diff --git a/src/BRepMesh/BRepMesh_CurveTessellator.cxx b/src/BRepMesh/BRepMesh_CurveTessellator.cxx
index 3779a52425..c8e942ec3a 100644
--- a/src/BRepMesh/BRepMesh_CurveTessellator.cxx
+++ b/src/BRepMesh/BRepMesh_CurveTessellator.cxx
@@ -101,9 +101,21 @@ void BRepMesh_CurveTessellator::init()
   myEdgeSqTol  = BRep_Tool::Tolerance (myEdge);
   myEdgeSqTol *= myEdgeSqTol;
 
-  const Standard_Integer aMinPntNb = Max(myMinPointsNb,
-    (myCurve.GetType() == GeomAbs_Circle) ? 4 : 2); //OCC287
+  Standard_Integer aMinPntThreshold = 2;
+  switch (myCurve.GetType())
+  {
+    case GeomAbs_Circle:
+    case GeomAbs_Ellipse:
+    case GeomAbs_Parabola:
+    case GeomAbs_Hyperbola:
+      aMinPntThreshold = 4;
+      break;
 
+    default:
+      break;
+  }
+
+  const Standard_Integer aMinPntNb = Max (myMinPointsNb, aMinPntThreshold); //OCC287
   myDiscretTool.Initialize (myCurve,
                             myCurve.FirstParameter(), myCurve.LastParameter(),
                             aPreciseAngDef, aPreciseLinDef, aMinPntNb,
diff --git a/tests/bugs/mesh/bug27845 b/tests/bugs/mesh/bug27845
index 6287701923..c9a337f75b 100644
--- a/tests/bugs/mesh/bug27845
+++ b/tests/bugs/mesh/bug27845
@@ -14,7 +14,7 @@ vdisplay result
 vfit 
 checkview -screenshot -3d -path ${imagedir}/${test_image}.png
 
-checktrinfo result -tri 3828 -nod 4771 -defl 3.0544822246414993 -tol_abs_defl 1e-6
+checktrinfo result -tri 3834 -nod 4777 -defl 3.0544822246414993 -tol_abs_defl 1e-6
 
 set log [tricheck result]
 if { [llength $log] != 0 } {
diff --git a/tests/bugs/mesh/bug31449_1 b/tests/bugs/mesh/bug31449_1
index f117e4a33c..05032627a6 100644
--- a/tests/bugs/mesh/bug31449_1
+++ b/tests/bugs/mesh/bug31449_1
@@ -15,6 +15,6 @@ vfit
 vsetdispmode 1
 
 XGetOneShape result Q
-checktrinfo result -tri 9025 -nod 6562
+checktrinfo result -tri 9097 -nod 6634
 
 checkview -screenshot -3d -path ${imagedir}/${test_image}.png
diff --git a/tests/bugs/mesh/bug31449_3 b/tests/bugs/mesh/bug31449_3
index f77c0646c0..b7dc0cdc22 100644
--- a/tests/bugs/mesh/bug31449_3
+++ b/tests/bugs/mesh/bug31449_3
@@ -14,6 +14,6 @@ XGetOneShape result Q
 
 incmesh result 0.00039624 -a 20 -min 0.0001 -parallel
 
-checktrinfo result -tri 7661 -nod 5810
+checktrinfo result -tri 7733 -nod 5882
 
 checkview -display result -3d -path ${imagedir}/${test_image}.png
diff --git a/tests/bugs/mesh/bug31865 b/tests/bugs/mesh/bug31865
new file mode 100644
index 0000000000..e9477fdc07
--- /dev/null
+++ b/tests/bugs/mesh/bug31865
@@ -0,0 +1,20 @@
+puts "========"
+puts "0031865: Mesh - triangulation fails with large deflection values due to unhandled Standard_OutOfRange, BRepMesh_PairOfIndex::Append()"
+puts "========"
+puts ""
+
+restore [locate_data_file bug31865.brep] result
+
+tclean result
+incmesh result 0.2
+
+vinit
+vdefaults -autoTriang 0
+checkview -display result -3d -path ${imagedir}/${test_image}.png
+
+set log [tricheck result]
+if { [llength $log] != 0 } {
+  puts "Error : Invalid mesh"
+} else {
+  puts "Mesh is OK"
+}
diff --git a/tests/bugs/modalg_2/bug264_6 b/tests/bugs/modalg_2/bug264_6
index 5285e22da3..08f75e8aa3 100755
--- a/tests/bugs/modalg_2/bug264_6
+++ b/tests/bugs/modalg_2/bug264_6
@@ -15,6 +15,6 @@ vclear
 isos result 0
 triangles result
 
-checktrinfo result -tri 18 -nod 20
+checktrinfo result -tri 20 -nod 22
 checkprops result -s 19.2399 
 checkview -display result -3d -path ${imagedir}/${test_image}.png
diff --git a/tests/bugs/modalg_2/bug264_7 b/tests/bugs/modalg_2/bug264_7
index 9197a0b3da..1a03acd2e1 100755
--- a/tests/bugs/modalg_2/bug264_7
+++ b/tests/bugs/modalg_2/bug264_7
@@ -15,6 +15,6 @@ vclear
 isos result 0
 triangles result
 
-checktrinfo result -tri 91 -nod 93
+checktrinfo result -tri 103 -nod 105
 checkprops result -s 150.283 
 checkview -display result -3d -path ${imagedir}/${test_image}.png
diff --git a/tests/de_mesh/shape_write_stl/A8 b/tests/de_mesh/shape_write_stl/A8
index b7880923c9..16bf1a7661 100644
--- a/tests/de_mesh/shape_write_stl/A8
+++ b/tests/de_mesh/shape_write_stl/A8
@@ -9,7 +9,7 @@ writestl m $imagedir/${casename}
 readstl res $imagedir/${casename}
 file delete $imagedir/${casename}
 
-checktrinfo res -tri 34 -nod 19
+checktrinfo res -tri 36 -nod 20
 
 # Visual check
 checkview -display res -2d -path ${imagedir}/${test_image}.png
diff --git a/tests/hlr/poly_hlr/A3 b/tests/hlr/poly_hlr/A3
index 804241eb90..df6a3bf924 100644
--- a/tests/hlr/poly_hlr/A3
+++ b/tests/hlr/poly_hlr/A3
@@ -4,7 +4,7 @@ puts "====================================="
 puts ""
 
 set viewname "vbottom"
-set length 8.40196
+set length 8.40188
 
 restore [locate_data_file bug27341_hlrsave.brep] a
 COMPUTE_HLR $viewname $algotype
diff --git a/tests/hlr/poly_hlr/C13 b/tests/hlr/poly_hlr/C13
index 6a502fbb74..69067419fd 100644
--- a/tests/hlr/poly_hlr/C13
+++ b/tests/hlr/poly_hlr/C13
@@ -1,5 +1,5 @@
 set viewname "vright"
-set length 9547.12
+set length 9546.37
 
 testreadstep [locate_data_file bug27341_Assembly_ABS_1_CAD.stp] a
 COMPUTE_HLR $viewname $algotype
diff --git a/tests/hlr/poly_hlr/C15 b/tests/hlr/poly_hlr/C15
index 804df0c48f..a90bd9554c 100644
--- a/tests/hlr/poly_hlr/C15
+++ b/tests/hlr/poly_hlr/C15
@@ -1,5 +1,5 @@
 set viewname "vright"
-set length 5101.26
+set length 5101.97
 
 testreadstep [locate_data_file bug27341_Assembly_GMS_Kurz_CAD.stp] a
 COMPUTE_HLR $viewname $algotype
diff --git a/tests/hlr/poly_hlr/C20 b/tests/hlr/poly_hlr/C20
index ae4a9db262..14bf820c30 100644
--- a/tests/hlr/poly_hlr/C20
+++ b/tests/hlr/poly_hlr/C20
@@ -1,5 +1,5 @@
 set viewname "vright"
-set length 1704.86
+set length 1705.48
 
 testreadstep [locate_data_file bug27341_Drehkopf_HSK_Gewinde_R_CAD.stp] a
 COMPUTE_HLR $viewname $algotype
diff --git a/tests/hlr/poly_hlr/C23 b/tests/hlr/poly_hlr/C23
index c1c2ed6118..7651d0d8fa 100644
--- a/tests/hlr/poly_hlr/C23
+++ b/tests/hlr/poly_hlr/C23
@@ -1,5 +1,5 @@
 set viewname "vright"
-set length 826.794
+set length 829.419
 
 testreadstep [locate_data_file bug27341_KMM_Adapter_CAD.stp] a
 COMPUTE_HLR $viewname $algotype
diff --git a/tests/hlr/poly_hlr/C24 b/tests/hlr/poly_hlr/C24
index 46a0173c34..0ea41a1855 100644
--- a/tests/hlr/poly_hlr/C24
+++ b/tests/hlr/poly_hlr/C24
@@ -1,5 +1,5 @@
 set viewname "vright"
-set length 217.803
+set length 239.091
 
 testreadstep [locate_data_file bug27341_MZX_01_1_CAD.stp] a
 COMPUTE_HLR $viewname $algotype
diff --git a/tests/hlr/poly_hlr/C3 b/tests/hlr/poly_hlr/C3
index b6806e8256..33f2568bd7 100644
--- a/tests/hlr/poly_hlr/C3
+++ b/tests/hlr/poly_hlr/C3
@@ -1,5 +1,5 @@
 set viewname "vright"
-set length 3057.35
+set length 3057.09
 
 testreadstep [locate_data_file bug27341_570-DWLNL-40-08-L_131LANG_16VERSATZ_DIN.stp] a
 COMPUTE_HLR $viewname $algotype
diff --git a/tests/hlr/poly_hlr/C6 b/tests/hlr/poly_hlr/C6
index e0ad37efcc..e3077f9d8e 100644
--- a/tests/hlr/poly_hlr/C6
+++ b/tests/hlr/poly_hlr/C6
@@ -1,5 +1,5 @@
 set viewname "vright"
-set length 2832.09
+set length 2832.15
 
 testreadstep [locate_data_file bug27341_ABS_Grundhalter_CAD.stp] a
 COMPUTE_HLR $viewname $algotype
diff --git a/tests/hlr/poly_hlr/C8 b/tests/hlr/poly_hlr/C8
index 0a1185bbfd..986c7614fe 100644
--- a/tests/hlr/poly_hlr/C8
+++ b/tests/hlr/poly_hlr/C8
@@ -1,5 +1,5 @@
 set viewname "vright"
-set length 1154.09
+set length 1154.43
 
 testreadstep [locate_data_file bug27341_AWN_Adapter_CAD.stp] a
 COMPUTE_HLR $viewname $algotype
diff --git a/tests/hlr/poly_hlr/bug27979_2 b/tests/hlr/poly_hlr/bug27979_2
index a5df15c581..48132c53ea 100644
--- a/tests/hlr/poly_hlr/bug27979_2
+++ b/tests/hlr/poly_hlr/bug27979_2
@@ -4,7 +4,7 @@ puts "========================================================================"
 puts ""
 
 set viewname "vtop"
-set length 13.0106
+set length 13.0091
 
 restore [locate_data_file bug27979_parsed.brep] a
 COMPUTE_HLR $viewname $algotype