1-u)^{m_{u}-i} \cdot C_{m_{v}}^{j} \cdot v^{j} \cdot (1-v)^{m_{v}-j}}, (u,v) \in [0,1] \times [0,1] @f]
|
||
|
||
where @f$ 0^{0} \equiv 1 @f$.
|
||
|
||
The example record is interpreted as a Bezier surface with a u rational flag *r<sub>u</sub>*=1, v rational flag *r<sub>v</sub>*=1, u degree *m<sub>u</sub>*=2, v degree *m<sub>v</sub>*=1, weight poles *B<sub>0,0</sub>*=(0, 0, 1), *h<sub>0,0</sub>*=7, *B<sub>0,1</sub>*=(1, 0, -4), *h<sub>0,1</sub>*=10, *B<sub>1,0</sub>*=(0, 1, -2), *h<sub>1,0</sub>*=8, *B<sub>1,1</sub>*=(1, 1, 5), *h<sub>1,1</sub>*=11, *B<sub>2,0</sub>*=(0, 2, 3), *h<sub>2,0</sub>*=9 and *B<sub>2,1</sub>*=(1, 2, 6), *h<sub>2,1</sub>*=12. The surface is defined by the following parametric equation:
|
||
|
||
@f[
|
||
\begin{align}
|
||
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 ]
|
||
\end{align}
|
||
@f]
|
||
|
||
|
||
@subsubsection specification__brep_format_5_2_9 B-spline Surface - \< surface record 9 \>
|
||
|
||
**Example**
|
||
|
||
@verbatim
|
||
9 1 1 0 0 1 1 3 2 5 4 0 0 1 7 1 0 -4 10
|
||
0 1 -2 8 1 1 5 11
|
||
0 2 3 9 1 2 6 12
|
||
|
||
0 1
|
||
0.25 1
|
||
0.5 1
|
||
0.75 1
|
||
1 1
|
||
|
||
0 1
|
||
0.3 1
|
||
0.7 1
|
||
1 1
|
||
@endverbatim
|
||
|
||
**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>;
|
||
@endverbatim
|
||
|
||
**Description**
|
||
|
||
\<surface record 9\> describes a B-spline surface. The surface data consist of a u rational flag *r<sub>u</sub>*, v rational flag *r<sub>v</sub>*, 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$, u multiplicity knot count *k<sub>u</sub>*, v multiplicity knot count *k<sub>v</sub>*, weight poles, u multiplicity knots, v multiplicity knots.
|
||
|
||
The weight poles are @f$ 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$.
|
||
|
||
The u multiplicity knots are *k<sub>u</sub>* pairs @f$ 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
|
||
|
||
@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 *k<sub>v</sub>* 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
|
||
|
||
@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:
|
||
|
||
@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 *N<sub>i,j</sub>* and *M<sub>i,j</sub>* have the following recursion definition by *j*:
|
||
|
||
@f[
|
||
\begin{align}
|
||
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);
|
||
\end{align}
|
||
@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]
|
||
|
||
The example record is interpreted as a B-spline surface with a u rational flag *r<sub>u</sub>*=1, v rational flag *r<sub>v</sub>*=1, u degree *m<sub>u</sub>*=1, v degree *m<sub>v</sub>*=1, u pole count *n<sub>u</sub>*=3, v pole count *n<sub>v</sub>*=2, u multiplicity knot count *k<sub>u</sub>*=5, v multiplicity knot count *k<sub>v</sub>*=4, weight poles *B<sub>1,1</sub>*=(0, 0, 1), *h<sub>1,1</sub>*=7, *B<sub>1,2</sub>*=(1, 0, -4), *h<sub>1,2</sub>*=10, *B<sub>2,1</sub>*=(0, 1, -2), *h<sub>2,1</sub>*=8, *B<sub>2,2</sub>*=(1, 1, 5), *h<sub>2,2</sub>*=11, *B<sub>3,1</sub>*=(0, 2, 3), *h<sub>3,1</sub>*=9 and *B<sub>3,2</sub>*=(1, 2, 6), *h<sub>3,2</sub>*=12, u multiplicity knots *u<sub>1</sub>*=0, *q<sub>1</sub>*=1, *u<sub>2</sub>*=0.25, *q<sub>2</sub>*=1, *u<sub>3</sub>*=0.5, *q<sub>3</sub>*=1, *u<sub>4</sub>*=0.75, *q<sub>4</sub>*=1 and *u<sub>5</sub>*=1, *q<sub>5</sub>*=1, v multiplicity knots *v<sub>1</sub>*=0, *r<sub>1</sub>*=1, *v<sub>2</sub>*=0.3, *r<sub>2</sub>*=1, *v<sub>3</sub>*=0.7, *r<sub>3</sub>*=1 and *v<sub>4</sub>*=1, *r<sub>4</sub>*=1. The B-spline surface is defined by the following parametric equation:
|
||
|
||
@f[
|
||
\begin{align}
|
||
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) ]
|
||
\end{align}
|
||
@f]
|
||
|
||
@subsubsection specification__brep_format_5_2_10 Rectangular Trim Surface - \< surface record 10 \>
|
||
|
||
**Example**
|
||
|
||
@verbatim
|
||
10 -1 2 -3 4
|
||
1 1 2 3 0 0 1 1 0 -0 -0 1 0
|
||
@endverbatim
|
||
|
||
**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>;
|
||
@endverbatim
|
||
|
||
**Description**
|
||
|
||
\<surface record 10\> describes a rectangular trim surface. The surface data consist of reals *u<sub>min</sub>*, *u<sub>max</sub>*, *v<sub>min</sub>* and *v<sub>max</sub>* and a \<surface record\> so that *u<sub>min</sub>* < *u<sub>max</sub>* and *v<sub>min</sub>* < *v<sub>max</sub>*. 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$. The rectangular trim surface is defined by the following parametric equation:
|
||
|
||
@f[ S(u,v)=B(u,v),\; (u,v) \in [u_{min},u_{max}] \times [v_{min},v_{max}] . @f]
|
||
|
||
The example record is interpreted as a rectangular trim surface to the set [-1, 2]x[-3, 4] for the base surface @f$ 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$.
|
||
|
||
|
||
@subsubsection specification__brep_format_5_2_11 Offset Surface - \< surface record 11 \>
|
||
|
||
**Example**
|
||
|
||
@verbatim
|
||
11 -2
|
||
1 1 2 3 0 0 1 1 0 -0 -0 1 0
|
||
@endverbatim
|
||
|
||
**BNF-like Definition**
|
||
|
||
@verbatim
|
||
<surface record 11> = "11" <_> <surface record distance> <_\n> <surface record>;
|
||
|
||
<surface record distance> = <real>;
|
||
@endverbatim
|
||
|
||
**Description**
|
||
|
||
\<surface record 11\> describes an offset surface.
|
||
The offset surface data consist of a distance *d* and a \<surface record\>. The offset surface is the result of offsetting the base surface *B* described in the record to the distance *d* along the normal *N* of surface *B*. The offset surface is defined by the following parametric equation:
|
||
|
||
@f[ S(u,v)=B(u,v)+d \cdot N(u,v),\; (u,v) \in domain(B) . \\
|
||
N(u,v) = [S'_{u}(u,v),S'_{v}(u,v)] @f]
|
||
if @f$ [S'_{u}(u,v),S'_{v}(u,v)] \neq \vec{0} @f$.
|
||
|
||
The example record is interpreted as an offset surface with a distance *d*=-2 and base surface @f$ 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$.
|
||
|
||
|
||
@subsection specification__brep_format_5_3 2D curves
|
||
|
||
**Example**
|
||
|
||
@verbatim
|
||
Curve2ds 24
|
||
1 0 0 1 0
|
||
1 0 0 1 0
|
||
1 3 0 0 -1
|
||
1 0 0 0 1
|
||
1 0 -2 1 0
|
||
1 0 0 1 0
|
||
1 0 0 0 -1
|
||
1 0 0 0 1
|
||
1 0 0 1 0
|
||
1 0 1 1 0
|
||
1 3 0 0 -1
|
||
1 1 0 0 1
|
||
1 0 -2 1 0
|
||
1 0 1 1 0
|
||
1 0 0 0 -1
|
||
1 1 0 0 1
|
||
1 0 0 0 1
|
||
1 0 0 1 0
|
||
1 3 0 0 1
|
||
1 0 0 1 0
|
||
1 0 0 0 1
|
||
1 0 2 1 0
|
||
1 3 0 0 1
|
||
1 0 2 1 0
|
||
@endverbatim
|
||
|
||
**BNF-like Definition**
|
||
|
||
@verbatim
|
||
<2D curves> = <2D curve header> <_\n> <2D curve records>;
|
||
|
||
<2D curve header> = "Curve2ds" <_> <2D curve count>;
|
||
|
||
<2D curve count> = <int>;
|
||
|
||
<2D curve records> = <2D curve record> ^ <2D curve count>;
|
||
|
||
<2D curve record> =
|
||
<2D curve record 1> |
|
||
<2D curve record 2> |
|
||
<2D curve record 3> |
|
||
<2D curve record 4> |
|
||
<2D curve record 5> |
|
||
<2D curve record 6> |
|
||
<2D curve record 7> |
|
||
<2D curve record 8> |
|
||
<2D curve record 9>;
|
||
@endverbatim
|
||
|
||
@subsubsection specification__brep_format_5_3_1 Line - \<2D curve record 1\>
|
||
|
||
**Example**
|
||
|
||
@verbatim
|
||
1 3 0 0 -1
|
||
@endverbatim
|
||
|
||
**BNF-like Definition**
|
||
|
||
@verbatim
|
||
<2D curve record 1> = "1" <_> <2D point> <_> <2D direction> <_\n>;
|
||
@endverbatim
|
||
|
||
**Description**
|
||
|
||
\<2D curve record 1\> describes a line. The line data consist of a 2D point *P* and a 2D direction *D*. The line passes through the point *P*, has the direction *D* and is defined by the following parametric equation:
|
||
|
||
@f[ C(u)=P+u \cdot D, \; u \in (-\infty,\; \infty). @f]
|
||
|
||
The example record is interpreted as a line which passes through a point *P*=(3,0), has a direction *D*=(0,-1) and is defined by the following parametric equation: @f$ C(u)=(3,0)+ u \cdot (0,-1) @f$.
|
||
|
||
|
||
@subsubsection specification__brep_format_5_3_2 Circle - \<2D curve record 2\>
|
||
|
||
**Example**
|
||
|
||
@verbatim
|
||
2 1 2 1 0 -0 1 3
|
||
@endverbatim
|
||
|
||
**BNF-like Definition**
|
||
|
||
~~~~{.cpp}
|
||
<2D curve record 2> = "2" <_> <2D circle center> <_> <2D circle Dx> <_> <2D circle Dy> <_> <2D circle radius> <_\n>;
|
||
|
||
<2D circle center> = <2D point>;
|
||
|
||
<2D circle Dx> = <2D direction>;
|
||
|
||
<2D circle Dy> = <2D direction>;
|
||
|
||
<2D circle radius> = <real>;
|
||
~~~~
|
||
|
||
**Description**
|
||
|
||
\<2D curve record 2\> describes a circle. The circle data consist of a 2D point *P*, orthogonal 2D directions *D<sub>x</sub>* and *D<sub>y</sub>* and a non-negative real *r*. The circle has a center *P*. The circle plane is parallel to directions *D<sub>x</sub>* and *D<sub>y</sub>*. The circle has a radius *r* and is defined by the following parametric equation:
|
||
|
||
@f[ C(u)=P+r \cdot (cos(u) \cdot D_{x} + sin(u) \cdot D_{y}),\; u \in [0,\; 2 \cdot \pi) . @f]
|
||
|
||
The example record is interpreted as a circle which has a center *P*=(1,2). The circle plane is parallel to directions *D<sub>x</sub>*=(1,0) and *D<sub>y</sub>*=(0,1). The circle has a radius *r*=3 and is defined by the following parametric equation: @f$ C(u)=(1,2)+3 \cdot (cos(u) \cdot (1,0) + sin(u) \cdot (0,1)) @f$.
|
||
|
||
|
||
@subsubsection specification__brep_format_5_3_3 Ellipse - \<2D curve record 3\>
|
||
|
||
**Example**
|
||
|
||
@verbatim
|
||
3 1 2 1 0 -0 1 4 3
|
||
@endverbatim
|
||
|
||
**BNF-like Definition**
|
||
|
||
@verbatim
|
||
<2D curve record 3> = "3" <_> <2D ellipse center> <_> <2D ellipse Dmaj> <_>
|
||
<2D ellipse Dmin> <_> <2D ellipse Rmaj> <_> <2D ellipse Rmin> <_\n>;
|
||
|
||
<2D ellipse center> = <2D point>;
|
||
|
||
<2D ellipse Dmaj> = <2D direction>;
|
||
|
||
<2D ellipse Dmin> = <2D direction>;
|
||
|
||
<2D ellipse Rmaj> = <real>;
|
||
|
||
<2D ellipse Rmin> = <real>;
|
||
@endverbatim
|
||
|
||
**Description**
|
||
|
||
\<2D curve record 3\> describes an ellipse. The ellipse data are 2D point *P*, orthogonal 2D directions *D<sub>maj</sub>* and *D<sub>min</sub>* and non-negative reals *r<sub>maj</sub>* and *r<sub>min</sub>* that *r<sub>maj</sub>* @f$ \leq @f$ *r<sub>min</sub>*. The ellipse has a center *P*, major and minor axis directions *D<sub>maj</sub>* and *D<sub>min</sub>*, major and minor radii *r<sub>maj</sub>* and *r<sub>min</sub>* and is defined by the following parametric equation:
|
||
|
||
@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 *P*=(1,2), major and minor axis directions *D<sub>maj</sub>*=(1,0) and *D<sub>min</sub>*=(0,1), major and minor radii *r<sub>maj</sub>*=4 and *r<sub>min</sub>*=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$.
|
||
|
||
|
||
@subsubsection specification__brep_format_5_3_4 Parabola - \<2D curve record 4\>
|
||
|
||
**Example**
|
||
|
||
@verbatim
|
||
4 1 2 1 0 -0 1 16
|
||
@endverbatim
|
||
|
||
**BNF-like Definition**
|
||
|
||
@verbatim
|
||
<2D curve record 4> = "4" <_> <2D parabola origin> <_> <2D parabola Dx> <_>
|
||
<2D parabola Dy> <_> <2D parabola focal length> <_\n>;
|
||
|
||
<2D parabola origin> = <2D point>;
|
||
|
||
<2D parabola Dx> = <2D direction>;
|
||
|
||
<2D parabola Dy> = <2D direction>;
|
||
|
||
<2D parabola focal length> = <real>;
|
||
@endverbatim
|
||
|
||
**Description**
|
||
|
||
\<2D curve record 4\> describes a parabola. The parabola data consist of a 2D point *P*, orthogonal 2D directions *D<sub>x</sub>* and *D<sub>y</sub>* and a non-negative real *f*. The parabola coordinate system has its origin *P* and axis directions *D<sub>x</sub>* and *D<sub>y</sub>*. The parabola has a focus length *f* and is defined by the following parametric equation:
|
||
|
||
@f[ C(u)=P+\frac{u^{2}}{4 \cdot f} \cdot D_{x}+u \cdot D_{y},\; u \in (-\infty,\; \infty) \Leftarrow f \neq 0;\\
|
||
C(u)=P+u \cdot D_{x},\; u \in (-\infty,\; \infty) \Leftarrow f = 0\; (degenerated\;case). @f]
|
||
|
||
The example record is interpreted as a parabola in plane which passes through a point *P*=(1,2) and is parallel to directions *D<sub>x</sub>*=(1,0) and *D<sub>y</sub>*=(0,1). The parabola has a focus length *f*=16 and is defined by the following parametric equation: @f$ C(u)=(1,2)+ \frac{u^{2}}{64} \cdot (1,0)+u \cdot (0,1) @f$.
|
||
|
||
|
||
@subsubsection specification__brep_format_5_3_5 Hyperbola - \<2D curve record 5\>
|
||
**Example**
|
||
|
||
5 1 2 1 0 -0 1 3 4
|
||
|
||
|
||
**BNF-like Definition**
|
||
|
||
@verbatim
|
||
<2D curve record 5> = "5" <_> <2D hyperbola origin> <_> <2D hyperbola Dx> <_>
|
||
<2D hyperbola Dy> <_> <2D hyperbola Kx> <_> <2D hyperbola Ky> <_\n>;
|
||
|
||
<2D hyperbola origin> = <2D point>;
|
||
|
||
<2D hyperbola Dx> = <2D direction>;
|
||
|
||
<2D hyperbola Dy> = <2D direction>;
|
||
|
||
<2D hyperbola Kx> = <real>;
|
||
|
||
<2D hyperbola Ky> = <real>;
|
||
@endverbatim
|
||
|
||
**Description**
|
||
|
||
\<2D curve record 5\> describes a hyperbola. The hyperbola data consist of a 2D point *P*, orthogonal 2D directions *D<sub>x</sub>* and *D<sub>y</sub>* and non-negative reals *k<sub>x</sub>* and *k<sub>y</sub>*. The hyperbola coordinate system has origin *P* and axis directions *D<sub>x</sub>* and *D<sub>y</sub>*. The hyperbola is defined by the following parametric equation:
|
||
|
||
@f[ C(u)=P+k_{x} \cdot cosh(u) D_{x}+k_{y} \cdot sinh(u) \cdot D_{y},\; u \in (-\infty,\; \infty). @f]
|
||
|
||
The example record is interpreted as a hyperbola with coordinate system which has origin *P*=(1,2) and axis directions *D<sub>x</sub>*=(1,0) and *D<sub>y</sub>*=(0,1). Other data for the hyperbola are *k<sub>x</sub>*=5 and *k<sub>y</sub>*=4. The hyperbola is defined by the following parametric equation: @f$ C(u)=(1,2)+3 \cdot cosh(u) \cdot (1,0)+4 \cdot sinh(u) \cdot (0,1) @f$.
|
||
|
||
|
||
@subsubsection specification__brep_format_5_3_6 Bezier Curve - \<2D curve record 6\>
|
||
|
||
**Example**
|
||
|
||
@verbatim
|
||
6 1 2 0 1 4 1 -2 5 2 3 6
|
||
@endverbatim
|
||
|
||
**BNF-like Definition**
|
||
|
||
@verbatim
|
||
<2D curve record 6> = "6" <_> <2D Bezier rational flag> <_> <2D Bezier degree>
|
||
<2D Bezier weight poles> <_\n>;
|
||
|
||
<2D Bezier rational flag> = <flag>;
|
||
|
||
<2D Bezier degree> = <int>;
|
||
|
||
<2D Bezier weight poles> = (<_> <2D Bezier weight pole>) ^ (<2D Bezier degree> <+> “1”);
|
||
|
||
<2D Bezier weight pole> = <2D point> [<_> <real>];
|
||
@endverbatim
|
||
|
||
**Description**
|
||
|
||
\<2D curve record 6\> describes a Bezier curve. The curve data consist of a rational flag *r*, a degree @f$ m \leq 25 @f$ and weight poles.
|
||
|
||
The weight poles are *m*+1 2D points *B<sub>0</sub> ... B<sub>m</sub>* if the flag *r* is 0. The weight poles are *m*+1 pairs *B<sub>0</sub>h<sub>0</sub> ... B<sub>m</sub>h<sub>m</sub>* if the flag *r* is 1. Here *B<sub>i</sub>* is a 2D point and *h<sub>i</sub>* is a positive real @f$ (0\leq i\leq m) @f$. *h<sub>i</sub>*=1 @f$(0\leq i\leq m) @f$ if the flag *r* is 0.
|
||
|
||
The Bezier curve is defined by the following parametric equation:
|
||
|
||
@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 @f$ 0^{0} \equiv 1 @f$.
|
||
|
||
The example record is interpreted as a Bezier curve with a rational flag *r*=1, a degree *m*=2 and weight poles *B<sub>0</sub>*=(0,1), *h<sub>0</sub>*=4, *B<sub>1</sub>*=(1,-2), *h<sub>1</sub>*=5 and *B<sub>2</sub>*=(2,3), *h<sub>2</sub>*=6. The Bezier curve is defined by the following parametric equation:
|
||
|
||
@f[ C(u)= \frac{(0,1) \cdot 4 \cdot (1-u)^{2}+(1,-2) \cdot 5 \cdot 2 \cdot u \cdot (1-u)+(2,3) \cdot 6 \cdot u^{2}}{ 4 \cdot (1-u)^{2}+5 \cdot 2 \cdot u \cdot (1-u)+6 \cdot u^{2}} . @f]
|
||
|
||
|
||
@subsubsection specification__brep_format_5_3_7 B-spline Curve - \<2D curve record 7\>
|
||
|
||
**Example**
|
||
|
||
@verbatim
|
||
7 1 0 1 3 5 0 1 4 1 -2 5 2 3 6
|
||
0 1 0.25 1 0.5 1 0.75 1 1 1
|
||
@endverbatim
|
||
|
||
**BNF-like Definition**
|
||
|
||
~~~~{.cpp}
|
||
<2D curve record 7> = "7" <_> <2D B-spline rational flag> <_> "0" <_> <2D B-spline degree> <_> <2D B-spline pole count> <_> <2D B-spline multiplicity knot count> <2D B-spline weight poles> <_\n> <2D B-spline multiplicity knots> <_\n>;
|
||
|
||
<2D B-spline rational flag> = <flag>;
|
||
|
||
<2D B-spline degree> = <int>;
|
||
|
||
<2D B-spline pole count> = <int>;
|
||
|
||
<2D B-spline multiplicity knot count> = <int>;
|
||
|
||
<2D B-spline weight poles> = <2D B-spline weight pole> ^ <2D B-spline pole count>;
|
||
|
||
<2D B-spline weight pole> = <_> <2D point> [<_> <real>];
|
||
|
||
<2D B-spline multiplicity knots> =
|
||
<2D B-spline multiplicity knot> ^ <2D B-spline multiplicity knot count>;
|
||
|
||
<2D B-spline multiplicity knot> = <_> <real> <_> <int>;
|
||
~~~~
|
||
|
||
**Description**
|
||
|
||
\<2D curve record 7\> describes a B-spline curve. The curve data consist of a rational flag *r*, a degree @f$ m \leq 25 @f$, a pole count @f$ n \geq 2 @f$, a multiplicity knot count *k*, weight poles and multiplicity knots.
|
||
|
||
The weight poles are *n* 2D points *B<sub>1</sub> ... B<sub>n</sub>* if the flag *r* is 0. The weight poles are *n* pairs *B<sub>1</sub>h<sub>1</sub> ... B<sub>n</sub>h<sub>n</sub>* if the flag *r* is 1. Here *B<sub>i</sub>* is a 2D point and *h<sub>i</sub>* is a positive real @f$ (1\leq i\leq n) @f$. *h<sub>i</sub>*=1 @f$(1\leq i\leq n) @f$ if the flag *r* is 0.
|
||
|
||
The multiplicity knots are *k* pairs *u<sub>1</sub>q<sub>1</sub> ... u<sub>k</sub>q<sub>k</sub>*. Here *u<sub>i</sub>* is a knot with multiplicity @f$ q_{i} \geq 1\; (1 \leq i \leq k) @f$ so that
|
||
|
||
@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:
|
||
|
||
@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 *N<sub>i,j</sub>* 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]
|
||
|
||
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]
|
||
|
||
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 *B<sub>1</sub>*=(0,1), *h<sub>1</sub>*=4, *B<sub>2</sub>*=(1,-2), *h<sub>2</sub>*=5 and *B<sub>3</sub>*=(2,3), *h<sub>3</sub>*=6 and multiplicity knots *u<sub>1</sub>*=0, *q<sub>1</sub>*=1, *u<sub>2</sub>*=0.25, *q<sub>2</sub>*=1, *u<sub>3</sub>*=0.5, *q<sub>3</sub>*=1, *u<sub>4</sub>*=0.75, *q<sub>4</sub>*=1 and *u<sub>5</sub>*=1, *q<sub>5</sub>*=1. The B-spline curve is defined by the following parametric equation:
|
||
|
||
@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 specification__brep_format_5_3_8 Trimmed Curve - \<2D curve record 8\>
|
||
|
||
**Example**
|
||
|
||
@verbatim
|
||
8 -4 5
|
||
1 1 2 1 0
|
||
@endverbatim
|
||
|
||
**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>;
|
||
@endverbatim
|
||
|
||
**Description**
|
||
|
||
\<2D curve record 8\> describes a trimmed curve. The trimmed curve data consist of reals *u<sub>min</sub>* and *u<sub>max</sub>* and a \<2D curve record\> so that *u<sub>min</sub>* < *u<sub>max</sub>*. 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$. The trimmed curve is defined by the following parametric equation:
|
||
|
||
@f[ C(u)=B(u),\; u \in [u_{min},\;u_{max}] . @f]
|
||
|
||
The example record is interpreted as a trimmed curve with *u<sub>min</sub>*=-4, *u<sub>max</sub>*=5 and base curve @f$ 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$.
|
||
|
||
|
||
@subsubsection specification__brep_format_5_3_9 Offset Curve - \<2D curve record 9\>
|
||
|
||
**Example**
|
||
|
||
@verbatim
|
||
9 2
|
||
1 1 2 1 0
|
||
@endverbatim
|
||
|
||
**BNF-like Definition**
|
||
|
||
@verbatim
|
||
<2D curve record 9> = "9" <_> <2D offset curve distance> <_\n> <2D curve record>;
|
||
|
||
<2D offset curve distance> = <real>;
|
||
@endverbatim
|
||
|
||
**Description**
|
||
|
||
\<2D curve record 9\> describes an offset curve. The offset curve data consist of a distance *d* and a \<2D curve record\>. The offset curve is the result of offsetting the base curve *B* described in the record to the distance *d* along the vector @f$ (B'_{Y}(u),\; -B'_{X}(u)) \neq \vec{0} @f$ where @f$ B(u)=(B'_{X}(u),\; B'_{Y}(u)) @f$. The offset curve is defined by the following parametric equation:
|
||
|
||
@f[ C(u)=B(u)+d \cdot (B'_{Y}(u),\; -B'_{X}(u)),\; u \in domain(B) . @f]
|
||
|
||
The example record is interpreted as an offset curve with a distance *d*=2 and base curve @f$ 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$.
|
||
|
||
@subsection specification__brep_format_5_4 3D polygons
|
||
|
||
**Example**
|
||
|
||
@verbatim
|
||
Polygon3D 1
|
||
2 1
|
||
0.1
|
||
1 0 0 2 0 0
|
||
0 1
|
||
@endverbatim
|
||
|
||
**BNF-like Definition**
|
||
|
||
@verbatim
|
||
<3D polygons> = <3D polygon header> <_\n> <3D polygon records>;
|
||
|
||
<3D polygon header> = "Polygon3D" <_> <3D polygon record count>;
|
||
|
||
<3D polygon records> = <3D polygon record> ^ <3D polygon record count>;
|
||
|
||
<3D polygon record> =
|
||
<3D polygon node count> <_> <3D polygon flag of parameter presence> <_\n>
|
||
<3D polygon deflection> <_\n>
|
||
<3D polygon nodes> <_\n>
|
||
[<3D polygon parameters> <_\n>];
|
||
|
||
<3D polygon node count> = <int>;
|
||
|
||
<3D polygon flag of parameter presence> = <flag>;
|
||
|
||
<3D polygon deflection> = <real>;
|
||
|
||
<3D polygon nodes> = (<3D polygon node> <_>) ^ <3D polygon node count>;
|
||
|
||
<3D polygon node> = <3D point>;
|
||
|
||
<3D polygon u parameters> = (<3D polygon u parameter> <_>) ^ <3D polygon node count>;
|
||
|
||
<3D polygon u parameter> = <real>;
|
||
@endverbatim
|
||
|
||
**Description**
|
||
|
||
\<3D polygons\> record describes a 3D polyline *L* which approximates a 3D curve *C*. The polyline data consist of a node count @f$ 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$. The parameters are present only if *p*=1. The polyline *L* passes through the nodes. The deflection *d* describes the deflection of polyline *L* from the curve *C*:
|
||
|
||
@f[ \underset{P \in C}{max}\; \underset{Q \in L}{min}|Q-P| \leq d . @f]
|
||
|
||
The parameter @f$ u_{i}\; (1\leq i \leq m) @f$ is the parameter of the node *N<sub>i</sub>* on the curve *C*:
|
||
|
||
@f[ C(u_{i})=N_{i} . @f]
|
||
|
||
The example record describes a polyline from *m*=2 nodes with a parameter presence flag *p*=1, a deflection *d*=0.1, nodes *N<sub>1</sub>*=(1,0,0) and *N<sub>2</sub>*=(2,0,0) and parameters *u<sub>1</sub>*=0 and *u<sub>2</sub>*=1.
|
||
|
||
|
||
@subsection specification__brep_format_6_4 Triangulations
|
||
|
||
**Example**
|
||
|
||
@verbatim
|
||
Triangulations 6
|
||
4 2 1 0
|
||
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
|
||
4 2 1 0
|
||
0 0 0 1 0 0 1 0 3 0 0 3 0 0 0 1 3 1 3 0 3 2 1 3 1 4
|
||
4 2 1 0
|
||
0 0 3 0 2 3 1 2 3 1 0 3 0 0 0 2 1 2 1 0 3 2 1 3 1 4
|
||
4 2 1 0
|
||
0 2 0 1 2 0 1 2 3 0 2 3 0 0 0 1 3 1 3 0 3 2 1 3 1 4
|
||
4 2 1 0
|
||
0 0 0 0 2 0 1 2 0 1 0 0 0 0 0 2 1 2 1 0 3 2 1 3 1 4
|
||
4 2 1 0
|
||
1 0 0 1 0 3 1 2 3 1 2 0 0 0 3 0 3 -2 0 -2 2 4 3 2 1 4
|
||
@endverbatim
|
||
|
||
**BNF-like Definition**
|
||
|
||
~~~~{.cpp}
|
||
<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> [<_> <need to write normals flag>] <_> <triangulation deflection> <_\n>
|
||
<triangulation nodes> [<_> <triangulation u v parameters>] <_> <triangulation triangles> [<_> <triangulation normals>] <_\n>;
|
||
|
||
<triangulation node count> = <int>;
|
||
|
||
<triangulation triangle count> = <int>;
|
||
|
||
<triangulation parameter presence flag> = <flag>;
|
||
|
||
<need to write normals 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>;
|
||
|
||
<triangulation normals> = (<triangulation normal> <_>) ^ <triangulation node count> ^ 3;
|
||
|
||
<triangulation normal> = <short real>.
|
||
~~~~
|
||
|
||
**Description**
|
||
|
||
\<triangulation u v parameters\> are used in version 2 or later.
|
||
\<need to write normals flag\> and \<triangulation normals\> are used in version 3.
|
||
|
||
\<triangulation record\> 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$. The parameters are present only if *p*=1. The deflection describes the triangulation deflection from the surface:
|
||
|
||
@f[ \underset{P \in S}{max}\; \underset{Q \in T}{min}|Q-P| \leq d . @f]
|
||
|
||
The parameter pair @f$ u_{i}\; v_{i}\; (1\leq i \leq m) @f$ describes the parameters of node *N<sub>i</sub>* on the surface:
|
||
|
||
@f[ S(u_{i},v_{i})=N_{i} . @f]
|
||
|
||
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
|
||
|
||
@verbatim
|
||
4 2 1 0
|
||
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 *m*=4 nodes, *k*=2 triangles, parameter presence flag *p*=1, deflection *d*=0, nodes *N<sub>1</sub>*=(0,0,0), *N<sub>2</sub>*=(0,0,3), *N<sub>3</sub>*=(0,2,3) and *N<sub>4</sub>*=(0,2,0), parameters (*u<sub>1</sub>*, *v<sub>1</sub>*)=(0,0), (*u<sub>2</sub>*, *v<sub>2</sub>*)=(3,0), (*u<sub>3</sub>*, *v<sub>3</sub>*)=(3,-2) and (*u<sub>4</sub>*, *v<sub>4</sub>*)=(0,-2), and triangles (*n<sub>1,1</sub>*, *n<sub>1,2</sub>*, *n<sub>1,3</sub>*)=(2,4,3) and (*n<sub>2,1</sub>*, *n<sub>2,2</sub>*, *n<sub>2,3</sub>*)=(2,1,4). From the point (1,0,0) ((-1,0,0)) the triangles have clockwise (counterclockwise) direction of the node circular traversal.
|
||
|
||
|
||
@subsection specification__brep_format_6_5 Polygons on triangulations
|
||
|
||
**Example**
|
||
|
||
@verbatim
|
||
PolygonOnTriangulations 24
|
||
2 1 2
|
||
p 0.1 1 0 3
|
||
2 1 4
|
||
p 0.1 1 0 3
|
||
2 2 3
|
||
p 0.1 1 0 2
|
||
2 1 2
|
||
p 0.1 1 0 2
|
||
2 4 3
|
||
p 0.1 1 0 3
|
||
2 1 4
|
||
p 0.1 1 0 3
|
||
2 1 4
|
||
p 0.1 1 0 2
|
||
2 1 2
|
||
p 0.1 1 0 2
|
||
2 1 2
|
||
p 0.1 1 0 3
|
||
2 2 3
|
||
p 0.1 1 0 3
|
||
2 2 3
|
||
p 0.1 1 0 2
|
||
2 4 3
|
||
p 0.1 1 0 2
|
||
2 4 3
|
||
p 0.1 1 0 3
|
||
2 2 3
|
||
p 0.1 1 0 3
|
||
2 1 4
|
||
p 0.1 1 0 2
|
||
2 4 3
|
||
p 0.1 1 0 2
|
||
2 1 2
|
||
p 0.1 1 0 1
|
||
2 1 4
|
||
p 0.1 1 0 1
|
||
2 4 3
|
||
p 0.1 1 0 1
|
||
2 1 4
|
||
p 0.1 1 0 1
|
||
2 1 2
|
||
p 0.1 1 0 1
|
||
2 2 3
|
||
p 0.1 1 0 1
|
||
2 4 3
|
||
p 0.1 1 0 1
|
||
2 2 3
|
||
p 0.1 1 0 1
|
||
@endverbatim
|
||
|
||
**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>;
|
||
@endverbatim
|
||
|
||
**Description**
|
||
|
||
\<polygons on triangulations\> 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$. The parameters are present only if *p*=1. The deflection *d* describes the deflection of polyline *L* from the curve *C*:
|
||
|
||
@f[ \underset{P \in C}{max}\; \underset{Q \in L}{min}|Q-P| \leq d . @f]
|
||
|
||
Parameter @f$ u_{i}\; (1\leq i\leq m) @f$ is *n<sub>i</sub>*-th node *C(u<sub>i</sub>)* parameter on curve *C*.
|
||
|
||
|
||
@subsection specification__brep_format_6_6 Geometric Sense of a Curve
|
||
|
||
Geometric sense of curve *C* described above is determined by the direction of parameter *u* increasing.
|
||
|
||
|
||
@section specification__brep_format_7 Shapes
|
||
|
||
An example of section shapes and a whole *.brep file are given in chapter 7 @ref specification__brep_format_8 "Appendix".
|
||
|
||
|
||
**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>);
|
||
@endverbatim
|
||
|
||
**Description**
|
||
|
||
\<shape flag word\> @f$ f_{1}\; f_{2}\; f_{3}\; f_{4}\; f_{5}\; f_{6}\; f_{7} @f$ \<flag\>s @f$ f_{i}\;(1\leq i \leq 7) @f$ are interpreted as shape flags in the following way:
|
||
|
||
* @f$ f_{1} @f$ -- free;
|
||
* @f$ f_{2} @f$ -- modified;
|
||
* @f$ f_{3} @f$ -- IGNORED(version 1 only) \\ checked (version 2 or later);
|
||
* @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:
|
||
|
||
* + -- forward;
|
||
* - -- reversed;
|
||
* i -- internal;
|
||
* e -- external.
|
||
|
||
\<shape subshape orientation\> 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.
|
||
|
||
\<shape subrecord\> types are interpreted in the following way:
|
||
|
||
* "Ve" -- vertex;
|
||
* "Ed" -- edge;
|
||
* "Wi" -- wire;
|
||
* "Fa" -- face;
|
||
* "Sh" -- shell;
|
||
* "So" -- solid;
|
||
* "CS" -- compsolid;
|
||
* "Co" -- compound.
|
||
|
||
\<shape final record\> determines the orientation and location for the whole model.
|
||
|
||
@subsection specification__brep_format_7_1 Common Terms
|
||
|
||
The terms below are used by \<vertex data\>, \<edge data\> and \<face data\>.
|
||
|
||
|
||
**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>;
|
||
@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.
|
||
|
||
\<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.
|
||
|
||
\<surface number\> is the number of a \<surface record\> from subsection \<surfaces\> of section \<geometry\>. \<surface record\> 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.
|
||
|
||
\<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.
|
||
|
||
\<triangulation number\> is the number of a \<triangulation record\> from subsection \<triangulations\> of section \<geometry\>. \<triangulation record\> 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.
|
||
|
||
\<curve parameter minimal and maximal values\> *u<sub>min</sub>* and *u<sub>max</sub>* are the curve parameter *u* bounds: *u<sub>min</sub>* @f$ \leq @f$ *u* @f$ \leq @f$ *u<sub>max</sub>*.
|
||
|
||
\<curve values for parameter minimal and maximal values\> *u<sub>min</sub>* and *u<sub>max</sub>* are real pairs *x<sub>min</sub> y<sub>min</sub>* and *x<sub>max</sub> y<sub>max</sub>* that (*x<sub>min</sub>*, *y<sub>min</sub>*)= *C* (*u<sub>min</sub>*) and (*x<sub>max</sub>*, *y<sub>max</sub>*)= *C* (*u<sub>max</sub>*) where *C* is a parametric equation of the curve.
|
||
|
||
|
||
@subsection specification__brep_format_7_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 1>) |
|
||
("2" <_> <vertex data representation data 2>) |
|
||
("3" <_> <vertex data representation data 3>);
|
||
|
||
<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>;
|
||
@endverbatim
|
||
|
||
**Description**
|
||
|
||
The usage of \<vertex data representation u parameter\> *U* is described below.
|
||
|
||
\<vertex data representation data 1\> 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 *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 *u* describe the position of the vertex *V* on a surface *S* through \<vertex data representation v parameter\> *v*: *S(u,v)=V*.
|
||
|
||
\<vertex data tolerance\> *t* describes the maximum distance from the vertex *V* to the set *R* of vertex *V* representations:
|
||
|
||
@f[ \underset{P \in R }{max} |P-V| \leq t . @f]
|
||
|
||
|
||
@subsection specification__brep_format_7_3 Edge data
|
||
|
||
**BNF-like Definition**
|
||
|
||
~~~~{.cpp}
|
||
<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>;
|
||
~~~~
|
||
|
||
**Description**
|
||
|
||
Flags \<edge data same parameter flag\>, \<edge data same range flag\> and \<edge data degenerated flag\> are used in a special way [1].
|
||
|
||
\<edge data representation data 1\> 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 in version 2 or later.
|
||
|
||
\<edge data representation data 3\> describes a 2D curve on a closed surface.
|
||
\<curve values for parameter minimal and maximal values\> are used in version 2 or later.
|
||
|
||
\<edge data representation data 5\> describes a 3D polyline.
|
||
|
||
\<edge data representation data 6\> describes a polyline on a triangulation.
|
||
|
||
\<edge data tolerance\> *t* describes the maximum distance from the edge *E* to the set *R* of edge *E* representations:
|
||
|
||
@f[ \underset{C \in R}{max}\;\underset{P \in E}{max}\;\underset{Q \in C}{min}|Q-P| \leq t @f]
|
||
|
||
|
||
@subsection specification__brep_format_7_4 Face data
|
||
|
||
**BNF-like Definition**
|
||
|
||
~~~~{.cpp}
|
||
<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>;
|
||
~~~~
|
||
|
||
**Description**
|
||
|
||
\<face data\> describes a surface *S* of face *F* and a triangulation *T* of face *F*. The surface *S* may be empty: \<surface number\> = 0.
|
||
|
||
\<face data tolerance\> *t* describes the maximum distance from the face *F* to the surface *S*:
|
||
|
||
@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].
|
||
|
||
|
||
@section specification__brep_format_8 Appendix
|
||
|
||
This chapter contains a *.brep file example.
|
||
|
||
@verbatim
|
||
DBRep_DrawableShape
|
||
|
||
CASCADE Topology V1, (c) Matra-Datavision
|
||
Locations 3
|
||
1
|
||
0 0 1 0
|
||
1 0 0 0
|
||
0 1 0 0
|
||
1
|
||
1 0 0 4
|
||
0 1 0 5
|
||
0 0 1 6
|
||
2 1 1 2 1 0
|
||
Curve2ds 24
|
||
1 0 0 1 0
|
||
1 0 0 1 0
|
||
1 3 0 0 -1
|
||
1 0 0 0 1
|
||
1 0 -2 1 0
|
||
1 0 0 1 0
|
||
1 0 0 0 -1
|
||
1 0 0 0 1
|
||
1 0 0 1 0
|
||
1 0 1 1 0
|
||
1 3 0 0 -1
|
||
1 1 0 0 1
|
||
1 0 -2 1 0
|
||
1 0 1 1 0
|
||
1 0 0 0 -1
|
||
1 1 0 0 1
|
||
1 0 0 0 1
|
||
1 0 0 1 0
|
||
1 3 0 0 1
|
||
1 0 0 1 0
|
||
1 0 0 0 1
|
||
1 0 2 1 0
|
||
1 3 0 0 1
|
||
1 0 2 1 0
|
||
Curves 13
|
||
1 0 0 0 0 0 1
|
||
1 0 0 3 -0 1 0
|
||
1 0 2 0 0 0 1
|
||
1 0 0 0 -0 1 0
|
||
1 1 0 0 0 0 1
|
||
1 1 0 3 0 1 0
|
||
1 1 2 0 0 0 1
|
||
1 1 0 0 -0 1 0
|
||
1 0 0 0 1 0 -0
|
||
1 0 0 3 1 0 -0
|
||
1 0 2 0 1 0 -0
|
||
1 0 2 3 1 0 -0
|
||
1 1 0 0 1 0 0
|
||
Polygon3D 1
|
||
2 1
|
||
0.1
|
||
1 0 0 2 0 0
|
||
0 1
|
||
PolygonOnTriangulations 24
|
||
2 1 2
|
||
p 0.1 1 0 3
|
||
2 1 4
|
||
p 0.1 1 0 3
|
||
2 2 3
|
||
p 0.1 1 0 2
|
||
2 1 2
|
||
p 0.1 1 0 2
|
||
2 4 3
|
||
p 0.1 1 0 3
|
||
2 1 4
|
||
p 0.1 1 0 3
|
||
2 1 4
|
||
p 0.1 1 0 2
|
||
2 1 2
|
||
p 0.1 1 0 2
|
||
2 1 2
|
||
p 0.1 1 0 3
|
||
2 2 3
|
||
p 0.1 1 0 3
|
||
2 2 3
|
||
p 0.1 1 0 2
|
||
2 4 3
|
||
p 0.1 1 0 2
|
||
2 4 3
|
||
p 0.1 1 0 3
|
||
2 2 3
|
||
p 0.1 1 0 3
|
||
2 1 4
|
||
p 0.1 1 0 2
|
||
2 4 3
|
||
p 0.1 1 0 2
|
||
2 1 2
|
||
p 0.1 1 0 1
|
||
2 1 4
|
||
p 0.1 1 0 1
|
||
2 4 3
|
||
p 0.1 1 0 1
|
||
2 1 4
|
||
p 0.1 1 0 1
|
||
2 1 2
|
||
p 0.1 1 0 1
|
||
2 2 3
|
||
p 0.1 1 0 1
|
||
2 4 3
|
||
p 0.1 1 0 1
|
||
2 2 3
|
||
p 0.1 1 0 1
|
||
Surfaces 6
|
||
1 0 0 0 1 0 -0 0 0 1 0 -1 0
|
||
1 0 0 0 -0 1 0 0 0 1 1 0 -0
|
||
1 0 0 3 0 0 1 1 0 -0 -0 1 0
|
||
1 0 2 0 -0 1 0 0 0 1 1 0 -0
|
||
1 0 0 0 0 0 1 1 0 -0 -0 1 0
|
||
1 1 0 0 1 0 -0 0 0 1 0 -1 0
|
||
Triangulations 6
|
||
4 2 1 0
|
||
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
|
||
4 2 1 0
|
||
0 0 0 1 0 0 1 0 3 0 0 3 0 0 0 1 3 1 3 0 3 2 1 3 1 4
|
||
4 2 1 0
|
||
0 0 3 0 2 3 1 2 3 1 0 3 0 0 0 2 1 2 1 0 3 2 1 3 1 4
|
||
4 2 1 0
|
||
0 2 0 1 2 0 1 2 3 0 2 3 0 0 0 1 3 1 3 0 3 2 1 3 1 4
|
||
4 2 1 0
|
||
0 0 0 0 2 0 1 2 0 1 0 0 0 0 0 2 1 2 1 0 3 2 1 3 1 4
|
||
4 2 1 0
|
||
1 0 0 1 0 3 1 2 3 1 2 0 0 0 3 0 3 -2 0 -2 2 4 3 2 1 4
|
||
|
||
TShapes 39
|
||
Ve
|
||
1e-007
|
||
0 0 3
|
||
0 0
|
||
|
||
0101101
|
||
*
|
||
Ve
|
||
1e-007
|
||
0 0 0
|
||
0 0
|
||
|
||
0101101
|
||
*
|
||
Ed
|
||
1e-007 1 1 0
|
||
1 1 0 0 3
|
||
2 1 1 0 0 3
|
||
2 2 2 0 0 3
|
||
6 1 1 0
|
||
6 2 2 0
|
||
0
|
||
|
||
0101000
|
||
-39 0 +38 0 *
|
||
Ve
|
||
1e-007
|
||
0 2 3
|
||
0 0
|
||
|
||
0101101
|
||
*
|
||
Ed
|
||
1e-007 1 1 0
|
||
1 2 0 0 2
|
||
2 3 1 0 0 2
|
||
2 4 3 0 0 2
|
||
6 3 1 0
|
||
6 4 3 0
|
||
0
|
||
|
||
0101000
|
||
-36 0 +39 0 *
|
||
Ve
|
||
1e-007
|
||
0 2 0
|
||
0 0
|
||
|
||
0101101
|
||
*
|
||
Ed
|
||
1e-007 1 1 0
|
||
1 3 0 0 3
|
||
2 5 1 0 0 3
|
||
2 6 4 0 0 3
|
||
6 5 1 0
|
||
6 6 4 0
|
||
0
|
||
|
||
0101000
|
||
-36 0 +34 0 *
|
||
Ed
|
||
1e-007 1 1 0
|
||
1 4 0 0 2
|
||
2 7 1 0 0 2
|
||
2 8 5 0 0 2
|
||
6 7 1 0
|
||
6 8 5 0
|
||
0
|
||
|
||
0101000
|
||
-34 0 +38 0 *
|
||
Wi
|
||
|
||
0101000
|
||
-37 0 -35 0 +33 0 +32 0 *
|
||
Fa
|
||
0 1e-007 1 0
|
||
2 1
|
||
0101000
|
||
+31 0 *
|
||
Ve
|
||
1e-007
|
||
1 0 3
|
||
0 0
|
||
|
||
0101101
|
||
*
|
||
Ve
|
||
1e-007
|
||
1 0 0
|
||
0 0
|
||
|
||
0101101
|
||
*
|
||
Ed
|
||
1e-007 1 1 0
|
||
1 5 0 0 3
|
||
2 9 6 0 0 3
|
||
2 10 2 0 0 3
|
||
6 9 6 0
|
||
6 10 2 0
|
||
0
|
||
|
||
0101000
|
||
-29 0 +28 0 *
|
||
Ve
|
||
1e-007
|
||
1 2 3
|
||
0 0
|
||
|
||
0101101
|
||
*
|
||
Ed
|
||
1e-007 1 1 0
|
||
1 6 0 0 2
|
||
2 11 6 0 0 2
|
||
2 12 3 0 0 2
|
||
6 11 6 0
|
||
6 12 3 0
|
||
0
|
||
|
||
0101000
|
||
-26 0 +29 0 *
|
||
Ve
|
||
1e-007
|
||
1 2 0
|
||
0 0
|
||
|
||
0101101
|
||
*
|
||
Ed
|
||
1e-007 1 1 0
|
||
1 7 0 0 3
|
||
2 13 6 0 0 3
|
||
2 14 4 0 0 3
|
||
6 13 6 0
|
||
6 14 4 0
|
||
0
|
||
|
||
0101000
|
||
-26 0 +24 0 *
|
||
Ed
|
||
1e-007 1 1 0
|
||
1 8 0 0 2
|
||
2 15 6 0 0 2
|
||
2 16 5 0 0 2
|
||
6 15 6 0
|
||
6 16 5 0
|
||
0
|
||
|
||
0101000
|
||
-24 0 +28 0 *
|
||
Wi
|
||
|
||
0101000
|
||
-27 0 -25 0 +23 0 +22 0 *
|
||
Fa
|
||
0 1e-007 6 0
|
||
2 6
|
||
0101000
|
||
+21 0 *
|
||
Ed
|
||
1e-007 1 1 0
|
||
1 9 0 0 1
|
||
2 17 2 0 0 1
|
||
2 18 5 0 0 1
|
||
6 17 2 0
|
||
6 18 5 0
|
||
0
|
||
|
||
0101000
|
||
-28 0 +38 0 *
|
||
Ed
|
||
1e-007 1 1 0
|
||
1 10 0 0 1
|
||
2 19 2 0 0 1
|
||
2 20 3 0 0 1
|
||
6 19 2 0
|
||
6 20 3 0
|
||
0
|
||
|
||
0101000
|
||
-29 0 +39 0 *
|
||
Wi
|
||
|
||
0101000
|
||
-19 0 -27 0 +18 0 +37 0 *
|
||
Fa
|
||
0 1e-007 2 0
|
||
2 2
|
||
0101000
|
||
+17 0 *
|
||
Ed
|
||
1e-007 1 1 0
|
||
1 11 0 0 1
|
||
2 21 4 0 0 1
|
||
2 22 5 0 0 1
|
||
6 21 4 0
|
||
6 22 5 0
|
||
0
|
||
|
||
0101000
|
||
-24 0 +34 0 *
|
||
Ed
|
||
1e-007 1 1 0
|
||
1 12 0 0 1
|
||
2 23 4 0 0 1
|
||
2 24 3 0 0 1
|
||
6 23 4 0
|
||
6 24 3 0
|
||
0
|
||
|
||
0101000
|
||
-26 0 +36 0 *
|
||
Wi
|
||
|
||
0101000
|
||
-15 0 -23 0 +14 0 +33 0 *
|
||
Fa
|
||
0 1e-007 4 0
|
||
2 4
|
||
0101000
|
||
+13 0 *
|
||
Wi
|
||
|
||
0101000
|
||
-32 0 -15 0 +22 0 +19 0 *
|
||
Fa
|
||
0 1e-007 5 0
|
||
2 5
|
||
0101000
|
||
+11 0 *
|
||
Wi
|
||
|
||
0101000
|
||
-35 0 -14 0 +25 0 +18 0 *
|
||
Fa
|
||
0 1e-007 3 0
|
||
2 3
|
||
0101000
|
||
+9 0 *
|
||
Sh
|
||
|
||
0101100
|
||
-30 0 +20 0 -16 0 +12 0 -10 0 +8 0 *
|
||
So
|
||
|
||
0100000
|
||
+7 0 *
|
||
CS
|
||
|
||
0101000
|
||
+6 3 *
|
||
Ve
|
||
1e-007
|
||
1 0 0
|
||
0 0
|
||
|
||
0101101
|
||
*
|
||
Ve
|
||
1e-007
|
||
2 0 0
|
||
0 0
|
||
|
||
0101101
|
||
*
|
||
Ed
|
||
1e-007 1 1 0
|
||
1 13 0 0 1
|
||
5 1 0
|
||
0
|
||
|
||
0101000
|
||
+4 0 -3 0 *
|
||
Co
|
||
|
||
1100000
|
||
+5 0 +2 0 *
|
||
|
||
+1 0
|
||
0
|
||
@endverbatim
|