- BRep Format Description {#occt_brep_format}
+ BRep Format {#occt_brep_format}
========================
@tableofcontents
* format file fragment to illustrate the part;
* BNF-like definition of the part;
* detailed description of the part.
+
+**Note** that the format is a part of Open CASCADE Technology (OCCT).
+
+Some data fields of the format have additional values, which are used in OCCT.
+
+Some data fields of the format are specific for OCCT.
@section occt_brep_format_2 Format Common Structure
BREP format uses the following BNF terms:
- * <\n>;
- * <_\n>;
- * <_>;
- * <flag>;
- * <int>;
- * <real>;
- * <2D point>;
- * <3D point>;
- * <2D direction>;
- * <3D direction>;
- * <+>;
-
- * \n is the operating-system-dependent ASCII character sequence which separates ASCII text strings in the operating system used.
- * _\n = " "* \n;
- * _ = " "+;
- * _ is a not empty sequence of space characters with ASCII code 21h.
- * flag = "0" | "1";
- * int is an integer number from @image html brep_wp_image003.gif to @image html brep_wp_image004.gif which is written in denary system.
- * real is a real from @image html brep_wp_image005.gif to @image html brep_wp_image006.gif
- which is written in decimal or E form with base 10.
-
- The point is used as a delimiter of the integer and fractional parts.
-
- * 2D point = real _ real;
- * 3D point = real (_ real) ^ 2;
- * 2D direction is a 2D point @image html brep_wp_image007.gif so that @image html brep_wp_image008.gif.
- * 3D direction is a 3D point @image html brep_wp_image009.gif so that @image html brep_wp_image010.gif.
- + is an arithmetic operation of addition.
-
+ * \<\\n\>: It is the operating-system-dependent ASCII character sequence which separates ASCII text strings in the operating system used;
+ * \<_\\n\>: = " "*\<\\n\>;
+ * \<_\>: = " "+; It is a not empty sequence of space characters with ASCII code 21h;
+ * \<flag\>: = "0" | "1";
+ * \<int\>: It is an integer number from -2<sup>31</sup> to 2<sup>31</sup>-1 which is written in denary system;
+ * \<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;
+ * \<2D point\>: = \<real\>\<_\>\<real\>;
+ * \<3D point\>: = \<real\>(\<_\>\<real)\><sup>2</sup>;
+ * \<2D direction\>: It is a \<2D point\> *x y* so that *x<sup>2</sup> + y<sup>2</sup>* = 1;
+ * \<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;
+ * \<+\>: It is an arithmetic operation of addition.
+
The format consists of the following sections:
- * content type;
- * version;
- * locations;
- * geometry;
- * shapes.
+ * \<content type\>;
+ * \<version\>;
+ * \<locations\>;
+ * \<geometry\>;
+ * \<shapes\>.
- content type = "DBRep_DrawableShape" _\n _\n;
- content type have other values [1].
+ \<content type\> = "DBRep_DrawableShape" \<_\\n\>\<_\\n\>;
+ \<content type\> have other values [1].
- version = ("CASCADE Topology V1, (c) Matra-Datavision" | "CASCADE Topology V2, (c) Matra-Datavision") _\n;
-
+ \<version\> = ("CASCADE Topology V1, (c) Matra-Datavision" | "CASCADE Topology V2, (c) Matra-Datavision")\<_\\n\>;
The difference of the versions is described in the document.
- Sections <locations>, <geometry> and <shapes> are described below in separate chapters of the document.
+ Sections \<locations\>, \<geometry\> and \<shapes\> are described below in separate chapters of the document.
-@section occt_brep_format_3 Section locations
+@section occt_brep_format_3 Locations
**Example**
**BNF-like Definition**
@verbatim
- locations = location header _\n location records;
- location header = "Locations" _ location record count;
- location record count = int;
- location records = location record ^ location record count;
- location record = location record 1 | location record 2;
- location record 1 = "1" _\n location data 1;
- location record 2 = "2" _ location data 2;
- location data 1 = ((_ real) ^ 4 _\n) ^ 3;
- location data 2 = (int _ int _)* "0" _\n;
+ <locations> = <location header> <_\n> <location records>;
+ <location header> = "Locations" <_> <location record count>;
+ <location record count> = <int>;
+ <location records> = <location record> ^ <location record count>;
+ <location record> = <location record 1> | <location record 2>;
+ <location record 1> = "1" <_\n> <location data 1>;
+ <location record 2> = "2" <_> <location data 2>;
+ <location data 1> = ((<_> <real>) ^ 4 <_\n>) ^ 3;
+ <location data 2> = (<int> <_> <int> <_>)* "0" <_\n>;
@endverbatim
-****Description****
+**Description**
-*location data 1* is interpreted as a 3 x 4 matrix @image html brep_wp_image011.gif
+\<location data 1\> is interpreted as a 3 x 4 matrix
+@f$Q =
+\begin{pmatrix}
+{q}_{1,1} &{q}_{1,2} &{q}_{1,3} &{q}_{1,4}\\
+{q}_{2,1} &{q}_{2,2} &{q}_{2,3} &{q}_{2,4}\\
+{q}_{3,1} &{q}_{3,2} &{q}_{3,3} &{q}_{3,4}
+\end{pmatrix}@f$
which describes transformation of 3 dimensional space and satisfies the following constraints:
-
- * @image html brep_wp_image012.gif where @image html brep_wp_image013.gif where @image html brep_wp_image014.gif;
- * @image html brep_wp_image015.gif where @image html brep_wp_image016.gif.
-
-The transformation transforms a point
-
-@image html brep_wp_image017.gif
-
-to another point
-
-@image html brep_wp_image018.gif
-
-by the rule:
-
-@image html brep_wp_image019.gif.
-@image html brep_wp_image020.gif
-
-may be a composition of matrices for the following elementary transformations:
-
- * parallel translation – @image html brep_wp_image021.gif;
- * rotation around an axis with a direction @image html brep_wp_image022.gif by an angle @image html brep_wp_image023.gif –
-
-@image html brep_wp_image024.gif;
-
- * scaling – @image html brep_wp_image025.gif where @image html brep_wp_image026.gif;
- * central symmetry – @image html brep_wp_image027.gif;
- * axis symmetry – @image html brep_wp_image028.gif;
- * plane symmetry – @image html brep_wp_image029.gif.
-
-*location data 2* is interpreted as a composition of locations raised to a power and placed above this location data 2 in the section locations. location data 2 is a sequence @image html brep_wp_image030.gif of @image html brep_wp_image031.gif integer pairs @image html brep_wp_image032.gif (@image html brep_wp_image033.gif). flag 0 is the indicator of the sequence end. The sequence is interpreted as a composition @image html brep_wp_image034.gif where @image html brep_wp_image035.gif is a location from @image html brep_wp_image036.gif-th location record in the section locations. location record numbering starts from1.
-
-
-@section occt_brep_format_4 Section geometry
-
-@verbatim
- geometry =
- 2D curves
- 3D curves
- 3D polygons
- polygons on triangulations
- surfaces
- triangulations;
+ * @f$ d \neq 0@f$ where @f$d = |Q_{2}|@f$ where
+ @f$ Q_{2} = \begin{pmatrix}
+ {q}_{1,1} &{q}_{1,2} &{q}_{1,3} &{q}_{1,4}\\
+ {q}_{2,1} &{q}_{2,2} &{q}_{2,3} &{q}_{2,4}\\
+ {q}_{3,1} &{q}_{3,2} &{q}_{3,3} &{q}_{3,4}
+ \end{pmatrix}; @f$
+ * @f$ Q_{3}^{T} = Q_{3}^{-1}@f$ where @f$Q_{3} = Q_{2}/d^{1/3}. @f$
+
+The transformation transforms a point (x, y, z) to another point (u, v, w) by the rule:
+@f[ \begin{pmatrix}
+u \\ v \\ w
+\end{pmatrix} =
+Q\cdot(x\;y\;z\;1)^{T} =
+\begin{pmatrix}
+{q}_{1,1}\cdot x +{q}_{1,2}\cdot y +{q}_{1,3}\cdot z +{q}_{1,4}\\
+{q}_{2,1}\cdot x +{q}_{2,2}\cdot y +{q}_{2,3}\cdot z +{q}_{2,4}\\
+{q}_{3,1}\cdot x +{q}_{3,2}\cdot y +{q}_{3,3}\cdot z +{q}_{3,4}
+\end{pmatrix} . @f]
+
+*Q* may be a composition of matrices for the following elementary transformations:
+
+ * parallel translation –
+ @f$ \begin{pmatrix}
+ 1 &0 &0 &{q}_{1,4}\\
+ 0 &1 &0 &{q}_{2,4}\\
+ 0 &0 &1 &{q}_{3,4}
+ \end{pmatrix}; @f$
+ * rotation around an axis with a direction *D(D<sub>x</sub>, D<sub>y</sub>, D<sub>z</sub>)* by an angle @f$ \varphi @f$ –
+
+ @f[ \begin{pmatrix}
+ D_{x}^{2} \cdot (1-cos(\varphi)) + cos(\varphi) &D_{x} \cdot D_{y} \cdot (1-cos(\varphi)) - D_{z} \cdot sin(\varphi) &D_{x} \cdot D_{z} \cdot (1-cos(\varphi)) + D_{y} \cdot sin(\varphi) &0\\
+ D_{x} \cdot D_{y} \cdot (1-cos(\varphi)) + D_{z} \cdot sin(\varphi) &D_{y}^{2} \cdot (1-cos(\varphi)) + cos(\varphi) &D_{y} \cdot D_{z} \cdot (1-cos(\varphi)) - D_{x} \cdot sin(\varphi) &0\\
+ D_{x} \cdot D_{z} \cdot (1-cos(\varphi)) - D_{y} \cdot sin(\varphi) &D_{y} \cdot D_{z} \cdot (1-cos(\varphi)) + D_{x} \cdot sin(\varphi) &D_{z}^{2} \cdot (1-cos(\varphi)) + cos(\varphi) &0
+ \end{pmatrix}; @f]
+
+ * scaling – @f$ \begin{pmatrix} s &0 &0 &0\\ 0 &s &0 &0\\ 0 &0 &s &0 \end{pmatrix} @f$ where @f$ S \in (-\infty,\; \infty)/\left \{ 0 \right \}; @f$
+ * central symmetry – @f$ \begin{pmatrix} -1 &0 &0 &0\\ 0 &-1 &0 &0\\ 0 &0 &-1 &0 \end{pmatrix}; @f$
+ * axis symmetry – @f$ \begin{pmatrix} -1 &0 &0 &0\\ 0 &-1 &0 &0\\ 0 &0 &1 &0 \end{pmatrix}; @f$
+ * plane symmetry – @f$ \begin{pmatrix} 1 &0 &0 &0\\ 0 &1 &0 &0\\ 0 &0 &-1 &0 \end{pmatrix}. @f$
+
+\<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.
+
+
+@section occt_brep_format_4 Geometry
+
+@verbatim
+ <geometry> =
+ <2D curves>
+ <3D curves>
+ <3D polygons>
+ <polygons on triangulations>
+ <surfaces>
+ <triangulations>;
@endverbatim
-@subsection occt_brep_format_4_1 Subsection 3D curves
+@subsection occt_brep_format_4_1 3D curves
**Example**
**BNF-like Definition**
@verbatim
- 3D curves = 3D curve header _\n 3D curve records;
+ <3D curves> = <3D curve header> <_\n> <3D curve records>;
- 3D curve header = "Curves" _ 3D curve count;
+ <3D curve header> = "Curves" <_> <3D curve count>;
- 3D curve count = int;
+ <3D curve count> = <int>;
- 3D curve records = 3D curve record ^ 3D curve count;
+ <3D curve records> = <3D curve record> ^ <3D curve count>;
- 3D curve record =
- 3D curve record 1 |
- 3D curve record 2 |
- 3D curve record 3 |
- 3D curve record 4 |
- 3D curve record 5 |
- 3D curve record 6 |
- 3D curve record 7 |
- 3D curve record 8 |
- 3D curve record 9;
-
+ <3D curve record> =
+ <3D curve record 1> |
+ <3D curve record 2> |
+ <3D curve record 3> |
+ <3D curve record 4> |
+ <3D curve record 5> |
+ <3D curve record 6> |
+ <3D curve record 7> |
+ <3D curve record 8> |
+ <3D curve record 9>;
+ @endverbatim
-@subsubsection occt_brep_format_4_1_1 3D curve record 1 – Line
+@subsubsection occt_brep_format_4_1_1 Line - \<3D curve record 1\>
**Example**
**BNF-like Definition**
@verbatim
- 3D curve record 1 = "1" _ 3D point _ 3D direction _\n;
+ <3D curve record 1> = "1" <_> <3D point> <_> <3D direction> <_\n>;
@endverbatim
**Description**
-3D curve record 1 describes a line. The line data consist of a 3D point @image html brep_wp_image037.gif and a 3D direction @image html brep_wp_image038.gif. The line passes through the point @image html brep_wp_image037.gif, has the direction @image html brep_wp_image038.gif and is defined by the following parametric equation:
+\<3D curve record 1\> describes a line. The line data consist of a 3D point *P* and a 3D direction *D*. The line passes through the point *P*, has the direction *D* and is defined by the following parametric equation:
-@image html brep_wp_image039.gif, @image html brep_wp_image040.gif.
+@f[ C(u)=P+u \cdot D, \; u \in (-\infty,\; \infty). @f]
-The **Example** record is interpreted as a line which passes through a point @image html brep_wp_image041.gif, has a direction @image html brep_wp_image042.gif and is defined by the following parametric equation: @image html brep_wp_image043.gif.
+The example record is interpreted as a line which passes through a point *P*=(1, 0, 3), has a direction *D*=(0, 1, 0) and is defined by the following parametric equation: @f$ C(u)=(1,0,3)+u \cdot (0,1,0) @f$.
-@subsubsection occt_brep_format_4_1_2 3D curve record 2 – Circle
+@subsubsection occt_brep_format_4_1_2 Circle - \<3D curve record 2\>
**Example**
**BNF-like Definition**
@verbatim
- 3D curve record 2 = "2" _ 3D circle center _ 3D circle N _ 3D circle Dx _ 3D circle Dy _ 3D circle radius _\n;
+ <3D curve record 2> = "2" <_> <3D circle center> <_> <3D circle N> <_> <3D circle Dx> <_> <3D circle Dy> <_> <3D circle radius> <_\n>;
- 3D circle center = 3D point;
+ <3D circle center> = <3D point>;
- 3D circle N = 3D direction;
+ <3D circle N> = <3D direction>;
- 3D circle Dx = 3D direction;
+ <3D circle Dx> = <3D direction>;
- 3D circle Dy = 3D direction;
+ <3D circle Dy> = <3D direction>;
- 3D circle radius = real;
+ <3D circle radius> = <real>;
@endverbatim
**Description**
-3D curve record 2 describes a circle. The circle data consist of a 3D point @image html brep_wp_image037.gif, pairwise orthogonal 3D directions @image html brep_wp_image044.gif, @image html brep_wp_image045.gif and @image html brep_wp_image046.gif and a non-negative real @image html brep_wp_image047.gif. The circle has a center @image html brep_wp_image037.gif and is located in a plane with a normal @image html brep_wp_image044.gif. The circle has a radius @image html brep_wp_image047.gif and is defined by the following parametric equation:
+\<3D curve record 2\> describes a circle. The circle data consist of a 3D point *P*, pairwise orthogonal 3D directions *N*, *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:
-@image html brep_wp_image048.gif, @image html brep_wp_image049.gif.
+@f[ C(u)=P+r \cdot (cos(u) \cdot D_{x} + sin(u) \cdot D_{y}), \; u \in [o,\;2 \cdot \pi). @f]
-The example record is interpreted as a circle which has its center @image html brep_wp_image050.gif, is located in plane with a normal @image html brep_wp_image051.gif. Directions for the circle are @image html brep_wp_image052.gif and @image html brep_wp_image053.gif. The circle has a radius @image html brep_wp_image054.gif and is defined by the following parametric equation: @image html brep_wp_image055.gif.
+The example record is interpreted as a circle which has its center *P*=(1, 2, 3), is located in plane with a normal *N*=(0, 0 ,1). Directions for the circle are *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$.
-@subsubsection occt_brep_format_4_1_3 3D curve record 3 – Ellipse
+@subsubsection occt_brep_format_4_1_3 Ellipse - \<3D curve record 3\>
**Example**
**BNF-like Definition**
@verbatim
- 3D curve record 3 = "3" _ 3D ellipse center _ 3D ellipse N _ 3D ellipse Dmaj _ 3D ellipse Dmin _ 3D ellipse Rmaj _ 3D ellipse Rmin _\n;
+ <3D curve record 3> = "3" <_> <3D ellipse center> <_> <3D ellipse N> <_> <3D ellipse Dmaj> <_> <3D ellipse Dmin> <_> <3D ellipse Rmaj> <_> <3D ellipse Rmin> <_\n>;
- 3D ellipse center = 3D point;
+ <3D ellipse center> = <3D point>;
- 3D ellipse N = 3D direction;
+ <3D ellipse N> = <3D direction>;
- 3D ellipse Dmaj = 3D direction;
+ <3D ellipse Dmaj> = <3D direction>;
- 3D ellipse Dmin = 3D direction;
+ <3D ellipse Dmin> = <3D direction>;
- 3D ellipse Rmaj = real;
+ <3D ellipse Rmaj> = <real>;
- 3D ellipse Rmin = real;
+ <3D ellipse Rmin> = <real>;
@endverbatim
**Description**
-3D curve record 3 describes an ellipse. The ellipse data consist of a 3D point @image html brep_wp_image037.gif, pairwise orthogonal 3D directions @image html brep_wp_image044.gif, @image html brep_wp_image056.gif and @image html brep_wp_image057.gif and non-negative reals @image html brep_wp_image058.gif and @image html brep_wp_image059.gif so that @image html brep_wp_image060.gif. The ellipse has its center @image html brep_wp_image037.gif, is located in plane with the normal @image html brep_wp_image044.gif, has major and minor axis directions @image html brep_wp_image056.gif and @image html brep_wp_image057.gif, major and minor radii @image html brep_wp_image058.gif and @image html brep_wp_image059.gif and is defined by the following parametric equation:
+\<3D curve record 3\> describes an ellipse. The ellipse data consist of a 3D point *P*, pairwise orthogonal 3D directions *N*, *D<sub>maj</sub>* and *D<sub>min</sub>* and non-negative reals *r<sub>maj</sub>* and *r<sub>min</sub>* so that *r<sub>min</sub>* @f$ \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:
-@image html brep_wp_image061.gif, @image html brep_wp_image049.gif.
+@f[ C(u)=P+r_{maj} \cdot cos(u) \cdot D_{maj} + r_{min} \cdot sin(u) \cdot D_{min}, u \in [0, 2 \cdot \pi). @f]
-The example record is interpreted as an ellipse which has its center @image html brep_wp_image050.gif, is located in plane with a normal @image html brep_wp_image051.gif, has major and minor axis directions @image html brep_wp_image062.gif and @image html brep_wp_image063.gif, major and minor radii @image html brep_wp_image064.gif and @image html brep_wp_image065.gif and is defined by the following parametric equation: @image html brep_wp_image066.gif.
+The example record is interpreted as an ellipse which has its center *P*=(1, 2, 3), is located in plane with a normal *N*=(0, 0, 1), has major and minor axis directions *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$.
-@subsubsection occt_brep_format_4_1_4 3D curve record 4 – Parabola
+@subsubsection occt_brep_format_4_1_4 Parabola - \<3D curve record 4\>
**Example**
**BNF-like Definition**
@verbatim
- 3D curve record 4 = "4" _ 3D parabola origin _ 3D parabola N _ 3D parabola Dx _ 3D parabola Dy _ 3D parabola focal length _\n;
+ <3D curve record 4> = "4" <_> <3D parabola origin> <_> <3D parabola N> <_> <3D parabola Dx> <_> <3D parabola Dy> <_> <3D parabola focal length> <_\n>;
- 3D parabola origin = 3D point;
+ <3D parabola origin> = <3D point>;
- 3D parabola N = 3D direction;
+ <3D parabola N> = <3D direction>;
- 3D parabola Dx = 3D direction;
+ <3D parabola Dx> = <3D direction>;
- 3D parabola Dy = 3D direction;
+ <3D parabola Dy> = <3D direction>;
- 3D parabola focal length = real;
+ <3D parabola focal length> = <real>;
@endverbatim
**Description**
-3D curve record 4 describes a parabola. The parabola data consist of a 3D point @image html brep_wp_image037.gif, pairwise orthogonal 3D directions @image html brep_wp_image044.gif, @image html brep_wp_image045.gif and @image html brep_wp_image046.gif and a non-negative real @image html brep_wp_image067.gif. The parabola is located in plane which passes through the point @image html brep_wp_image037.gif and has the normal @image html brep_wp_image044.gif. The parabola has a focus length @image html brep_wp_image067.gif and is defined by the following parametric equation:
+\<3D curve record 4\> describes a parabola. The parabola data consist of a 3D point *P*, pairwise orthogonal 3D directions *N*, *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:
-@image html brep_wp_image068.gif, @image html brep_wp_image069.gif Ü @image html brep_wp_image070.gif;
+@f[ C(u)=P+\frac{u^{2}}{4 \cdot f} \cdot D_{x} + u \cdot D_{y}, u \in (-\infty,\; \infty) \Leftarrow f \neq 0; @f]
+@f[ C(u)=P+u \cdot D_{x}, u \in (-\infty,\; \infty) \Leftarrow f = 0\;(degenerated\;case). @f]
-@image html brep_wp_image071.gif, @image html brep_wp_image069.gif Ü @image html brep_wp_image072.gif (degenerated case).
+The example record is interpreted as a parabola in plane which passes through a point *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$.
-The example record is interpreted as a parabola in plane which passes through a point @image html brep_wp_image073.gif and has a normal @image html brep_wp_image074.gif. Directions for the parabola are @image html brep_wp_image075.gif and @image html brep_wp_image076.gif. The parabola has a focus length @image html brep_wp_image077.gif and is defined by the following parametric equation: @image html brep_wp_image078.gif.
-
-@subsubsection occt_brep_format_4_1_5 3D curve record 5 – Hyperbola
+@subsubsection occt_brep_format_4_1_5 Hyperbola - \<3D curve record 5\>
**Example**
@verbatim
5 1 2 3 0 0 1 1 0 -0 -0 1 0 5 4
-@verbatim
+@endverbatim
**BNF-like Definition**
@verbatim
- 3D curve record 5 = "5" _ 3D hyperbola origin _ 3D hyperbola N _ 3D hyperbola Dx _ 3D hyperbola Dy _ 3D hyperbola Kx _ 3D hyperbola Ky _\n;
+ <3D curve record 5> = "5" <_> <3D hyperbola origin> <_> <3D hyperbola N> <_> <3D hyperbola Dx> <_> <3D hyperbola Dy> <_> <3D hyperbola Kx> <_> <3D hyperbola Ky> <_\n>;
- 3D hyperbola origin = 3D point;
+ <3D hyperbola origin> = <3D point>;
- 3D hyperbola N = 3D direction;
+ <3D hyperbola N> = <3D direction>;
- 3D hyperbola Dx = 3D direction;
+ <3D hyperbola Dx> = <3D direction>;
- 3D hyperbola Dy = 3D direction;
+ <3D hyperbola Dy> = <3D direction>;
- 3D hyperbola Kx = real;
+ <3D hyperbola Kx> = <real>;
- 3D hyperbola Ky = real;
+ <3D hyperbola Ky> = <real>;
@endverbatim
-Descripton
+**Descripton**
-3D curve record 5 describes a hyperbola. The hyperbola data consist of a 3D point @image html brep_wp_image037.gif, pairwise orthogonal 3D directions @image html brep_wp_image044.gif, @image html brep_wp_image045.gif and @image html brep_wp_image046.gif and non-negative reals @image html brep_wp_image079.gif and @image html brep_wp_image080.gif. The hyperbola is located in plane which passes through the point @image html brep_wp_image037.gif and has the normal @image html brep_wp_image044.gif. The hyperbola is defined by the following parametric equation:
+\<3D curve record 5\> describes a hyperbola. The hyperbola data consist of a 3D point *P*, pairwise orthogonal 3D directions *N*, *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:
-@image html brep_wp_image081.gif, @image html brep_wp_image069.gif.
+@f[ C(u)=P+k_{x} \cdot cosh(u) \cdot D_{x}+k_{y} \cdot sinh(u) \cdot D_{y} , u \in (-\infty,\; \infty). @f]
-The example record is interpreted as a hyperbola in plane which passes through a point @image html brep_wp_image073.gif and has a normal @image html brep_wp_image074.gif. Other hyperbola data are @image html brep_wp_image075.gif, @image html brep_wp_image076.gif, @image html brep_wp_image082.gif and @image html brep_wp_image083.gif. The hyperbola is defined by the following parametric equation: @image html brep_wp_image084.gif.
+The example record is interpreted as a hyperbola in plane which passes through a point *P*=(1, 2, 3) and has a normal *N*=(0, 0, 1). Other hyperbola data are *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$.
-@subsubsection occt_brep_format_4_1_6 3D curve record 6 – Bezier Curve
+@subsubsection occt_brep_format_4_1_6 Bezier Curve - \<3D curve record 6\>
**Example**
@verbatim
6 1 2 0 1 0 4 1 -2 0 5 2 3 0 6
-@verbatim
+@endverbatim
**BNF-like Definition**
@verbatim
- 3D curve record 6 = "6" _ 3D Bezier rational flag _ 3D Bezier degree 3D Bezier weight poles _\n;
-
- 3D Bezier rational flag = flag;
-
- 3D Bezier degree = int;
-
- 3D Bezier weight poles = (_ 3D Bezier weight pole) ^ (3D Bezier degree + "1");
-
- 3D Bezier weight pole = 3D point [_ real];
-@verbatim
+ <3D curve record 6> = "6" <_> <3D Bezier rational flag> <_> <3D Bezier degree>
+ <3D Bezier weight poles> <_\n>;
+
+ <3D Bezier rational flag> = <flag>;
+
+ <3D Bezier degree> = <int>;
+
+ 3D Bezier weight poles> = (<_> <3D Bezier weight pole>) ^ (<3D Bezier degree> <+> "1");
+
+ <3D Bezier weight pole> = <3D point> [<_> <real>];
+@endverbatim
**Description**
-3D curve record 6 describes a Bezier curve. The curve data consist of a rational flag @image html brep_wp_image047.gif, a degree @image html brep_wp_image085.gif and weight poles.
+\<3D curve record 6\> describes a Bezier curve. The curve data consist of a rational *r*, a degree @f$ m \leq 25 @f$ and weight poles.
-The weight poles are @image html brep_wp_image086.gif 3D points @image html brep_wp_image087.gif if the flag @image html brep_wp_image047.gif is 0. The weight poles are @image html brep_wp_image086.gif pairs @image html brep_wp_image088.gif if flag @image html brep_wp_image047.gif is 1. Here @image html brep_wp_image089.gif is a 3D point and @image html brep_wp_image090.gif is a positive real (@image html brep_wp_image091.gif). @image html brep_wp_image092.gif (@image html brep_wp_image093.gif) if the flag @image html brep_wp_image047.gif is 0.
+The weight poles are *m*+1 3D points *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:
-@image html brep_wp_image094.gif, @image html brep_wp_image095.gif
+@f[ C(u) = \frac{\sum_{i=0}^{m}B_{i} \cdot h_{i} \cdot C_{m}^{i} \cdot u^{i} \cdot (1-u)^{m-i}}{\sum_{i=0}^{m}h_{i} \cdot C_{m}^{i} \cdot u^{i} \cdot (1-u)^{m-i}},\;u \in [0,\; 1] @f]
-where @image html brep_wp_image096.gif.
+where @f$ 0^{0} \equiv 1 @f$.
-The example record is interpreted as a Bezier curve with a rational flag @image html brep_wp_image097.gif, degree @image html brep_wp_image098.gif and weight poles @image html brep_wp_image099.gif, @image html brep_wp_image100.gif, @image html brep_wp_image101.gif, @image html brep_wp_image102.gif and @image html brep_wp_image103.gif, @image html brep_wp_image104.gif. The Bezier curve is defined by the following parametric equation:
+The example record is interpreted as a Bezier curve with a rational flag *r*=1, degree *m*=2 and weight poles *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:
-@image html brep_wp_image105.gif.
+@f[ C(u)=\frac{(0,1,0) \cdot 4 \cdot (1-u)^{2}+(1,-2,0) \cdot 5 \cdot 2 \cdot u \cdot (1-u) + (2,3,0) \cdot 6 \cdot u^{2} )}{4 \cdot (1-u)^{2}+5 \cdot 2 \cdot u \cdot (1-u)+6 \cdot u^{2}}. @f]
-@subsubsection occt_brep_format_4_1_7 3D curve record 7 – B-spline Curve
+@subsubsection occt_brep_format_4_1_7 B-spline Curve - \<3D curve record 7\>
**Example**
**BNF-like Definition**
@verbatim
- 3D curve record 7 = "7" _ 3D B-spline rational flag _ "0" _ 3D B-spline degree _ 3D B-spline pole count _ 3D B-spline multiplicity knot count 3D B-spline weight poles _\n 3D B-spline multiplicity knots _\n;
-
- 3D B-spline rational flag = flag;
-
- 3D B-spline degree = int;
-
- 3D B-spline pole count = int;
-
- 3D B-spline multiplicity knot count = int;
-
- 3D B-spline weight poles = (_ 3D B-spline weight pole) ^ 3D B-spline pole count;
-
- 3D B-spline weight pole = 3D point [_ real];
-
- 3D B-spline multiplicity knots =
- (_ 3D B-spline multiplicity knot) ^ 3D B-spline multiplicity knot count;
-
- 3D B-spline multiplicity knot = real _ int;
+ <3D curve record 7> = "7" <_> <3D B-spline rational flag> <_> "0" <_> <3D B-spline degree> <_>
+ <3D B-spline pole count> <_> <3D B-spline multiplicity knot count> <3D B-spline weight poles>
+ <_\n> <3D B-spline multiplicity knots> <_\n>;
+
+ <3D B-spline rational flag> = <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>;
@endverbatim
**Description**
-3D curve record 7 describes a B-spline curve. The curve data consist of a rational flag @image html brep_wp_image047.gif, a degree @image html brep_wp_image085.gif, pole count @image html brep_wp_image106.gif, multiplicity knot count @image html brep_wp_image107.gif, weight poles and multiplicity knots.
+\<3D curve record 7\> describes a B-spline curve. The curve data consist of a rational flag *r*, a degree @f$ m \leq 25 @f$, pole count @f$ n \geq 2 @f$, multiplicity knot count *k*, weight poles and multiplicity knots.
-The weight poles are @image html brep_wp_image108.gif 3D points @image html brep_wp_image109.gif if the flag @image html brep_wp_image047.gif is 0. The weight poles are @image html brep_wp_image108.gif pairs @image html brep_wp_image110.gif if the flag @image html brep_wp_image047.gif is 1. Here @image html brep_wp_image089.gif is a 3D point and @image html brep_wp_image090.gif is a positive real (@image html brep_wp_image111.gif). @image html brep_wp_image092.gif (@image html brep_wp_image111.gif) if the flag @image html brep_wp_image047.gif is 0.
+The weight poles are *n* 3D points *B<sub>1</sub> ... B<sub>n</sub>* if the flag *r* is 0. The weight poles are *n* pairs *B<sub>1</sub>h<sub>1</sub> ... B<sub>n</sub>h<sub>n</sub>* if the flag *r* is 1. Here *B<sub>i</sub>* is a 3D point and *h<sub>i</sub>* is a positive real @f$ (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 @image html brep_wp_image107.gif pairs @image html brep_wp_image112.gif. Here @image html brep_wp_image113.gif is a knot with a multiplicity @image html brep_wp_image114.gif (@image html brep_wp_image115.gif) so that
+The multiplicity knots are *k* pairs *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
-@image html brep_wp_image116.gif (@image html brep_wp_image117.gif),
-@image html brep_wp_image118.gif, @image html brep_wp_image119.gif, @image html brep_wp_image120.gif (@image html brep_wp_image121.gif), @image html brep_wp_image122.gif.
+@f[ u_{i} < u_{i+1} (1 \leq i \leq k-1),@f]
+@f[ q_{1} \leq m+1,\; q_{k} \leq m+1,\; q_{i} \leq m\; (2 \leq i \leq k-1), \sum_{i=1}^{k}q_{i}=m+n+1. @f]
The B-spline curve is defined by the following parametric equation:
-@image html brep_wp_image123.gif, @image html brep_wp_image124.gif
-
-where functions @image html brep_wp_image125.gif have the following recursion definition by @image html brep_wp_image126.gif
+@f[ C(u) = \frac{\sum_{i=1}^{n}B_{i} \cdot h_{i} \cdot N_{i,m+1}(u)}{\sum_{i=1}^{n}h_{i} \cdot N_{i,m+1}(u)},\;u \in [u_{1},\; u_{k}] @f]
-@image html brep_wp_image127.gif, @image html brep_wp_image128.gif (@image html brep_wp_image129.gif)
+where functions @f$ N_{i,j} @f$ have the following recursion definition by *j*:
+
+@f[ N_{i,1}(u)=\left\{\begin{matrix}
+1\Leftarrow \bar{u}_{i} \leq u \leq \bar{u}_{i+1}\\
+0\Leftarrow u < \bar{u}_{i} \vee \bar{u}_{i+1} \leq u \end{matrix} \right.,\;
+N_{i,j}(u)=\frac{(u-\bar{u}_{i}) \cdot N_{i,j-1}(u) }{\bar{u}_{i+j-1}-\bar{u}_{i}}+ \frac{(\bar{u}_{i+j}-u) \cdot N_{i+1,j-1}(u)}{\bar{u}_{i+j}-\bar{u}_{i+1}},\;(2 \leq j \leq m+1) @f]
-where
+where
-@image html brep_wp_image130.gif (@image html brep_wp_image131.gif, @image html brep_wp_image132.gif).
+@f[ \bar{u}_{i} = u_{j},\; (1 \leq j \leq k,\; \sum_{l=1}^{j-1}q_{l}+1 \leq i \leq \sum_{l=1}^{j}q_{l} ). @f]
-The example record is interpreted as a B-spline curve with a rational flag @image html brep_wp_image097.gif, a degree @image html brep_wp_image133.gif, pole count @image html brep_wp_image134.gif, multiplicity knot count @image html brep_wp_image135.gif, weight poles @image html brep_wp_image136.gif, @image html brep_wp_image137.gif, @image html brep_wp_image138.gif, @image html brep_wp_image139.gif and @image html brep_wp_image140.gif, @image html brep_wp_image141.gif, multiplicity knots @image html brep_wp_image142.gif, @image html brep_wp_image143.gif, @image html brep_wp_image144.gif, @image html brep_wp_image145.gif, @image html brep_wp_image146.gif, @image html brep_wp_image147.gif, @image html brep_wp_image148.gif, @image html brep_wp_image149.gif and @image html brep_wp_image150.gif, @image html brep_wp_image151.gif. The B-spline curve is defined by the following parametric equation:
+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:
-@image html brep_wp_image152.gif.
+@f[ C(u)=\frac{(0,1,0) \cdot 4 \cdot N_{1,2}(u) + (1,-2,0) \cdot 5 \cdot N_{2,2}(u)+(2,3,0) \cdot 6 \cdot N_{3,2}(u)}{4 \cdot N_{1,2}(u)+5 \cdot N_{2,2}(u)+6 \cdot N_{3,2}(u)}. @f]
-@subsubsection occt_brep_format_4_1_8 3D curve record 8 – Trimmed Curve
+@subsubsection occt_brep_format_4_1_8 Trimmed Curve - \<3D curve record 8\>
**Example**
**BNF-like Definition**
@verbatim
- 3D curve record 8 = "8" _ 3D trimmed curve u min _ 3D trimmed curve u max _\n 3D curve record;
-
- 3D trimmed curve u min = real;
-
- 3D trimmed curve u max = real;
+ <3D curve record 8> = "8" <_> <3D trimmed curve u min> <_> <3D trimmed curve u max> <_\n> <3D curve record>;
+
+ <3D trimmed curve u min> = <real>;
+
+ <3D trimmed curve u max> = <real>;
@endverbatim
**Description**
-3D curve record 8 describes a trimmed curve. The trimmed curve data consist of reals @image html brep_wp_image153.gif and @image html brep_wp_image154.gif and 3D curve record so that @image html brep_wp_image155.gif. The trimmed curve is a restriction of the base curve @image html brep_wp_image156.gif described in the record to the segment @image html brep_wp_image157.gif. The trimmed curve is defined by the following parametric equation:
+\<3D curve record 8\> describes a trimmed curve. The trimmed curve data consist of reals *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$. The trimmed curve is defined by the following parametric equation:
-@image html brep_wp_image158.gif, @image html brep_wp_image159.gif.
+@f[ C(u)=B(u),\; u \in [u_{min},\;u_{max}]. @f]
-The example record is interpreted as a trimmed curve with @image html brep_wp_image160.gif and @image html brep_wp_image161.gif for the base curve @image html brep_wp_image162.gif. The trimmed curve is defined by the following parametric equation: @image html brep_wp_image163.gif, @image html brep_wp_image164.gif.
+The example record is interpreted as a trimmed curve with *u<sub>min</sub>*=-4 and *u<sub>max</sub>*=5 for the base curve @f$ 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$.
-@subsubsection occt_brep_format_4_1_9 3D curve record 9 – Offset Curve
+@subsubsection occt_brep_format_4_1_9 Offset Curve - \<3D curve record 9\>
**Example**
**BNF-like Definition**
@verbatim
- 3D curve record 9 = "9" _ 3D offset curve distance _\n
- 3D offset curve direction _\n
- 3D curve record;
-
- 3D offset curve distance = real;
-
- 3D offset curve direction = 3D direction;
+ <3D curve record 9> = "9" <_> <3D offset curve distance> <_\n>;
+ <3D offset curve direction> <_\n>;
+ <3D curve record>;
+
+ <3D offset curve distance> = <real>;
+
+ <3D offset curve direction> = <3D direction>;
@endverbatim
**Description**
-3D curve record 9 describes an offset curve. The offset curve data consist of a distance @image html brep_wp_image165.gif, a 3D direction @image html brep_wp_image038.gif and a 3D curve record. The offset curve is the result of offsetting the base curve @image html brep_wp_image156.gif described in the record to the distance @image html brep_wp_image165.gif along the vector @image html brep_wp_image166.gif. The offset curve is defined by the following parametric equation:
+\<3D curve record 9\> describes an offset curve. The offset curve data consist of a distance *d*, a 3D direction *D* and a \<3D curve record\>. The offset curve is the result of offsetting the base curve *B* described in the record to the distance *d* along the vector @f$ [B'(u),\; D] \neq \vec{0} @f$. The offset curve is defined by the following parametric equation:
-@image html brep_wp_image167.gif, @image html brep_wp_image168.gif.
+@f[ C(u)=B(u)+d \cdot \frac{[B'(u),\; D]}{|[B'(u),\; D]|},\; u \in domain(B) . @f]
-The example record is interpreted as an offset curve with a distance @image html brep_wp_image169.gif, direction @image html brep_wp_image170.gif, base curve @image html brep_wp_image162.gif and defined by the following parametric equation: @image html brep_wp_image171.gif.
+The example record is interpreted as an offset curve with a distance *d*=2, direction *D*=(0, 1, 0), base curve @f$ B(u)=(1,2,3)+u \cdot (1,0,0) @f$ and defined by the following parametric equation: @f$ C(u)=(1,2,3)+u \cdot (1,0,0)+2 \cdot (0,0,1) @f$.
-@subsection occt_brep_format_4_2 Subsection surfaces
+@subsection occt_brep_format_4_2 Surfaces
**Example**
**BNF-like Definition**
@verbatim
- surfaces = surface header _\n surface records;
-
- surface header = "Surfaces" _ surface count;
-
- surface records = surface record ^ surface count;
-
- surface record =
- surface record 1 |
- surface record 2 |
- surface record 3 |
- surface record 4 |
- surface record 5 |
- surface record 6 |
- surface record 7 |
- surface record 8 |
- surface record 9 |
- surface record 10 |
- surface record 11;
-@verbatim
-
-@subsubsection occt_brep_format_4_2_1 surface record 1 – Plane
+ <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 occt_brep_format_4_2_1 Plane - \< surface record 1 \>
+
**Example**
@verbatim
**BNF-like Definition**
@verbatim
-surface record 1 = "1" _ 3D point (_ 3D direction) ^ 3 _\n;
+ <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 @image html brep_wp_image037.gif and pairwise orthogonal 3D directions @image html brep_wp_image044.gif, @image html brep_wp_image172.gif and @image html brep_wp_image173.gif. The plane passes through the point @image html brep_wp_image037.gif, has the normal @image html brep_wp_image174.gif and is defined by the following parametric equation:
+\<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:
-@image html brep_wp_image175.gif, @image html brep_wp_image176.gif.
+@f[ S(u,v)=P+u \cdot D_{u}+v \cdot D_{v},\; (u,\;v) \in (-\infty,\; \infty) \times (-\infty,\; \infty). @f]
-The example record is interpreted as a plane which passes through a point @image html brep_wp_image177.gif, has a normal @image html brep_wp_image074.gif and is defined by the following parametric equation: @image html brep_wp_image178.gif.
+The example record is interpreted as a plane which passes through a point *P*=(0, 0, 3), has a normal *N*=(0, 0, 1) and is defined by the following parametric equation: @f$ S(u,v)=(0,0,3)+u \cdot (1,0,0) + v \cdot (0,1,0) @f$.
-@subsubsection occt_brep_format_4_2_2 surface record 2 – Cylinder
+@subsubsection occt_brep_format_4_2_2 Cylinder - \< surface record 2 \>
**Example**
**BNF-like Definition**
@verbatim
- surface record 2 = "2" _ 3D point (_ 3D direction) ^ 3 _ real _\n;
+ <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 @image html brep_wp_image037.gif, pairwise orthogonal 3D directions @image html brep_wp_image173.gif, @image html brep_wp_image045.gif and @image html brep_wp_image046.gif and a non-negative real @image html brep_wp_image047.gif. The cylinder axis passes through the point @image html brep_wp_image037.gif and has the direction @image html brep_wp_image173.gif. The cylinder has the radius @image html brep_wp_image179.gif and is defined by the following parametric equation:
+\<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:
-@image html brep_wp_image180.gif, @image html brep_wp_image181.gif.
+@f[ S(u,v)=P+r \cdot (cos(u) \cdot D_{x}+sin(u) \cdot D_{y} )+v \cdot D_{v},\; (u,v) \in [0,\; 2 \cdot \pi) \times (-\infty,\; \infty) . @f]
-The example record is interpreted as a cylinder which axis passes through a point @image html brep_wp_image073.gif and has a direction @image html brep_wp_image182.gif. Directions for the cylinder are @image html brep_wp_image075.gif and @image html brep_wp_image076.gif. The cylinder has a radius @image html brep_wp_image054.gif and is defined by the following parametric equation: @image html brep_wp_image183.gif.
+The example record is interpreted as a cylinder which axis passes through a point *P*=(1, 2, 3) and has a direction *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 occt_brep_format_4_2_3 surface record 3 – Cone
+@subsubsection occt_brep_format_4_2_3 Cone - \< surface record 3 \>
**Example**
**BNF-like Definition**
@verbatim
- surface record 3 = "3" _ 3D point (_ 3D direction) ^ 3 (_ real) ^ 2 _\n;
+ <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 @image html brep_wp_image037.gif, pairwise orthogonal 3D directions @image html brep_wp_image184.gif, @image html brep_wp_image045.gif and @image html brep_wp_image046.gif, a non-negative real @image html brep_wp_image047.gif and a real @image html brep_wp_image185.gif. The cone axis passes through the point @image html brep_wp_image037.gif and has the direction @image html brep_wp_image184.gif. The plane which passes through the point @image html brep_wp_image037.gif and is parallel to directions @image html brep_wp_image045.gif and @image html brep_wp_image046.gif is the cone referenced plane. The cone section by the plane is a circle with the radius @image html brep_wp_image047.gif. The direction from the point @image html brep_wp_image037.gif to the cone apex is @image html brep_wp_image186.gif. The cone has a half-angle @image html brep_wp_image187.gif and is defined by the following parametric equation:
+\<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:
-@image html brep_wp_image188.gif, @image html brep_wp_image189.gif.
+@f[ S(u,v)=P+(r+v \cdot sin(\varphi)) \cdot (cos(u) \cdot D_{X}+sin(u) \cdot D_{Y})+v \cdot cos(\varphi) \cdot D_{Z}, (u,v) \in [0,\; 2 \cdot \pi) \times (-\infty,\; \infty) . @f]
-The example record is interpreted as a cone with an axis which passes through a point @image html brep_wp_image073.gif and has a direction @image html brep_wp_image190.gif. Other cone data are @image html brep_wp_image075.gif, @image html brep_wp_image076.gif, @image html brep_wp_image191.gif and @image html brep_wp_image192.gif. The cone is defined by the following parametric equation:
-@image html brep_wp_image193.gif.
+The example record is interpreted as a cone with an axis which passes through a point *P*=(1, 2, 3) and has a direction *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 occt_brep_format_4_2_4 surface record 4 – Sphere
+@subsubsection occt_brep_format_4_2_4 Sphere - \< surface record 4 \>
**Example**
**BNF-like Definition**
@verbatim
- surface record 4 = "4" _ 3D point (_ 3D direction) ^ 3 _ real _\n;
+ <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 @image html brep_wp_image037.gif, pairwise orthogonal 3D directions @image html brep_wp_image184.gif, @image html brep_wp_image045.gif and @image html brep_wp_image046.gif and a non-negative real @image html brep_wp_image047.gif. The sphere has the center @image html brep_wp_image194.gif, radius @image html brep_wp_image179.gif and is defined by the following parametric equation:
+\<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:
-@image html brep_wp_image195.gif, @image html brep_wp_image196.gif.
+@f[ S(u,v)=P+r \cdot cos(v) \cdot (cos(u) \cdot D_{x}+sin(u) \cdot D_{y} ) +r \cdot sin(v) \cdot D_{Z},\; (u,v) \in [0,\;2 \cdot \pi) \times [-\pi /2,\; \pi /2] . @f]
-The example record is interpreted as a sphere with its center @image html brep_wp_image073.gif. Directions for the sphere are @image html brep_wp_image190.gif, @image html brep_wp_image075.gif and @image html brep_wp_image076.gif. The sphere has a radius @image html brep_wp_image191.gif and is defined by the following parametric equation:
-@image html brep_wp_image197.gif.
+The example record is interpreted as a sphere with its center *P*=(1, 2, 3). Directions for the sphere are *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 occt_brep_format_4_2_5 surface record 5 – Torus
+@subsubsection occt_brep_format_4_2_5 Torus - \< surface record 5 \>
**Example**
**BNF-like Definition**
@verbatim
- surface record 5 = "5" _ 3D point (_ 3D direction) ^ 3 (_ real) ^ 2 _\n;
+ <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 @image html brep_wp_image037.gif, pairwise orthogonal 3D directions @image html brep_wp_image184.gif, @image html brep_wp_image045.gif and @image html brep_wp_image046.gif and non-negative reals @image html brep_wp_image198.gif and @image html brep_wp_image199.gif. The torus axis passes through the point @image html brep_wp_image037.gif and has the direction @image html brep_wp_image184.gif. @image html brep_wp_image198.gif is the distance from the torus circle center to the axis. The torus circle has the radius @image html brep_wp_image199.gif. The torus is defined by the following parametric equation:
+\<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:
-@image html brep_wp_image200.gif, @image html brep_wp_image201.gif.
+@f[ S(u,v)=P+(r_{1}+r_{2} \cdot cos(v)) \cdot (cos(u) \cdot D_{x}+sin(u) \cdot D_{y} ) +r_{2} \cdot sin(v) \cdot D_{Z},\; (u,v) \in [0,\;2 \cdot \pi) \times [0,\; 2 \cdot \pi) . @f]
-The example record is interpreted as a torus with an axis which passes through a point @image html brep_wp_image073.gif and has a direction @image html brep_wp_image190.gif. @image html brep_wp_image075.gif, @image html brep_wp_image076.gif, @image html brep_wp_image202.gif and @image html brep_wp_image203.gif for the torus. The torus is defined by the following parametric equation:
-@image html brep_wp_image204.gif.
+The example record is interpreted as a torus with an axis which passes through a point *P*=(1, 2, 3) and has a direction *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 occt_brep_format_4_2_6 surface record 6 – Linear Extrusion
+@subsubsection occt_brep_format_4_2_6 Linear Extrusion - \< surface record 6 \>
**Example**
**BNF-like Definition**
@verbatim
- surface record 6 = "6" _ 3D direction _\n 3D curve record;
+ <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 @image html brep_wp_image173.gif and a 3D curve record. The linear extrusion surface has the direction @image html brep_wp_image173.gif, the base curve @image html brep_wp_image205.gif described in the record and is defined by the following parametric equation:
+\<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:
-@image html brep_wp_image206.gif, @image html brep_wp_image207.gif.
+@f[ S(u,v)=C(u)+v \cdot D_{v},\; (u,v) \in domain(C) \times (-\infty,\; \infty) . @f]
-The example record is interpreted as a linear extrusion surface with a direction @image html brep_wp_image208.gif. The base curve is a circle for the surface. The surface is defined by the following parametric equation: @image html brep_wp_image209.gif, @image html brep_wp_image189.gif.
+The example record is interpreted as a linear extrusion surface with a direction *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 occt_brep_format_4_2_7 surface record 7 – Revolution Surface
+@subsubsection occt_brep_format_4_2_7 Revolution Surface - \< surface record 7 \>
**Example**
**BNF-like Definition**
@verbatim
- surface record 7 = "7" _ 3D point _ 3D direction _\n 3D curve record;
+ <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 @image html brep_wp_image037.gif, a 3D direction @image html brep_wp_image038.gif and a 3D curve record. The surface axis passes through the point @image html brep_wp_image037.gif and has the direction @image html brep_wp_image038.gif. The base curve @image html brep_wp_image210.gif described by the record and the axis are coplanar. The surface is defined by the following parametric equation:
+\<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:
-@image html brep_wp_image211.gif, @image html brep_wp_image212.gif
+@f[ S(u,v)= P+V_{D}(v)+cos(u) \cdot (V(v)-V_{D}(v))+sin(u) \cdot [D,V(v)],\;(u,v) \in [0,\; 2 \cdot \pi)\times domain(C) @f]
-where @image html brep_wp_image213.gif, @image html brep_wp_image214.gif.
+where @f$ V(v)=C(v)-P, V_{D}(v)=(D,V(v)) \cdot D @f$.
-The example record is interpreted as a revolution surface with an axis which passes through a point @image html brep_wp_image215.gif and has a direction @image html brep_wp_image170.gif. The base curve is a circle for the surface. The surface is defined by the following parametric equation:
-@image html brep_wp_image216.gif, @image html brep_wp_image201.gif where @image html brep_wp_image217.gif, @image html brep_wp_image218.gif.
+The example record is interpreted as a revolution surface with an axis which passes through a point *P*=(-4, 0, 3) and has a direction *D*=(0, 1, 0). The base curve is a circle for the surface. The surface is defined by the following parametric equation:
+
+@f[ S(u,v)= (-4,0,3)+V_{D}(v)+cos(u) \cdot (V(v)-V_{D}(v))+sin(u) \cdot [(0,1,0),V(v)],\;(u,v) \in [0,\; 2 \cdot \pi)\times [0,\; 2 \cdot \pi) @f]
+
+where @f$ V(v)=(5,2,0)+4 \cdot (cos(v) \cdot (1,0,0)+sin(v) \cdot (0,1,0)), V_{D}(v)=((0,1,0),V(v)) \cdot (0,1,0) @f$.
-@subsubsection occt_brep_format_4_2_8 surface record 8 – Bezier Surface
+@subsubsection occt_brep_format_4_2_8 Bezier Surface - \< surface record 8 \>
**Example**
**BNF-like Definition**
@verbatim
- surface record 8 = "8" _ Bezier surface u rational flag _ Bezier surface v rational flag _ Bezier surface u degree _ Bezier surface v degree _ Bezier surface weight poles;
-
- Bezier surface u rational flag = flag;
-
- Bezier surface v rational flag = flag;
-
- Bezier surface u degree = int;
-
- Bezier surface v degree = int;
-
- Bezier surface weight poles =
- (Bezier surface weight pole group _\n) ^ (Bezier surface u degree + "1");
-
- Bezier surface weight pole group = Bezier surface weight pole
- (_ Bezier surface weight pole) ^ Bezier surface v degree;
-
- Bezier surface weight pole = 3D point [_ real];
- @endverbatim
+ <surface record 8> = "8" <_> <Bezier surface u rational flag> <_> <Bezier surface v rational flag> <_> <Bezier surface u degree> <_> <Bezier surface v degree> <_>
+ <Bezier surface weight poles>;
+
+ <Bezier surface u rational flag> = <flag>;
+
+ <Bezier surface v rational flag> = <flag>;
+
+ <Bezier surface u degree> = <int>;
+
+ <Bezier surface v degree> = <int>;
+
+ <Bezier surface weight poles> =
+ (<Bezier surface weight pole group> <_\n>) ^ (<Bezier surface u degree> <+> "1");
+
+ <Bezier surface weight pole group> = <Bezier surface weight pole>
+ (<_> <Bezier surface weight pole>) ^ <Bezier surface v degree>;
+
+ <Bezier surface weight pole> = <3D point> [<_> <real>];
+@endverbatim
**Description**
-surface record 8 describes a Bezier surface. The surface data consist of a u rational flag @image html brep_wp_image219.gif, v rational flag @image html brep_wp_image220.gif, u degree @image html brep_wp_image221.gif, v degree @image html brep_wp_image222.gif and weight poles.
+\<surface record 8\> describes a Bezier surface. The surface data consist of a u rational flag *r<sub>u</sub>*, v rational flag *r<sub>v</sub>*, u degree @f$ m_{u} \leq 25 @f$, v degree @f$ m_{v} \leq 25 @f$ and weight poles.
-The weight poles are @image html brep_wp_image223.gif 3D points @image html brep_wp_image224.gif (@image html brep_wp_image225.gif) if @image html brep_wp_image226.gif. The weight poles are @image html brep_wp_image223.gif pairs @image html brep_wp_image227.gif (@image html brep_wp_image225.gif) if @image html brep_wp_image228.gif. Here @image html brep_wp_image224.gif is a 3D point and @image html brep_wp_image229.gif is a positive real (@image html brep_wp_image225.gif). @image html brep_wp_image230.gif (@image html brep_wp_image225.gif) if @image html brep_wp_image226.gif.
+The weight poles are @f$ (m_{u}+1) \cdot (m_{v}+1) @f$ 3D points @f$ B_{i,j}\; ((i,j) \in \left \{ 0,...,m_{u} \right \} \times \left \{ 0,...,m_{v} \right \}) @f$ if @f$ r_{u}+r_{v}=0 @f$. The weight poles are @f$ (m_{u}+1) \cdot (m_{v}+1) @f$ pairs @f$ B_{i,j}h_{i,j}\; ((i,j) \in \left \{ 0,...,m_{u} \right \} \times \left \{ 0,...,m_{v} \right \}) @f$ if @f$ r_{u}+r_{v} \neq 0 @f$. Here @f$ B_{i,j} @f$ is a 3D point and @f$ h_{i,j} @f$ is a positive real @f$ ((i,j) \in \left \{ 0,...,m_{u} \right \} \times \left \{ 0,...,m_{v} \right \}) @f$. @f$ h_{i,j}=1\; ((i,j) \in \left \{ 0,...,m_{u} \right \} \times \left \{ 0,...,m_{v} \right \}) @f$ if @f$ r_{u}+r_{v} = 0 @f$.
The Bezier surface is defined by the following parametric equation:
-@image html brep_wp_image231.gif, @image html brep_wp_image232.gif
+@f[ S(u,v)=\frac{\sum_{i=0}^{m_{u}} \sum_{j=0}^{m_{v}} B_{i,j} \cdot h_{i,j} \cdot C_{m_{u}}^{i} \cdot u^{i} \cdot (1-u)^{m_{u}-i} \cdot C_{m_{v}}^{j} \cdot v^{j} \cdot (1-v)^{m_{v}-j}}{\sum_{i=0}^{m_{u}} \sum_{j=0}^{m_{v}} h_{i,j} \cdot C_{m_{u}}^{i} \cdot u^{i} \cdot (1-u)^{m_{u}-i} \cdot C_{m_{v}}^{j} \cdot v^{j} \cdot (1-v)^{m_{v}-j}}, (u,v) \in [0,1] \times [0,1] @f]
-where @image html brep_wp_image096.gif.
+where @f$ 0^{0} \equiv 1 @f$.
-The example record is interpreted as a Bezier surface with a u rational flag @image html brep_wp_image233.gif, v rational flag @image html brep_wp_image234.gif, u degree @image html brep_wp_image235.gif, v degree @image html brep_wp_image236.gif, weight poles @image html brep_wp_image237.gif, @image html brep_wp_image238.gif, @image html brep_wp_image239.gif, @image html brep_wp_image240.gif, @image html brep_wp_image241.gif, @image html brep_wp_image242.gif, @image html brep_wp_image243.gif, @image html brep_wp_image244.gif, @image html brep_wp_image245.gif, @image html brep_wp_image246.gif and @image html brep_wp_image247.gif, @image html brep_wp_image248.gif. The surface is defined by the following parametric equation:
+The example record is interpreted as a Bezier surface with a u rational flag *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:
-@image html brep_wp_image249.gif
+@f[ S(u,v)= [ (0,0,1) \cdot 7 \cdot (1-u)^{2} \cdot (1-v)+(1,0,-4) \cdot 10 \cdot (1-u)^{2} \cdot v+ \\
+(0,1,-2) \cdot 8 \cdot 2 \cdot u \cdot (1-u) \cdot (1-v)+(1,1,5) \cdot 11 \cdot 2 \cdot u \cdot (1-u) \cdot v+ \\
+(0,2,3) \cdot 9 \cdot u^{2} \cdot (1-v)+(1,2,6) \cdot 12 \cdot u^{2} \cdot v] \div \\
+[7 \cdot (1-u)^{2} \cdot (1-v)+10 \cdot (1-u)^{2} \cdot v+ \\
+8 \cdot 2 \cdot u \cdot (1-u) \cdot (1-v)+11 \cdot 2 \cdot u \cdot (1-u) \cdot v+ \\
+9 \cdot u^{2} \cdot (1-v)+12 \cdot u^{2} \cdot v ] @f]
-@subsubsection occt_brep_format_4_2_9 surface record 9 – B-spline Surface
+@subsubsection occt_brep_format_4_2_9 B-spline Surface - \< surface record 9 \>
**Example**
**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;
+ <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 @image html brep_wp_image219.gif, v rational flag @image html brep_wp_image220.gif, u degree @image html brep_wp_image221.gif, v degree @image html brep_wp_image222.gif, u pole count @image html brep_wp_image250.gif, v pole count @image html brep_wp_image251.gif, u multiplicity knot count @image html brep_wp_image252.gif, v multiplicity knot count @image html brep_wp_image253.gif, weight poles, u multiplicity knots, v multiplicity knots.
+\<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$, u multiplicity knot count *k<sub>u</sub>*, v multiplicity knot count *k<sub>v</sub>*, weight poles, u multiplicity knots, v multiplicity knots.
-The weight poles are @image html brep_wp_image254.gif 3D points @image html brep_wp_image224.gif (@image html brep_wp_image255.gif) if @image html brep_wp_image226.gif. The weight poles are @image html brep_wp_image254.gif pairs @image html brep_wp_image227.gif (@image html brep_wp_image255.gif) if @image html brep_wp_image228.gif. Here @image html brep_wp_image224.gif is a 3D point and @image html brep_wp_image229.gif is a positive real (@image html brep_wp_image255.gif). @image html brep_wp_image230.gif (@image html brep_wp_image255.gif) if @image html brep_wp_image226.gif.
+The weight poles are @f$ n_{u} \cdot n_{v} @f$ 3D points @f$ B_{i,j}\; ((i,j) \in \left \{ 1,...,n_{u} \right \} \times \left \{ 1,...,n_{v} \right \}) @f$ if @f$ r_{u}+r_{v}=0 @f$. The weight poles are @f$ n_{u} \cdot n_{v} @f$ pairs @f$ B_{i,j}h_{i,j}\; ((i,j) \in \left \{ 1,...,n_{u} \right \} \times \left \{ 1,...,n_{v} \right \}) @f$ if @f$ r_{u}+r_{v} \neq 0 @f$. Here @f$ B_{i,j} @f$ is a 3D point and @f$ h_{i,j} @f$ is a positive real @f$ ((i,j) \in \left \{ 1,...,n_{u} \right \} \times \left \{ 1,...,n_{v} \right \}) @f$. @f$ h_{i,j}=1\; ((i,j) \in \left \{ 1,...,n_{u} \right \} \times \left \{ 1,...,n_{v} \right \}) @f$ if @f$ r_{u}+r_{v} = 0 @f$.
-The u multiplicity knots are @image html brep_wp_image252.gif pairs @image html brep_wp_image256.gif. Here @image html brep_wp_image113.gif is a knot with multiplicity @image html brep_wp_image114.gif (@image html brep_wp_image257.gif) so that
+The u multiplicity knots are *k<sub>u</sub>* pairs @f$ 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
-@image html brep_wp_image116.gif (@image html brep_wp_image258.gif),
-@image html brep_wp_image259.gif, @image html brep_wp_image260.gif, @image html brep_wp_image261.gif (@image html brep_wp_image262.gif), @image html brep_wp_image263.gif.
+@f[ u_{i} < u_{i+1} \; (1\leq i\leq k_{u}-1), \\
+q_{1} \leq m_{u}+1,\; q_{k_{u}} \leq m_{u}+1,\; q_{i} \leq m_{u}\; (2\leq i\leq k_{u}-1),\; \sum_{i=1}^{k_{u}}q_{i}=m_{u}+n_{u}+1. @f]
-The v multiplicity knots are @image html brep_wp_image253.gif pairs @image html brep_wp_image264.gif. Here @image html brep_wp_image265.gif is a knot with multiplicity @image html brep_wp_image266.gif (@image html brep_wp_image267.gif) so that
+The v multiplicity knots are *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
-@image html brep_wp_image268.gif (@image html brep_wp_image269.gif),
-@image html brep_wp_image270.gif, @image html brep_wp_image271.gif, @image html brep_wp_image272.gif (@image html brep_wp_image273.gif), @image html brep_wp_image274.gif.
+@f[ v_{j} < v_{j+1} \; (1\leq j\leq k_{v}-1), \\
+t_{1} \leq m_{v}+1,\; t_{k_{v}} \leq m_{v}+1,\; t_{j} \leq m_{v}\; (2\leq j\leq k_{v}-1),\; \sum_{j=1}^{k_{v}}t_{j}=m_{v}+n_{v}+1. @f]
The B-spline surface is defined by the following parametric equation:
-@image html brep_wp_image275.gif, @image html brep_wp_image276.gif
-
-where functions @image html brep_wp_image277.gif and @image html brep_wp_image278.gif have the following recursion definition by @image html brep_wp_image279.gif
-
-@image html brep_wp_image127.gif, @image html brep_wp_image128.gif (@image html brep_wp_image280.gif);
-
-@image html brep_wp_image281.gif, @image html brep_wp_image282.gif (@image html brep_wp_image283.gif);
-where
+@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]
-@image html brep_wp_image284.gif (@image html brep_wp_image285.gif,@image html brep_wp_image286.gif),
+where functions *N<sub>i,j</sub>* and *M<sub>i,j</sub>* have the following recursion definition by *j*:
-@image html brep_wp_image287.gif (@image html brep_wp_image288.gif,@image html brep_wp_image289.gif).
+@f[ N_{i,1}(u)=\left\{\begin{matrix}
+1\Leftarrow \bar{u}_{i} \leq u \leq \bar{u}_{i+1}\\
+0\Leftarrow u < \bar{u}_{i} \vee \bar{u}_{i+1} \leq u \end{matrix} \right.,\;
+N_{i,j}(u)=\frac{(u-\bar{u}_{i}) \cdot N_{i,j-1}(u) }{\bar{u}_{i+j-1}-\bar{u}_{i}}+ \frac{(\bar{u}_{i+j}-u) \cdot N_{i+1,j-1}(u)}{\bar{u}_{i+j}-\bar{u}_{i+1}},\;(2 \leq j \leq m_{u}+1); \\
+M_{i,1}(v)=\left\{\begin{matrix}
+1\Leftarrow \bar{v}_{i} \leq v \leq \bar{v}_{i+1}\\
+0\Leftarrow v < \bar{v}_{i} \vee \bar{v}_{i+1} \leq v \end{matrix} \right.,\;
+M_{i,j}(v)=\frac{(v-\bar{v}_{i}) \cdot M_{i,j-1}(v) }{\bar{v}_{i+j-1}-\bar{v}_{i}}+ \frac{(\bar{v}_{i+j}-v) \cdot M_{i+1,j-1}(v)}{\bar{v}_{i+j}-\bar{v}_{i+1}},\;(2 \leq j \leq m_{v}+1); @f]
+
+where
+@f[ \bar{u}_{i}=u_{j}\; (1 \leq j \leq k_{u},\; \sum_{l=1}^{j-1}q_{l} \leq i \leq \sum_{l=1}^{j}q_{l}), \\
+ \bar{v}_{i}=v_{j}\; (1 \leq j \leq k_{v},\; \sum_{l=1}^{j-1}t_{l} \leq i \leq \sum_{l=1}^{j}t_{l}); @f]
-The example record is interpreted as a B-spline surface with a u rational flag @image html brep_wp_image233.gif, v rational flag @image html brep_wp_image234.gif, u degree @image html brep_wp_image290.gif, v degree @image html brep_wp_image236.gif, u pole count @image html brep_wp_image291.gif, v pole count @image html brep_wp_image292.gif, u multiplicity knot count @image html brep_wp_image293.gif, v multiplicity knot count @image html brep_wp_image294.gif, weight poles @image html brep_wp_image295.gif, @image html brep_wp_image296.gif, @image html brep_wp_image297.gif, @image html brep_wp_image298.gif, @image html brep_wp_image299.gif, @image html brep_wp_image300.gif, @image html brep_wp_image301.gif, @image html brep_wp_image302.gif, @image html brep_wp_image303.gif, @image html brep_wp_image304.gif and @image html brep_wp_image305.gif, @image html brep_wp_image306.gif, u multiplicity knots @image html brep_wp_image142.gif, @image html brep_wp_image143.gif, @image html brep_wp_image144.gif, @image html brep_wp_image145.gif, @image html brep_wp_image146.gif, @image html brep_wp_image147.gif, @image html brep_wp_image148.gif, @image html brep_wp_image149.gif and @image html brep_wp_image150.gif, @image html brep_wp_image151.gif, v multiplicity knots @image html brep_wp_image307.gif, @image html brep_wp_image308.gif, @image html brep_wp_image309.gif, @image html brep_wp_image310.gif, @image html brep_wp_image311.gif, @image html brep_wp_image312.gif and @image html brep_wp_image313.gif, @image html brep_wp_image314.gif. The B-spline surface is defined by the following parametric equation:
+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:
-@image html brep_wp_image315.gif
+@f[ S(u,v)= [ (0,0,1) \cdot 7 \cdot N_{1,2}(u) \cdot M_{1,2}(v)+(1,0,-4) \cdot 10 \cdot N_{1,2}(u) \cdot M_{2,2}(v)+ \\
+(0,1,-2) \cdot 8 \cdot N_{2,2}(u) \cdot M_{1,2}(v)+(1,1,5) \cdot 11 \cdot N_{2,2}(u) \cdot M_{2,2}(v)+ \\
+(0,2,3) \cdot 9 \cdot N_{3,2}(u) \cdot M_{1,2}(v)+(1,2,6) \cdot 12 \cdot N_{3,2}(u) \cdot M_{2,2}(v)] \div \\
+[7 \cdot N_{1,2}(u) \cdot M_{1,2}(v)+10 \cdot N_{1,2}(u) \cdot M_{2,2}(v)+ \\
+8 \cdot N_{2,2}(u) \cdot M_{1,2}(v)+11 \cdot N_{2,2}(u) \cdot M_{2,2}(v)+ \\
+9 \cdot N_{3,2}(u) \cdot M_{1,2}(v)+12 \cdot N_{3,2}(u) \cdot M_{2,2}(v) ] @f]
-@subsubsection occt_brep_format_4_2_10 surface record 10 – Rectangular Trim Surface
+@subsubsection occt_brep_format_4_2_10 Rectangular Trim Surface - \< surface record 10 \>
**Example**
**BNF-like Definition**
@verbatim
- surface record 10 = "10" _ trim surface u min _ trim surface u max _ trim surface v min _ trim surface v max _\n surface record;
-
- trim surface u min = real;
-
- trim surface u max = real;
-
- trim surface v min = real;
-
- trim surface v max = real;
-@verbatim
+ <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 @image html brep_wp_image153.gif, @image html brep_wp_image154.gif, @image html brep_wp_image316.gif and @image html brep_wp_image317.gif and a surface record so that @image html brep_wp_image155.gif and @image html brep_wp_image318.gif. The rectangular trim surface is a restriction of the base surface @image html brep_wp_image156.gif described in the record to the set @image html brep_wp_image319.gif. The rectangular trim surface is defined by the following parametric equation:
+\<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$. The rectangular trim surface is defined by the following parametric equation:
-@image html brep_wp_image320.gif, @image html brep_wp_image321.gif.
+@f[ S(u,v)=B(u,v),\; (u,v) \in [u_{min},u_{max}] \times [v_{min},v_{max}] . @f]
-The example record is interpreted as a rectangular trim surface to the set @image html brep_wp_image322.gif for the base surface @image html brep_wp_image323.gif. The rectangular trim surface is defined by the following parametric equation: @image html brep_wp_image324.gif, @image html brep_wp_image325.gif.
+The example record is interpreted as a rectangular trim surface to the set [-1, 2]x[-3, 4] for the base surface @f$ B(u,v)=(1,2,3)+u \cdot (1,0,0)+v \cdot (0,1,0) @f$. The rectangular trim surface is defined by the following parametric equation: @f$ B(u,v)=(1,2,3)+u \cdot (1,0,0)+ v \cdot (0,1,0),\; (u,v) \in [-1,2] \times [-3,4] @f$.
-@subsubsection occt_brep_format_4_2_11 surface record 11 – Offset Surface
+@subsubsection occt_brep_format_4_2_11 Offset Surface - \< surface record 11 \>
**Example**
@verbatim
11 -2
1 1 2 3 0 0 1 1 0 -0 -0 1 0
-@verbatim
+@endverbatim
**BNF-like Definition**
@verbatim
- surface record 11 = "11" _ surface record distance _\n surface record;
-
- surface record distance = real;
-@verbatim
+ <surface record 11> = "11" <_> <surface record distance> <_\n> <surface record>;
+
+ <surface record distance> = <real>;
+@endverbatim
**Description**
-surface record 11 describes an offset surface.
-The offset surface data consist of a distance @image html brep_wp_image165.gif and a surface record. The offset surface is the result of offsetting the base surface @image html brep_wp_image326.gif described in the record to the distance @image html brep_wp_image165.gif along the normal @image html brep_wp_image044.gif of surface @image html brep_wp_image156.gif. The offset surface is defined by the following parametric equation:
-
-@image html brep_wp_image327.gif, @image html brep_wp_image328.gif.
+\<surface record 11\> describes an offset surface.
+The offset surface data consist of a distance *d* and a \<surface record\>. The offset surface is the result of offsetting the base surface *B* described in the record to the distance *d* along the normal *N* of surface *B*. The offset surface is defined by the following parametric equation:
-@image html brep_wp_image329.gif
-if @image html brep_wp_image330.gif.
+@f[ S(u,v)=B(u,v)+d \cdot N(u,v),\; (u,v) \in domain(B) . \\
+N(u,v) = [S'_{u}(u,v),S'_{v}(u,v)] @f]
+if @f$ [S'_{u}(u,v),S'_{v}(u,v)] \neq \vec{0} @f$.
-The example record is interpreted as an offset surface with a distance @image html brep_wp_image331.gif and base surface @image html brep_wp_image323.gif. The offset surface is defined by the following parametric equation: @image html brep_wp_image332.gif.
+The example record is interpreted as an offset surface with a distance *d*=-2 and base surface @f$ B(u,v)=(1,2,3)+u \cdot (1,0,0)+v \cdot (0,1,0) @f$. The offset surface is defined by the following parametric equation: @f$ S(u,v)=(1,2,3)+u \cdot (1,0,0)+v \cdot (0,1,0)-2 \cdot (0,0,1) @f$.
-@subsection occt_brep_format_4_3 Subsection 2D curves
+@subsection occt_brep_format_4_3 2D curves
**Example**
**BNF-like Definition**
@verbatim
- 2D curves = 2D curve header _\n 2D curve records;
-
- 2D curve header = "Curve2ds" _ 2D curve count;
-
- 2D curve count = int;
-
- 2D curve records = 2D curve record ^ 2D curve count;
-
- 2D curve record =
- 2D curve record 1 |
- 2D curve record 2 |
- 2D curve record 3 |
- 2D curve record 4 |
- 2D curve record 5 |
- 2D curve record 6 |
- 2D curve record 7 |
- 2D curve record 8 |
- 2D curve record 9;
+ <2D curves> = <2D curve header> <_\n> <2D curve records>;
+
+ <2D curve header> = "Curve2ds" <_> <2D curve count>;
+
+ <2D curve count> = <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 occt_brep_format_4_3_1 2D curve record 1 – Line
+@subsubsection occt_brep_format_4_3_1 Line - \<2D curve record 1\>
**Example**
**BNF-like Definition**
@verbatim
- 2D curve record 1 = "1" _ 2D point _ 2D direction _\n;
+ <2D curve record 1> = "1" <_> <2D point> <_> <2D direction> <_\n>;
@endverbatim
**Description**
-2D curve record 1 describes a line. The line data consist of a 2D point @image html brep_wp_image037.gif and a 2D direction @image html brep_wp_image038.gif. The line passes through the point @image html brep_wp_image037.gif, has the direction @image html brep_wp_image038.gif and is defined by the following parametric equation:
+\<2D curve record 1\> describes a line. The line data consist of a 2D point *P* and a 2D direction *D*. The line passes through the point *P*, has the direction *D* and is defined by the following parametric equation:
-@image html brep_wp_image333.gif, @image html brep_wp_image069.gif.
+@f[ C(u)=P+u \cdot D, \; u \in (-\infty,\; \infty). @f]
-The example record is interpreted as a line which passes through a point@image html brep_wp_image334.gif, has a direction @image html brep_wp_image335.gif and is defined by the following parametric equation: @image html brep_wp_image336.gif.
+The example record is interpreted as a line which passes through a point *P*=(3,0), has a direction *D*=(0,-1) and is defined by the following parametric equation: @f$ C(u)=(3,0)+ u \cdot (0,-1) @f$.
-@subsubsection occt_brep_format_4_3_2 2D curve record 2 – Circle
+@subsubsection occt_brep_format_4_3_2 Circle - \<2D curve record 2\>
**Example**
**BNF-like Definition**
@verbatim
- 2D curve record 2 = "2" _ 2D circle center _ 2D circle Dx _ 2D circle Dy _ 2D circle radius _\n;
-
- 2D circle center = 2D point;
-
- 2D circle Dx = 2D direction;
-
- 2D circle Dy = 2D direction;
-
- 2D circle radius = real;
+ <2D curve record 2> = "2" <_> <2D circle center> <_> <2D circle Dx> <_> <2D circle Dy> <_> <2D circle radius> <_\n>;
+
+ <2D circle center> = <2D point>;
+
+ <2D circle Dx> = <2D direction>;
+
+ <2D circle Dy> = <2D direction>;
+
+ <2D circle radius> = <real>;
@endverbatim
**Description**
-2D curve record 2 describes a circle. The circle data consist of a 2D point @image html brep_wp_image037.gif, orthogonal 2D directions @image html brep_wp_image045.gif and @image html brep_wp_image046.gif and a non-negative real @image html brep_wp_image047.gif. The circle has a center @image html brep_wp_image037.gif. The circle plane is parallel to directions @image html brep_wp_image045.gif and @image html brep_wp_image046.gif. The circle has a radius @image html brep_wp_image047.gif and is defined by the following parametric equation:
+\<2D curve record 2\> describes a circle. The circle data consist of a 2D point *P*, orthogonal 2D directions *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:
-@image html brep_wp_image337.gif, @image html brep_wp_image338.gif.
+@f[ C(u)=P+r \cdot (cos(u) \cdot D_{x} + sin(u) \cdot D_{y}),\; u \in [0,\; 2 \cdot \pi) . @f]
-The example record is interpreted as a circle which has a center @image html brep_wp_image339.gif. The circle plane is parallel to directions @image html brep_wp_image340.gif and @image html brep_wp_image341.gif. The circle has a radius @image html brep_wp_image342.gif and is defined by the following parametric equation: @image html brep_wp_image343.gif.
+The example record is interpreted as a circle which has a center *P*=(1,2). The circle plane is parallel to directions *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$.
-@subsubsection occt_brep_format_4_3_3 2D curve record 3 – Ellipse
+@subsubsection occt_brep_format_4_3_3 Ellipse - \<2D curve record 3\>
**Example**
**BNF-like Definition**
@verbatim
- 2D curve record 3 = "3" _ 2D ellipse center _ 2D ellipse Dmaj _ 2D ellipse Dmin _ 2D ellipse Rmaj _ 2D ellipse Rmin _\n;
-
- 2D ellipse center = 2D point;
-
- 2D ellipse Dmaj = 2D direction;
-
- 2D ellipse Dmin = 2D direction;
-
- 2D ellipse Rmaj = real;
-
- 2D ellipse Rmin = real;
+ <2D curve record 3> = "3" <_> <2D ellipse center> <_> <2D ellipse Dmaj> <_>
+ <2D ellipse Dmin> <_> <2D ellipse Rmaj> <_> <2D ellipse Rmin> <_\n>;
+
+ <2D ellipse center> = <2D point>;
+
+ <2D ellipse Dmaj> = <2D direction>;
+
+ <2D ellipse Dmin> = <2D direction>;
+
+ <2D ellipse Rmaj> = <real>;
+
+ <2D ellipse Rmin> = <real>;
@endverbatim
**Description**
-2D curve record 3 describes an ellipse. The ellipse data are 2D point @image html brep_wp_image037.gif, orthogonal 2D directions @image html brep_wp_image056.gif and @image html brep_wp_image057.gif and non-negative reals @image html brep_wp_image344.gif and @image html brep_wp_image345.gif that @image html brep_wp_image346.gif. The ellipse has a center @image html brep_wp_image037.gif, major and minor axis directions @image html brep_wp_image056.gif and @image html brep_wp_image057.gif, major and minor radii @image html brep_wp_image058.gif and @image html brep_wp_image345.gif and is defined by the following parametric equation:
+\<2D curve record 3\> describes an ellipse. The ellipse data are 2D point *P*, orthogonal 2D directions *D<sub>maj</sub>* and *D<sub>min</sub>* and non-negative reals *r<sub>maj</sub>* and *r<sub>min</sub>* that *r<sub>maj</sub>* @f$ \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:
-@image html brep_wp_image347.gif, @image html brep_wp_image338.gif.
+@f[ C(u)=P+r_{maj} \cdot cos(u) \cdot D_{maj}+r_{min} \cdot sin(u) \cdot D_{min},\; u \in [0,\; 2 \cdot \pi) . @f]
-The example record is interpreted as an ellipse which has a center @image html brep_wp_image339.gif, major and minor axis directions @image html brep_wp_image348.gif and @image html brep_wp_image349.gif, major and minor radii @image html brep_wp_image350.gif and @image html brep_wp_image351.gif and is defined by the following parametric equation: @image html brep_wp_image352.gif.
+The example record is interpreted as an ellipse which has a center *P*=(1,2), major and minor axis directions *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 occt_brep_format_4_3_4 2D curve record 4 – Parabola
+@subsubsection occt_brep_format_4_3_4 Parabola - \<2D curve record 4\>
**Example**
**BNF-like Definition**
@verbatim
- 2D curve record 4 = "4" _ 2D parabola origin _ 2D parabola Dx _ 2D parabola Dy _ 2D parabola focal length _\n;
-
- 2D parabola origin = 2D point;
-
- 2D parabola Dx = 2D direction;
-
- 2D parabola Dy = 2D direction;
-
- 2D parabola focal length = real;
+ <2D curve record 4> = "4" <_> <2D parabola origin> <_> <2D parabola Dx> <_>
+ <2D parabola Dy> <_> <2D parabola focal length> <_\n>;
+
+ <2D parabola origin> = <2D point>;
+
+ <2D parabola Dx> = <2D direction>;
+
+ <2D parabola Dy> = <2D direction>;
+
+ <2D parabola focal length> = <real>;
@endverbatim
**Description**
-2D curve record 4 describes a parabola. The parabola data consist of a 2D point @image html brep_wp_image037.gif, orthogonal 2D directions @image html brep_wp_image045.gif and @image html brep_wp_image046.gif and a non-negative real @image html brep_wp_image067.gif. The parabola coordinate system has its origin @image html brep_wp_image037.gif and axis directions @image html brep_wp_image045.gif and @image html brep_wp_image046.gif. The parabola has a focus length @image html brep_wp_image067.gif and is defined by the following parametric equation:
-
-@image html brep_wp_image068.gif, @image html brep_wp_image069.gif Ü @image html brep_wp_image070.gif;
-
-@image html brep_wp_image071.gif, @image html brep_wp_image069.gif Ü @image html brep_wp_image072.gif (degenerated case).
+\<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:
-The example record is interpreted as a parabola in plane which passes through a point @image html brep_wp_image339.gif and is parallel to directions @image html brep_wp_image340.gif and @image html brep_wp_image341.gif. The parabola has a focus length @image html brep_wp_image077.gif and is defined by the following parametric equation: @image html brep_wp_image353.gif.
+@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 occt_brep_format_4_3_5 2D curve record 5 – Hyperbola
+@subsubsection occt_brep_format_4_3_5 Hyperbola - \<2D curve record 5\>
**Example**
5 1 2 1 0 -0 1 3 4
**BNF-like Definition**
@verbatim
- 2D curve record 5 = "5" _ 2D hyperbola origin _ 2D hyperbola Dx _ 2D hyperbola Dy _ 2D hyperbola Kx _ 2D hyperbola Ky _\n;
-
- 2D hyperbola origin = 2D point;
-
- 2D hyperbola Dx = 2D direction;
-
- 2D hyperbola Dy = 2D direction;
-
- 2D hyperbola Kx = real;
-
- 2D hyperbola Ky = real;
+ <2D curve record 5> = "5" <_> <2D hyperbola origin> <_> <2D hyperbola Dx> <_>
+ <2D hyperbola Dy> <_> <2D hyperbola Kx> <_> <2D hyperbola Ky> <_\n>;
+
+ <2D hyperbola origin> = <2D point>;
+
+ <2D hyperbola Dx> = <2D direction>;
+
+ <2D hyperbola Dy> = <2D direction>;
+
+ <2D hyperbola Kx> = <real>;
+
+ <2D hyperbola Ky> = <real>;
@endverbatim
**Description**
-2D curve record 5 describes a hyperbola. The hyperbola data consist of a 2D point @image html brep_wp_image037.gif, orthogonal 2D directions @image html brep_wp_image045.gif and @image html brep_wp_image046.gif and non-negative reals @image html brep_wp_image079.gif and @image html brep_wp_image080.gif. The hyperbola coordinate system has origin @image html brep_wp_image037.gif and axis directions @image html brep_wp_image045.gif and @image html brep_wp_image046.gif. The hyperbola is defined by the following parametric equation:
+\<2D curve record 5\> describes a hyperbola. The hyperbola data consist of a 2D point *P*, orthogonal 2D directions *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:
-@image html brep_wp_image081.gif, @image html brep_wp_image069.gif.
+@f[ C(u)=P+k_{x} \cdot cosh(u) D_{x}+k_{y} \cdot sinh(u) \cdot D_{y},\; u \in (-\infty,\; \infty). @f]
-The example record is interpreted as a hyperbola with coordinate system which has origin @image html brep_wp_image339.gif and axis directions @image html brep_wp_image354.gif and @image html brep_wp_image341.gif. Other data for the hyperbola are @image html brep_wp_image082.gif and @image html brep_wp_image083.gif. The hyperbola is defined by the following parametric equation: @image html brep_wp_image355.gif.
+The example record is interpreted as a hyperbola with coordinate system which has origin *P*=(1,2) and axis directions *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$.
-@subsubsection occt_brep_format_4_3_6 2D curve record 6 – Bezier Curve
+@subsubsection occt_brep_format_4_3_6 Bezier Curve - \<2D curve record 6\>
**Example**
**BNF-like Definition**
@verbatim
-2D curve record 6 = "6" _ 2D Bezier rational flag _ 2D Bezier degree 2D Bezier weight poles _\n;
-
-2D Bezier rational flag = flag;
-
-2D Bezier degree = int;
-
-2D Bezier weight poles = (_ 2D Bezier weight pole) ^ (2D Bezier degree + "1");
-
-2D Bezier weight pole = 2D point [_ real];
+ <2D curve record 6> = "6" <_> <2D Bezier rational flag> <_> <2D Bezier degree>
+ <2D Bezier weight poles> <_\n>;
+
+ <2D Bezier rational flag> = <flag>;
+
+ <2D Bezier degree> = <int>;
+
+ <2D Bezier weight poles> = (<_> <2D Bezier weight pole>) ^ (<2D Bezier degree> <+> “1”);
+
+ <2D Bezier weight pole> = <2D point> [<_> <real>];
@endverbatim
**Description**
-2D curve record 6 describes a Bezier curve. The curve data consist of a rational flag @image html brep_wp_image047.gif, a degree @image html brep_wp_image085.gif and weight poles.
+\<2D curve record 6\> describes a Bezier curve. The curve data consist of a rational flag *r*, a degree @f$ m \leq 25 @f$ and weight poles.
-The weight poles are @image html brep_wp_image086.gif 2D points @image html brep_wp_image087.gif if the flag @image html brep_wp_image047.gif is 0. The weight poles are @image html brep_wp_image086.gif pairs @image html brep_wp_image088.gif if the flag @image html brep_wp_image047.gif is 1. Here @image html brep_wp_image089.gif is a 2D point and @image html brep_wp_image090.gif is a positive real (@image html brep_wp_image093.gif). @image html brep_wp_image092.gif (@image html brep_wp_image093.gif) if the flag @image html brep_wp_image047.gif is 0.
+The weight poles are *m*+1 2D points *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:
-@image html brep_wp_image094.gif, @image html brep_wp_image095.gif
+@f[ C(u)= \frac{\sum_{i=0}^{m} B_{i} \cdot h_{i} \cdot C_{m}^{i} \cdot u^{i} \cdot (1-u)^{m-i}}{\sum_{i=0}^{m} h_{i} \cdot C_{m}^{i} \cdot u^{i} \cdot (1-u)^{m-i}},\; u \in [0,1] @f]
-where @image html brep_wp_image096.gif.
+where @f$ 0^{0} \equiv 1 @f$.
-The example record is interpreted as a Bezier curve with a rational flag @image html brep_wp_image097.gif, a degree @image html brep_wp_image098.gif and weight poles @image html brep_wp_image356.gif, @image html brep_wp_image100.gif, @image html brep_wp_image357.gif, @image html brep_wp_image102.gif and @image html brep_wp_image358.gif, @image html brep_wp_image104.gif. The Bezier curve is defined by the following parametric equation:
+The example record is interpreted as a Bezier curve with a rational flag *r*=1, a degree *m*=2 and weight poles *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:
-@image html brep_wp_image359.gif.
+@f[ C(u)= \frac{(0,1) \cdot 4 \cdot (1-u)^{2}+(1,-2) \cdot 5 \cdot 2 \cdot u \cdot (1-u)+(2,3) \cdot 6 \cdot u^{2}}{ 4 \cdot (1-u)^{2}+5 \cdot 2 \cdot u \cdot (1-u)+6 \cdot u^{2}} . @f]
-@subsubsection occt_brep_format_4_3_7 2D curve record 7 – B-spline Curve
+@subsubsection occt_brep_format_4_3_7 B-spline Curve - \<2D curve record 7\>
**Example**
**BNF-like Definition**
@verbatim
- 2D curve record 7 = "7" _ 2D B-spline rational flag _ "0" _ 2D B-spline degree _ 2D B-spline pole count _ 2D B-spline multiplicity knot count 2D B-spline weight poles _\n 2D B-spline multiplicity knots _\n;
-
- 2D B-spline rational flag = flag;
-
- 2D B-spline degree = int;
-
- 2D B-spline pole count = int;
-
- 2D B-spline multiplicity knot count = int;
-
- 2D B-spline weight poles = 2D B-spline weight pole ^ 2D B-spline pole count;
-
- 2D B-spline weight pole = _ 2D point [_ real];
-
- 2D B-spline multiplicity knots =
- 2D B-spline multiplicity knot ^ 2D B-spline multiplicity knot count;
-
- 2D B-spline multiplicity knot = _ real _ int;
+ <2D curve record 7> = "7" <_> <2D B-spline rational flag> <_> "0" <_> <2D B-spline degree> <_> <2D B-spline pole count> <_> <2D B-spline multiplicity knot count> <2D B-spline weight poles> <_\n> <2D B-spline multiplicity knots> <_\n>;
+
+ <2D B-spline rational flag> = <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>;
@endverbatim
**Description**
-2D curve record 7 describes a B-spline curve. The curve data consist of a rational flag @image html brep_wp_image047.gif, a degree @image html brep_wp_image085.gif, a pole count @image html brep_wp_image106.gif, a multiplicity knot count @image html brep_wp_image107.gif, weight poles and multiplicity knots.
+\<2D curve record 7\> describes a B-spline curve. The curve data consist of a rational flag *r*, a degree @f$ m \leq 25 @f$, a pole count @f$ n \geq 2 @f$, a multiplicity knot count *k*, weight poles and multiplicity knots.
-The weight poles are @image html brep_wp_image108.gif 2D points @image html brep_wp_image109.gif if the flag @image html brep_wp_image047.gif is 0. The weight poles are @image html brep_wp_image108.gif pairs @image html brep_wp_image110.gif if the flag @image html brep_wp_image047.gif is 1. Here @image html brep_wp_image089.gif is a 2D point and @image html brep_wp_image090.gif is a positive real (@image html brep_wp_image111.gif). @image html brep_wp_image092.gif (@image html brep_wp_image111.gif) if the flag @image html brep_wp_image047.gif is 0.
+The weight poles are *n* 2D points *B<sub>1</sub> ... B<sub>n</sub>* if the flag *r* is 0. The weight poles are *n* pairs *B<sub>1</sub>h<sub>1</sub> ... B<sub>n</sub>h<sub>n</sub>* if the flag *r* is 1. Here *B<sub>i</sub>* is a 2D point and *h<sub>i</sub>* is a positive real @f$ (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 @image html brep_wp_image107.gif pairs @image html brep_wp_image112.gif. Here @image html brep_wp_image113.gif is a knot with multiplicity @image html brep_wp_image114.gif (@image html brep_wp_image115.gif) so that
+The multiplicity knots are *k* pairs *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
-@image html brep_wp_image360.gif (@image html brep_wp_image361.gif),
-@image html brep_wp_image362.gif, @image html brep_wp_image363.gif, @image html brep_wp_image364.gif (@image html brep_wp_image365.gif), @image html brep_wp_image366.gif.
+@f[ u_{i} < u_{i+1}\; (1 \leq i \leq k-1), \\
+q_{1} \leq m+1,\; q_{k} \leq m+1,\; q_{i} \leq m\; (2 \leq i \leq k-1),\; \sum_{i=1}^{k}q_{i}=m+n+1 . @f]
The B-spline curve is defined by the following parametric equation:
-@image html brep_wp_image367.gif, @image html brep_wp_image368.gif
+@f[ C(u)= \frac{\sum_{i=1}^{n} B_{i} \cdot h_{i} \cdot N_{i,m+1}(u) }{\sum_{i=1}^{n} h_{i} \cdot N_{i,m+1}(u)},\; u \in [u_{1},\; u_{k}] @f]
-where functions @image html brep_wp_image125.gif have the following recursion definition by @image html brep_wp_image126.gif
+where functions *N<sub>i,j</sub>* have the following recursion definition by *j*
-@image html brep_wp_image127.gif, @image html brep_wp_image128.gif (@image html brep_wp_image129.gif)
+@f[ N_{i,1}(u)=\left\{\begin{matrix}
+1\Leftarrow \bar{u}_{i} \leq u \leq \bar{u}_{i+1}\\
+0\Leftarrow u < \bar{u}_{i} \vee \bar{u}_{i+1} \leq u \end{matrix} \right.,\;
+N_{i,j}(u)=\frac{(u-\bar{u}_{i}) \cdot N_{i,j-1}(u) }{\bar{u}_{i+j-1}-\bar{u}_{i}}+ \frac{(\bar{u}_{i+j}-u) \cdot N_{i+1,j-1}(u)}{\bar{u}_{i+j}-\bar{u}_{i+1}},\;(2 \leq j \leq m+1) @f]
where
-@image html brep_wp_image284.gif (@image html brep_wp_image369.gif,@image html brep_wp_image286.gif).
+@f[ \bar{u}_{i}=u_{j}\; (1\leq j\leq k,\; \sum_{l=1}^{j-1}q_{l}+1 \leq i \leq \sum_{l=1}^{j}q_{l}) . @f]
-The example record is interpreted as a B-spline curve with a rational flag @image html brep_wp_image097.gif, a degree @image html brep_wp_image133.gif, a pole count @image html brep_wp_image134.gif, a multiplicity knot count @image html brep_wp_image135.gif, weight poles @image html brep_wp_image370.gif, @image html brep_wp_image137.gif, @image html brep_wp_image371.gif, @image html brep_wp_image139.gif and @image html brep_wp_image372.gif, @image html brep_wp_image141.gif and multiplicity knots @image html brep_wp_image373.gif, @image html brep_wp_image374.gif, @image html brep_wp_image375.gif, @image html brep_wp_image376.gif, @image html brep_wp_image377.gif, @image html brep_wp_image378.gif, @image html brep_wp_image379.gif, @image html brep_wp_image380.gif and @image html brep_wp_image381.gif, @image html brep_wp_image382.gif. The B-spline curve is defined by the following parametric equation:
+The example record is interpreted as a B-spline curve with a rational flag *r*=1, a degree *m*=1, a pole count *n*=3, a multiplicity knot count *k*=5, weight poles *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:
-@image html brep_wp_image383.gif.
+@f[ C(u)= \frac{(0,1) \cdot 4 \cdot N_{1,2}(u)+(1,-2) \cdot 5 \cdot N_{2,2}(u)+(2,3) \cdot 6 \cdot N_{3,2}(u)}{ 4 \cdot N_{1,2}(u)+5 \cdot N_{2,2}(u)+6 \cdot N_{3,2}(u)} . @f]
-@subsubsection occt_brep_format_4_3_8 2D curve record 8 – Trimmed Curve
+@subsubsection occt_brep_format_4_3_8 Trimmed Curve - \<2D curve record 8\>
**Example**
**BNF-like Definition**
@verbatim
- 2D curve record 8 = "8" _ 2D trimmed curve u min _ 2D trimmed curve u max _\n 2D curve record;
-
- 2D trimmed curve u min = real;
-
- 2D trimmed curve u max = real;
+ <2D curve record 8> = "8" <_> <2D trimmed curve u min> <_> <2D trimmed curve u max> <_\n>
+ <2D curve record>;
+
+ <2D trimmed curve u min> = <real>;
+
+ <2D trimmed curve u max> = <real>;
@endverbatim
-**Description**
+**Description**
-2D curve record 8 describes a trimmed curve. The trimmed curve data consist of reals @image html brep_wp_image153.gif and @image html brep_wp_image154.gif and a 2D curve record so that @image html brep_wp_image155.gif. The trimmed curve is a restriction of the base curve @image html brep_wp_image326.gif described in the record to the segment @image html brep_wp_image157.gif. The trimmed curve is defined by the following parametric equation:
+\<2D curve record 8\> describes a trimmed curve. The trimmed curve data consist of reals *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$. The trimmed curve is defined by the following parametric equation:
-@image html brep_wp_image158.gif, @image html brep_wp_image159.gif.
+@f[ C(u)=B(u),\; u \in [u_{min},\;u_{max}] . @f]
-The example record is interpreted as a trimmed curve with @image html brep_wp_image160.gif, @image html brep_wp_image161.gif and base curve @image html brep_wp_image384.gif. The trimmed curve is defined by the following parametric equation: @image html brep_wp_image385.gif, @image html brep_wp_image164.gif.
+The example record is interpreted as a trimmed curve with *u<sub>min</sub>*=-4, *u<sub>max</sub>*=5 and base curve @f$ 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$.
-@subsubsection occt_brep_format_4_3_9 2D curve record 9 – Offset Curve
+@subsubsection occt_brep_format_4_3_9 Offset Curve - \<2D curve record 9\>
**Example**
**BNF-like Definition**
@verbatim
-2D curve record 9 = "9" _ 2D offset curve distance _\n 2D curve record;
-
-2D offset curve distance = real;
+<2D curve record 9> = "9" <_> <2D offset curve distance> <_\n> <2D curve record>;
+
+<2D offset curve distance> = <real>;
@endverbatim
**Description**
-2D curve record 9 describes an offset curve. The offset curve data consist of a distance @image html brep_wp_image165.gif and a 2D curve record. The offset curve is the result of offsetting the base curve @image html brep_wp_image156.gif described in the record to the distance @image html brep_wp_image165.gif along the vector @image html brep_wp_image386.gif where @image html brep_wp_image387.gif. The offset curve is defined by the following parametric equation:
+\<2D curve record 9\> describes an offset curve. The offset curve data consist of a distance *d* and a \<2D curve record\>. The offset curve is the result of offsetting the base curve *B* described in the record to the distance *d* along the vector @f$ (B'_{Y}(u),\; -B'_{X}(u)) \neq \vec{0} @f$ where @f$ B(u)=(B'_{X}(u),\; B'_{Y}(u)) @f$. The offset curve is defined by the following parametric equation:
-@image html brep_wp_image388.gif, @image html brep_wp_image168.gif.
+@f[ C(u)=B(u)+d \cdot (B'_{Y}(u),\; -B'_{X}(u)),\; u \in domain(B) . @f]
-The example record is interpreted as an offset curve with a distance
-@image html brep_wp_image169.gif and base curve @image html brep_wp_image384.gif and
-is defined by the following parametric equation: @image html brep_wp_image389.gif.
+The example record is interpreted as an offset curve with a distance *d*=2 and base curve @f$ B(u)=(1,2)+u \cdot (1,0) @f$ and is defined by the following parametric equation: @f$ C(u)=(1,2)+u \cdot (1,0)+2 \cdot (0,-1) @f$.
-@subsection occt_brep_format_4_4 Subsection 3D polygons
+@subsection occt_brep_format_4_4 3D polygons
**Example**
**BNF-like Definition**
@verbatim
- 3D polygons = 3D polygon header _\n 3D polygon records;
-
- 3D polygon header = "Polygon3D" _ 3D polygon record count;
-
- 3D polygon records = 3D polygon record ^ 3D polygon record count;
-
- 3D polygon record =
- 3D polygon node count _ 3D polygon flag of parameter presence _\n
- 3D polygon deflection _\n
- 3D polygon nodes _\n
- [3D polygon parameters _\n];
-
- 3D polygon node count = int;
-
- 3D polygon flag of parameter presence = flag;
-
- 3D polygon deflection = real;
-
- 3D polygon nodes = (3D polygon node _) ^ 3D polygon node count;
-
- 3D polygon node = 3D point;
-
- 3D polygon u parameters = (3D polygon u parameter _) ^ 3D polygon node count;
-
- 3D polygon u parameter = real;
+ <3D polygons> = <3D polygon header> <_\n> <3D polygon records>;
+
+ <3D polygon header> = "Polygon3D" <_> <3D polygon record count>;
+
+ <3D polygon records> = <3D polygon record> ^ <3D polygon record count>;
+
+ <3D polygon record> =
+ <3D polygon node count> <_> <3D polygon flag of parameter presence> <_\n>
+ <3D polygon deflection> <_\n>
+ <3D polygon nodes> <_\n>
+ [<3D polygon parameters> <_\n>];
+
+ <3D polygon node count> = <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 polygon record describes a 3D polyline @image html brep_wp_image390.gif which approximates a 3D curve @image html brep_wp_image205.gif. The polyline data consist of a node count @image html brep_wp_image391.gif, a parameter presence flag @image html brep_wp_image392.gif, a deflection @image html brep_wp_image393.gif, nodes @image html brep_wp_image394.gif (@image html brep_wp_image395.gif) and parameters @image html brep_wp_image113.gif (@image html brep_wp_image395.gif). The parameters are present only if@image html brep_wp_image396.gif. The polyline @image html brep_wp_image390.gif passes through the nodes. The deflection @image html brep_wp_image165.gif describes the deflection of polyline @image html brep_wp_image390.gif from the curve @image html brep_wp_image205.gif:
+\<3D polygons\> record describes a 3D polyline *L* which approximates a 3D curve *C*. The polyline data consist of a node count @f$ m \geq 2 @f$, a parameter presence flag *p*, a deflection @f$ d \geq 0 @f$, nodes @f$ N_{i}\; (1\leq i \leq m) @f$ and parameters @f$ u_{i}\; (1\leq i \leq m) @f$. The parameters are present only if *p*=1. The polyline *L* passes through the nodes. The deflection *d* describes the deflection of polyline *L* from the curve *C*:
-@image html brep_wp_image397.gif.
+@f[ \underset{P \in C}{max}\; \underset{Q \in L}{min}|Q-P| \leq d . @f]
-The parameter @image html brep_wp_image113.gif (@image html brep_wp_image395.gif) is the parameter of the node @image html brep_wp_image394.gif on the curve @image html brep_wp_image205.gif:
+The parameter @f$ u_{i}\; (1\leq i \leq m) @f$ is the parameter of the node *N<sub>i</sub>* on the curve *C*:
-@image html brep_wp_image398.gif.
+@f[ C(u_{i})=N_{i} . @f]
-The example record describes a polyline from @image html brep_wp_image098.gif nodes with a parameter presence flag @image html brep_wp_image396.gif, a deflection @image html brep_wp_image399.gif, nodes @image html brep_wp_image400.gif and @image html brep_wp_image401.gif and parameters @image html brep_wp_image142.gif and @image html brep_wp_image402.gif.
+The example record describes a polyline from *m*=2 nodes with a parameter presence flag *p*=1, a deflection *d*=0.1, nodes *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 occt_brep_format_4_5 Subsection triangulations
+@subsection occt_brep_format_4_5 Triangulations
**Example**
**BNF-like Definition**
@verbatim
- triangulations = triangulation header _\n triangulation records;
-
- triangulation header = "Triangulations" _ triangulation count;
-
- triangulation records = triangulation record ^ triangulation count;
-
- triangulation record = triangulation node count _ triangulation triangle count _ triangulation parameter presence flag _ triangulation deflection _\n triangulation nodes [_ triangulation u v parameters] _ triangulation triangles _\n;
-
- triangulation node count = int;
-
- triangulation triangle count = int;
-
- triangulation parameter presence flag = flag;
-
- triangulation deflection = real;
-
- triangulation nodes = (triangulation node _) ^ triangulation node count;
-
- triangulation node = 3D point;
-
- triangulation u v parameters = (triangulation u v parameter pair _) ^ triangulation node count;
-
- triangulation u v parameter pair = real _ real;
-
- triangulation triangles = (triangulation triangle _) ^ triangulation triangle count;
-
- triangulation triangle = int _ int _ int.
+ <triangulations> = <triangulation header> <_\n> <triangulation records>;
+
+ <triangulation header> = "Triangulations" <_> <triangulation count>;
+
+ <triangulation records> = <triangulation record> ^ <triangulation count>;
+
+ <triangulation record> = <triangulation node count> <_> <triangulation triangle count> <_> <triangulation parameter presence flag> <_> <triangulation deflection> <_\n>
+ <triangulation nodes> [<_> <triangulation u v parameters>] <_> <triangulation triangles> <_\n>;
+
+ <triangulation node count> = <int>;
+
+ <triangulation triangle count> = <int>;
+
+ <triangulation parameter presence flag> = <flag>;
+
+ <triangulation deflection> = <real>;
+
+ <triangulation nodes> = (<triangulation node> <_>) ^ <triangulation node count>;
+
+ <triangulation node> = <3D point>;
+
+ <triangulation u v parameters> =
+ (<triangulation u v parameter pair> <_>) ^ <triangulation node count>;
+
+ <triangulation u v parameter pair> = <real> <_> <real>;
+
+ <triangulation triangles> = (<triangulation triangle> <_>) ^ <triangulation triangle count>;
+
+ <triangulation triangle> = <int> <_> <int> <_> <int>.
@endverbatim
**Description**
-triangulation record describes a triangulation @image html brep_wp_image403.gif which approximates a surface @image html brep_wp_image404.gif. The triangulation data consist of a node count @image html brep_wp_image405.gif, a triangle count @image html brep_wp_image406.gif, a parameter presence flag @image html brep_wp_image392.gif, a deflection @image html brep_wp_image393.gif, nodes @image html brep_wp_image394.gif (@image html brep_wp_image395.gif), parameter pairs @image html brep_wp_image407.gif (@image html brep_wp_image395.gif), triangles @image html brep_wp_image408.gif (@image html brep_wp_image131.gif, @image html brep_wp_image409.gif (@image html brep_wp_image410.gif)). The parameters are present only if @image html brep_wp_image396.gif. The deflection describes the triangulation deflection from the surface:
+\<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$. The parameters are present only if *p*=1. The deflection describes the triangulation deflection from the surface:
-@image html brep_wp_image411.gif.
+@f[ \underset{P \in S}{max}\; \underset{Q \in T}{min}|Q-P| \leq d . @f]
-The parameter pair @image html brep_wp_image407.gif (@image html brep_wp_image395.gif) describes the parameters of node @image html brep_wp_image394.gif on the surface:
+The parameter pair @f$ u_{i}\; v_{i}\; (1\leq i \leq m) @f$ describes the parameters of node *N<sub>i</sub>* on the surface:
-@image html brep_wp_image412.gif.
+@f[ S(u_{i},v_{i})=N_{i} . @f]
-The triangle @image html brep_wp_image408.gif (@image html brep_wp_image131.gif) is interpreted as a triangle of nodes @image html brep_wp_image413.gif, @image html brep_wp_image414.gif and @image html brep_wp_image415.gif with circular traversal of the nodes in the order @image html brep_wp_image413.gif, @image html brep_wp_image414.gif and @image html brep_wp_image415.gif. From any side of the triangulation @image html brep_wp_image403.gif all its triangles have the same direction of the node circular traversal: either clockwise or counterclockwise.
+The triangle @f$ n_{j,1}\; n_{j,2}\; n_{j,3}\; (1\leq j \leq k) @f$ is interpreted as a triangle of nodes @f$ N_{n_{j},1}\; N_{n_{j},2}@f$ and @f$ N_{n_{j},3} @f$ with circular traversal of the nodes in the order @f$ N_{n_{j},1}\; N_{n_{j},2}@f$ and @f$ N_{n_{j},3} @f$. From any side of the triangulation *T* all its triangles have the same direction of the node circular traversal: either clockwise or counterclockwise.
Triangulation record
0 0 0 0 0 3 0 2 3 0 2 0 0 0 3 0 3 -2 0 -2 2 4 3 2 1 4
@endverbatim
-describes a triangulation with @image html brep_wp_image416.gif nodes, @image html brep_wp_image417.gif triangles, parameter presence flag @image html brep_wp_image396.gif, deflection @image html brep_wp_image418.gif, nodes @image html brep_wp_image419.gif, @image html brep_wp_image420.gif, @image html brep_wp_image421.gif and @image html brep_wp_image422.gif, parameters @image html brep_wp_image423.gif, @image html brep_wp_image424.gif, @image html brep_wp_image425.gif and @image html brep_wp_image426.gif, and triangles @image html brep_wp_image427.gif and @image html brep_wp_image428.gif. From the point @image html brep_wp_image429.gif (@image html brep_wp_image430.gif) the triangles have clockwise (counterclockwise) direction of the node circular traversal.
+describes a triangulation with *m*=4 nodes, *k*=2 triangles, parameter presence flag *p*=1, deflection *d*=0, nodes *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 occt_brep_format_4_6 Subsection polygons on triangulations
+@subsection occt_brep_format_4_6 Polygons on triangulations
**Example**
**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;
+ <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 @image html brep_wp_image390.gif on a triangulation which approximates a curve @image html brep_wp_image205.gif. The polyline data consist of a node count @image html brep_wp_image391.gif, node numbers @image html brep_wp_image431.gif, deflection @image html brep_wp_image393.gif, a parameter presence flag @image html brep_wp_image392.gif and parameters @image html brep_wp_image432.gif (@image html brep_wp_image395.gif). The parameters are present only if @image html brep_wp_image396.gif. The deflection @image html brep_wp_image165.gif describes the deflection of polyline @image html brep_wp_image390.gif from the curve @image html brep_wp_image205.gif:
+\<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$. The parameters are present only if *p*=1. The deflection *d* describes the deflection of polyline *L* from the curve *C*:
-@image html brep_wp_image397.gif.
+@f[ \underset{P \in C}{max}\; \underset{Q \in L}{min}|Q-P| \leq d . @f]
-Parameter @image html brep_wp_image113.gif (@image html brep_wp_image395.gif) is @image html brep_wp_image433.gif-th node @image html brep_wp_image434.gif parameter on curve @image html brep_wp_image205.gif.
+Parameter @f$ u_{i}\; (1\leq i\leq m) @f$ is *n<sub>i</sub>*-th node *C(u<sub>i</sub>)* parameter on curve *C*.
@subsection occt_brep_format_4_7 Geometric Sense of a Curve
-Geometric sense of curve @image html brep_wp_image210.gif described above is determined by the direction of parameter @image html brep_wp_image435.gif increasing.
+Geometric sense of curve *C* described above is determined by the direction of parameter *u* increasing.
-@section occt_brep_format_5 Section shapes
+@section occt_brep_format_5 Shapes
An example of section shapes and a whole *.brep file are given in chapter 7 "Appendix".
**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);
+ <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 @image html brep_wp_image436.gif flags @image html brep_wp_image437.gif
-(@image html brep_wp_image438.gif) are interpreted as shape flags in the following way:
+\<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:
- * @image html brep_wp_image439.gif – free;
- * @image html brep_wp_image440.gif – modified;
- * @image html brep_wp_image441.gif – IGNORED (version * / checked (version * ;
- * @image html brep_wp_image442.gif – orientable;
- * @image html brep_wp_image443.gif – closed;
- * @image html brep_wp_image444.gif – infinite;
- * @image html brep_wp_image445.gif – convex.
+ * @f$ f_{1} @f$ – free;
+ * @f$ f_{2} @f$ – modified;
+ * @f$ f_{3} @f$ – IGNORED(version 1) \\ checked (version 2);
+ * @f$ f_{4} @f$ – orientable;
+ * @f$ f_{5} @f$ – closed;
+ * @f$ f_{6} @f$ – infinite;
+ * @f$ f_{7} @f$ – convex.
The flags are used in a special way [1].
-shape subshape orientation is interpreted in the following way:
+\<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 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 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:
+\<shape subrecord\> types are interpreted in the following way:
* "Ve" – vertex;
* "Ed" – edge;
* "CS" – compsolid;
* "Co" – compound.
-shape final record determines the orientation and location for the whole model.
+\<shape final record\> determines the orientation and location for the whole model.
@subsection occt_brep_format_5_1 Common Terms
-The terms below are used by vertex data, edge data and face data.
+The terms below are used by \<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;
+ <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.
+\<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.
+\<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.
+\<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.
+\<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.
+\<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.
+\<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.
+\<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 @image html brep_wp_image446.gif and @image html brep_wp_image447.gif are the curve parameter@image html brep_wp_image448.gif bounds: @image html brep_wp_image449.gif.
+\<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 @image html brep_wp_image446.gif and @image html brep_wp_image447.gif are real pairs @image html brep_wp_image450.gif and @image html brep_wp_image451.gif that @image html brep_wp_image452.gif and @image html brep_wp_image453.gif where @image html brep_wp_image205.gif is a parametric equation of the curve.
+\<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 occt_brep_format_5_2 vertex data
+@subsection occt_brep_format_5_2 Vertex data
**BNF-like Definition**
@verbatim
- vertex data = vertex data tolerance _\n vertex data 3D representation _\n vertex data representations;
-
- vertex data tolerance = real;
-
- vertex data 3D representation = 3D point;
-
- vertex data representations = (vertex data representation _\n)* "0 0";
-
- vertex data representation = vertex data representation u parameter _
- vertex data representation data _ location number;
-
- vertex data representation u parameter = real;
-
- vertex data representation data =
- ("1" _ vertex data representation data * |
- ("2" _ vertex data representation data * |
- ("3" _ vertex data representation data * ;
-
- vertex data representation data 1 = 3D curve number;
-
- vertex data representation data 2 = 2D curve number _ surface number;
-
- vertex data representation data 3 =
- vertex data representation v parameter _ surface number;
-
- vertex data representation v parameter = real;
+ <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 @image html brep_wp_image448.gif is described below.
+The usage of \<vertex data representation u parameter\> *U* is described below.
-vertex data representation data 1 and parameter @image html brep_wp_image448.gif describe the position of the vertex @image html brep_wp_image454.gif on a 3D curve @image html brep_wp_image205.gif. Parameter @image html brep_wp_image448.gif is a parameter of the vertex @image html brep_wp_image454.gif on the curve @image html brep_wp_image205.gif: @image html brep_wp_image455.gif.
+\<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 @image html brep_wp_image448.gif describe the position of the vertex @image html brep_wp_image454.gif on a 2D curve @image html brep_wp_image205.gif which is located on a surface. Parameter @image html brep_wp_image448.gif is a parameter of the vertex @image html brep_wp_image454.gif on the curve @image html brep_wp_image205.gif: @image html brep_wp_image455.gif.
+\<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 @image html brep_wp_image435.gif describe the position of the vertex @image html brep_wp_image454.gif on a surface @image html brep_wp_image404.gif through vertex data representation v parameter @image html brep_wp_image456.gif: @image html brep_wp_image457.gif.
+\<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 @image html brep_wp_image458.gif describes the maximum distance from the vertex @image html brep_wp_image454.gif to the set @image html brep_wp_image459.gif* *of vertex @image html brep_wp_image454.gif representations:
+\<vertex data tolerance\> *t* describes the maximum distance from the vertex *V* to the set *R* of vertex *V* representations:
-@image html brep_wp_image460.gif.
+@f[ \underset{P \in R }{max} |P-V| \leq t . @f]
-@subsection occt_brep_format_5_3 edge data
+@subsection occt_brep_format_5_3 Edge data
**BNF-like Definition**
@verbatim
- edge data = _ edge data tolerance _ edge data same parameter flag _ edge data same range flag _ edge data degenerated flag _\n edge data representations;
-
- edge data tolerance = real;
-
- edge data same parameter flag = flag;
-
- edge data same range flag = flag;
-
- edge data degenerated flag = flag;
-
- edge data representations = (edge data representation _\n)* "0";
-
- edge data representation =
- "1" _ edge data representation data 1
- "2" _ edge data representation data 2
- "3" _ edge data representation data 3
- "4" _ edge data representation data 4
- "5" _ edge data representation data 5
- "6" _ edge data representation data 6
- "7" _ edge data representation data 7;
-
- edge data representation data 1 = 3D curve number _ location number _
- curve parameter minimal and maximal values;
-
- edge data representation data 2 = 2D curve number _ surface number _ location number _ curve parameter minimal and maximal values
- [_\n curve values for parameter minimal and maximal values];
-
- edge data representation data 3 = (2D curve number _) ^ 2 continuity order _ surface number _ location number _ curve parameter minimal and maximal values \n curve values for parameter minimal and maximal values];
-
- continuity order = "C0" | "C1" | "C2" | "C3" | "CN" | "G1" | "G2".
-
- edge data representation data 4 =
- continuity order (_ surface number _ location number) ^ 2;
-
- edge data representation data 5 = 3D polygon number _ location number;
-
- edge data representation data 6 =
- polygon on triangulation number _ triangulation number _ location number;
-
- edge data representation data 7 = (polygon on triangulation number _) ^ 2 triangulation number _ location number;
+ <edge data> = <_> <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>;
@endverbatim
**Description**
-Flags edge data same parameter flag, edge data same range flag and edge data degenerated flag are used in a special way [1].
+Flags \<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 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 only in version 2.
+\<edge data representation data 2\> describes a 2D curve on a surface.
+\<curve values for parameter minimal and maximal values\> are used only in version 2.
-edge data representation data 3 describes a 2D curve on a closed surface.
-curve values for parameter minimal and maximal values are used only in version 2.
+\<edge data representation data 3\> describes a 2D curve on a closed surface.
+\<curve values for parameter minimal and maximal values\> are used only in version 2.
-edge data representation data 5 describes a 3D polyline.
+\<edge data representation data 5\> describes a 3D polyline.
-edge data representation data 6 describes a polyline on a triangulation.
+\<edge data representation data 6\> describes a polyline on a triangulation.
-edge data tolerance @image html brep_wp_image458.gif describes the maximum distance from the edge @image html brep_wp_image461.gif to the set @image html brep_wp_image459.gif of edge @image html brep_wp_image461.gif representations:
+\<edge data tolerance\> *t* describes the maximum distance from the edge *E* to the set *R* of edge *E* representations:
-@image html brep_wp_image462.gif.
+@f[ \underset{C \in R}{max}\;\underset{P \in E}{max}\;\underset{Q \in C}{min}|Q-P| \leq t @f]
-@subsection occt_brep_format_5_4 face data
+@subsection occt_brep_format_5_4 Face data
**BNF-like Definition**
@verbatim
- face data = face data natural restriction flag _ face data tolerance _ surface number _ location number \n ["2" _ triangulation number];
-
- face data natural restriction flag = flag;
-
- face data tolerance = real;
+ <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>;
@endverbatim
**Description**
-face data describes a surface @image html brep_wp_image404.gif of face @image html brep_wp_image463.gif and a triangulation @image html brep_wp_image403.gif of face @image html brep_wp_image463.gif. The surface @image html brep_wp_image404.gif may be empty: surface number = 0.
+\<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 @image html brep_wp_image458.gif describes the maximum distance from the face @image html brep_wp_image463.gif to the surface @image html brep_wp_image404.gif:
+\<face data tolerance\> *t* describes the maximum distance from the face *F* to the surface *S*:
-@image html brep_wp_image464.gif.
+@f[ \underset{P \in F}{max}\;\underset{Q \in S}{min}|Q-P| \leq t @f]
- Flag face data natural restriction flag is used in a special way [1].
+Flag \<face data natural restriction flag\> is used in a special way [1].
-@section occt_brep_format_6 References
-
- The format is part of Open CASCADE Technology (OCCT).
- Some data fields of the format have additional values which are used in OCCT.
- Some data fields of the format are specific for OCCT.
-
-@section occt_brep_format_7 Appendix
+
+@section occt_brep_format_6 Appendix
This chapter contains a *.brep file example.
projects / occt.git / blobdiff