.. role:: blue :class: blue MbSurface Surface ================= .. _MbSurface_Surface_: .. version: 1180ХХ The :blue:`MbSurface` class is declared in the *surface.h* file. The :blue:`MbSurface` surface is an inheritor of the :ref:`MbSpaceItem ` class, Fig. 1. .. figure:: /MODELER/SURFACES/pictures/O.5.1.1.png :width: 12cm :align: center Fig. 1 The surface is an abstract class. The C3D geometric kernel implements the following surfaces, which are inheritors of the :blue:`MbSurface` class: - :ref:`MbPlane ` — plane, - :ref:`MbCylinderSurface ` — cylindrical surface, - :ref:`MbConeSurface ` — conical surface, - :ref:`MbSphereSurface ` — spherical surface, - :ref:`MbTorusSurface ` — toroidal surface, - :ref:`MbExtrusionSurface ` — extrusion surface, - :ref:`MbRevolutionSurface ` — revolution surface, - :ref:`MbExpansionSurface ` — plane-parallel kinematic surface, - :ref:`MbSpiralSurface ` — spiral surface, - :ref:`MbEvolutionSurface ` — swept surface, - :ref:`MbExactionSurface ` — swept surface with adaptation, - :ref:`MbSectorSurface ` — sectorial surface, - :ref:`MbRuledSurface ` — ruled surface, - :ref:`MbLoftedSurface ` — surface based on a family of curves, - :ref:`MbElevationSurface ` — surface based on a family of curves and a guiding curve, - :ref:`MbCornerSurface ` — surface based on three curves, - :ref:`MbCoverSurface ` — surface based on four curves, - :ref:`MbCoonsPatchSurface ` — bicubic Coons surface, - :ref:`MbMeshSurface ` — surface based on a network of curves, - :ref:`MbJoinSurface ` — joint surface, - :ref:`MbSplineSurface ` — NURBS surface (NonUniform Rational B-Spline surface), - :ref:`MbOffsetSurface ` — equidistant surface, - :ref:`MbChamferSurface ` — chamfer surface, - :ref:`MbFilletSurface ` — fillet surface, - :ref:`MbChannelSurface ` — fillet surface with variable radius, - :ref:`MbCurveBoundedSurface ` — surface with arbitrary borders. The :blue:`MbSurface` surface is a vector function .. math:: \mathbf{surface}(u, v) = \begin{bmatrix} x(u,v) & y(u,v) & z(u,v) \end{bmatrix} of two scalar parameters *u* and *v* taking values on a two-dimensional connected domain :math:`\Omega`. The surface is a continuous mapping of the two-dimensional connected domain :math:`\Omega` into three-dimensional space. The domain :math:`\Omega` is described in a two-dimensional Cartesian coordinate system. In the particular case, the domain :math:`\Omega` is a rectangle, and the surface parameters take values within *u*\ :sub:`min` :math:`\leq` *u* :math:`\leq` *u*\ :sub:`max`, *v*\ :sub:`min` :math:`\leq` *v* :math:`\leq` *v*\ :sub:`max`. In the general case, the domain :math:`\Omega` is described by two-dimensional curves. The coordinates *x*\ (*u*, *v*), *y*\ (*u*, *v*), *z*\ (*u*, *v*) of the surface point **surface**\ (*u*, *v*) are single-valued continuous functions of the parameters *u* and *v*. The boundary values *u*\ :sub:`min`, *u*\ :sub:`max`, *v*\ :sub:`min`, *v*\ :sub:`max` of the parameter domain are returned by the surface methods double :blue:`GetUMin`\(), double :blue:`GetUMax`\(), double :blue:`GetVMin`\(), double :blue:`GetVMax`\() respectively. A surface is called periodic with respect to the first parameter if there exists *p*\ :sub:`u` > 0 such that **surface**\ (*u* ± *kp*\ :sub:`u`, *v*) = **surface**\ (*u*, *v*), where *k* is an integer. A surface is called periodic with respect to the second parameter if there exists *p*\ :sub:`v` > 0 such that **surface**\ (*u*, *v* ± *kp*\ :sub:`v`) = **surface**\ (*u*, *v*), where *k* is an integer. The domain of a periodic surface parameter lies within one period for the corresponding parameter. The method bool :blue:`IsUClosed`\() of a surface periodic with respect to the first parameter returns *true*. The method bool :blue:`IsVClosed`\() of a surface periodic with respect to the second parameter returns *true*. The method double :blue:`GetUPeriod`\() of a surface periodic with respect to the first parameter, or of a surface that can be extended to a periodic one, returns the period *p*\ :sub:`u`. The method double :blue:`GetVPeriod`\() of a surface periodic with respect to the second parameter, or of a surface that can be extended to a periodic one, returns the period *p*\ :sub:`v`. The domain of a periodic surface parameter always lies within one period. Let us introduce the notation .. math:: s_{u} = \frac{\partial\mathbf{surface}(u,v)}{\partial u}; .. math:: s_{v} = \frac{\partial\mathbf{surface}(u,v)}{\partial v}; .. math:: s_{uu} = \frac{\partial^{2}\mathbf{surface}(u,v)}{\partial u^{2}}; .. math:: s_{vv} = \frac{\partial^{2}\mathbf{surface}(u,v)}{\partial v^{2}}; .. math:: s_{uv} = \frac{\partial^{2}\mathbf{surface}(u,v)}{\partial u \partial v} = \frac{\partial^{2}\mathbf{surface}(u,v)}{\partial v \partial u}; .. math:: s_{uuu} = \frac{\partial^{3}\mathbf{surface}(u,v)}{\partial u^{3}}; .. math:: s_{uuv} = \frac{\partial^{3}\mathbf{surface}(u,v)}{\partial u \partial u \partial v}; .. math:: s_{uvv} = \frac{\partial^{3}\mathbf{surface}(u,v)}{\partial u \partial v \partial v}; .. math:: s_{vvv} = \frac{\partial^{3}\mathbf{surface}(u,v)}{\partial v^{3}} for the partial derivatives of the surface with respect to its parameters. The primary method of the surface is void :blue:`PointOn`\( double \& *u*, double \& *v*, :ref:`MbCartPoint3D ` \& **s** ). It returns the radius vector **s**\ (*u*, *v*) of the surface point for the given parameters *u* and *v*. The methods void :blue:`DeriveU`\( double \& *u*, double \& *v*, :ref:`MbVector3D ` \& **s**\ :sub:`u` ), void :blue:`DeriveV`\( double \& *u*, double \& *v*, :ref:`MbVector3D ` \& **s**\ :sub:`v` ), void :blue:`DeriveUU`\( double \& *u*, double \& *v*, :ref:`MbVector3D ` \& **s**\ :sub:`uu` ), void :blue:`DeriveUV`\( double \& *u*, double \& *v*, :ref:`MbVector3D ` \& **s**\ :sub:`uv` ), void :blue:`DeriveVV`\( double \& *u*, double \& *v*, :ref:`MbVector3D ` \& **s**\ :sub:`vv` ), void :blue:`DeriveUUU`\( double \& *u*, double \& *v*, :ref:`MbVector3D ` \& **s**\ :sub:`uuu` ), void :blue:`DeriveUUV`\( double \& *u*, double \& *v*, :ref:`MbVector3D ` \& **s**\ :sub:`uuv` ), void :blue:`DeriveUVV`\( double \& *u*, double \& *v*, :ref:`MbVector3D ` \& **s**\ :sub:`uvv` ), void :blue:`DeriveVVV`\( double \& *u*, double \& *v*, :ref:`MbVector3D ` \& **s**\ :sub:`vvv` ) return the derivatives **s**\ :sub:`u`, **s**\ :sub:`v`, **s**\ :sub:`uu`, **s**\ :sub:`uv`, **s**\ :sub:`vv`, **s**\ :sub:`uuu`, **s**\ :sub:`uuv`, **s**\ :sub:`uvv`, **s**\ :sub:`vvv` of the surface radius vector for the given parameters *u* and *v*, respectively. The listed methods adjust the surface parameters when they go beyond the domain (the exception is the plane :ref:`MbPlane `). When the parameter *u* goes beyond the interval [*u*\ :sub:`min`, *u*\ :sub:`max`], surfaces that are non-periodic with respect to the first parameter shift the parameter *u* to the nearest boundary *u*\ :sub:`min` or *u*\ :sub:`max`, while surfaces that are periodic with respect to the first parameter add or subtract the required number of periods. When the parameter *v* goes beyond the interval [*v*\ :sub:`min`, *v*\ :sub:`max`], surfaces that are non-periodic with respect to the second parameter shift the parameter *v* to the nearest boundary *v*\ :sub:`min` or *v*\ :sub:`max`, while surfaces that are periodic with respect to the second parameter add or subtract the required number of periods. The method void :blue:`_PointOn`\( double *u*, double *v*, :blue:`MbCartPoint3D` \& **s** ) returns the radius vector **s**\ (*u*, *v*) of the surface point for the given parameters *u* and *v* both within the surface parameter domain and beyond it. Each non-periodic surface is extended beyond the parameter domain according to its own law. In the absence of such a law (in the general case), a non-periodic surface is extended beyond the parameter domain along the tangent to the corresponding boundary point of the surface. The methods void :blue:`_DeriveU`\( double *u*, double *v*, :ref:`MbVector3D ` \& **s**\ :sub:`u` ), void :blue:`_DeriveV`\( double *u*, double *v*, :ref:`MbVector3D ` \& **s**\ :sub:`v` ), void :blue:`_DeriveUU`\( double *u*, double *v*, :ref:`MbVector3D ` \& **s**\ :sub:`uu` ), void :blue:`_DeriveUV`\( double *u*, double *v*, :ref:`MbVector3D ` \& **s**\ :sub:`uv` ), void :blue:`_DeriveVV`\( double *u*, double *v*, :ref:`MbVector3D ` \& **s**\ :sub:`vv` ), void :blue:`_DeriveUUU`\( double *u*, double *v*, :ref:`MbVector3D ` \& **s**\ :sub:`uuu` ), void :blue:`_DeriveUUV`\( double *u*, double *v*, :ref:`MbVector3D ` \& **s**\ :sub:`uuv` ), void :blue:`_DeriveUVV`\( double *u*, double *v*, :ref:`MbVector3D ` \& **s**\ :sub:`uvv` ), void :blue:`_DeriveVVV`\( double *u*, double *v*, :ref:`MbVector3D ` \& **s**\ :sub:`vvv` ) return the derivatives **s**\ :sub:`u`, **s**\ :sub:`v`, **s**\ :sub:`uu`, **s**\ :sub:`uv`, **s**\ :sub:`vv`, **s**\ :sub:`uuu`, **s**\ :sub:`uuv`, **s**\ :sub:`uvv`, **s**\ :sub:`vvv` of the surface radius vector for the given parameters *u* and *v* both within the surface domain and beyond it. Surfaces override the following methods of a three-dimensional geometric object: methods for geometric object transformation, void :blue:`Move`\( const :ref:`MbVector3D ` \& **v**, MbRegTransform \* iReg = NULL ), void :blue:`Rotate`\( const MbAxis3D \& **axis**, double **angle**, MbRegTransform \* iReg = NULL ), void :blue:`Transform`\( const :ref:`MbMatrix3D ` \& **m**, MbRegTransform \* iReg = NULL ), methods for copying, checking for coincidence, checking for the possibility of making objects coincident, and making objects coincident, :blue:`MbSpaceItem` \& :blue:`Duplicate`\( MbRegDuplicate \* iReg = NULL ), bool :blue:`IsSame`\( const :ref:`MbSpaceItem ` \& **item** ), bool :blue:`IsSimilar`\( const :ref:`MbSpaceItem ` \& **item** ), bool :blue:`SetEqual`\( const :ref:`MbSpaceItem ` \& **item** ), methods returning a type from the geometric object enumeration, MbeSpaceType :blue:`IsA`\(), MbeSpaceType :blue:`Type`\(), MbeSpaceType :blue:`Family`\(), methods for retrieving and editing the internal data of an object, MbProperty \& :blue:`CreateProperty`\( MbePrompt name ), void :blue:`GetProperties`\( MbProperties \& *properties* ), void :blue:`SetProperties`\( MbProperties \& *properties* ), a method that fills the polygonal copy of a geometric object, void :blue:`CalculateWire`\( double sag, :ref:`MbMesh ` \& **mesh** ). Most surfaces have a rectangular parameter domain. Among all surfaces, :ref:`MbCurveBoundedSurface ` stands out as a universal surface. :ref:`MbCurveBoundedSurface ` has curvilinear boundaries and may have arbitrary cutouts inside. :blue:`MbCurveBoundedSurface` is built on the basis of any surface with a rectangular parameter domain.