MbSurface Surface

The MbSurface class is declared in the surface.h file.

The MbSurface surface is an inheritor of the MbSpaceItem class, Fig. 1.

Fig. 1

The surface is an abstract class. The C3D geometric kernel implements the following surfaces, which are inheritors of the MbSurface class:

• MbPlane — plane,

• MbCylinderSurface — cylindrical surface,

• MbConeSurface — conical surface,

• MbSphereSurface — spherical surface,

• MbTorusSurface — toroidal surface,

• MbExtrusionSurface — extrusion surface,

• MbRevolutionSurface — revolution surface,

• MbExpansionSurface — plane-parallel kinematic surface,

• MbSpiralSurface — spiral surface,

• MbEvolutionSurface — swept surface,

• MbExactionSurface — swept surface with adaptation,

• MbSectorSurface — sectorial surface,

• MbRuledSurface — ruled surface,

• MbLoftedSurface — surface based on a family of curves,

• MbElevationSurface — surface based on a family of curves and a guiding curve,

• MbCornerSurface — surface based on three curves,

• MbCoverSurface — surface based on four curves,

• MbCoonsPatchSurface — bicubic Coons surface,

• MbMeshSurface — surface based on a network of curves,

• MbJoinSurface — joint surface,

• MbSplineSurface — NURBS surface (NonUniform Rational B-Spline surface),

• MbOffsetSurface — equidistant surface,

• MbChamferSurface — chamfer surface,

• MbFilletSurface — fillet surface,

• MbChannelSurface — fillet surface with variable radius,

• MbCurveBoundedSurface — surface with arbitrary borders.

The MbSurface surface is a vector function

\[\mathbf{surface}(u, v) = \begin{bmatrix} x(u,v) & y(u,v) & z(u,v) \end{bmatrix}\]

of two scalar parameters u and v taking values on a two-dimensional connected domain \(\Omega\). The surface is a continuous mapping of the two-dimensional connected domain \(\Omega\) into three-dimensional space. The domain \(\Omega\) is described in a two-dimensional Cartesian coordinate system. In the particular case, the domain \(\Omega\) is a rectangle, and the surface parameters take values within u_min \(\leq\) u \(\leq\) u_max, v_min \(\leq\) v \(\leq\) v_max. In the general case, the domain \(\Omega\) 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_min, u_max, v_min, v_max of the parameter domain are returned by the surface methods double GetUMin(), double GetUMax(), double GetVMin(), double GetVMax() respectively.

A surface is called periodic with respect to the first parameter if there exists p_u > 0 such that surface(u ± kp_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_v > 0 such that surface(u, v ± kp_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 IsUClosed() of a surface periodic with respect to the first parameter returns true.

The method bool IsVClosed() of a surface periodic with respect to the second parameter returns true.

The method double 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_u. The method double 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_v. The domain of a periodic surface parameter always lies within one period.

Let us introduce the notation

\[s_{u} = \frac{\partial\mathbf{surface}(u,v)}{\partial u};\]

\[s_{v} = \frac{\partial\mathbf{surface}(u,v)}{\partial v};\]

\[s_{uu} = \frac{\partial^{2}\mathbf{surface}(u,v)}{\partial u^{2}};\]

\[s_{vv} = \frac{\partial^{2}\mathbf{surface}(u,v)}{\partial v^{2}};\]

\[s_{uv} = \frac{\partial^{2}\mathbf{surface}(u,v)}{\partial u \partial v} = \frac{\partial^{2}\mathbf{surface}(u,v)}{\partial v \partial u};\]

\[s_{uuu} = \frac{\partial^{3}\mathbf{surface}(u,v)}{\partial u^{3}};\]

\[s_{uuv} = \frac{\partial^{3}\mathbf{surface}(u,v)}{\partial u \partial u \partial v};\]

\[s_{uvv} = \frac{\partial^{3}\mathbf{surface}(u,v)}{\partial u \partial v \partial v};\]

\[s_{vvv} = \frac{\partial^{3}\mathbf{surface}(u,v)}{\partial v^{3}}\]

for the partial derivatives of the surface with respect to its parameters.

The primary method of the surface is

void PointOn( double & u, double & v, MbCartPoint3D & s ).

It returns the radius vector s(u, v) of the surface point for the given parameters u and v.

The methods

void DeriveU( double & u, double & v, MbVector3D & s_u ),

void DeriveV( double & u, double & v, MbVector3D & s_v ),

void DeriveUU( double & u, double & v, MbVector3D & s_uu ),

void DeriveUV( double & u, double & v, MbVector3D & s_uv ),

void DeriveVV( double & u, double & v, MbVector3D & s_vv ),

void DeriveUUU( double & u, double & v, MbVector3D & s_uuu ),

void DeriveUUV( double & u, double & v, MbVector3D & s_uuv ),

void DeriveUVV( double & u, double & v, MbVector3D & s_uvv ),

void DeriveVVV( double & u, double & v, MbVector3D & s_vvv )

return the derivatives s_u, s_v, s_uu, s_uv, s_vv, s_uuu, s_uuv, s_uvv, s_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 MbPlane). When the parameter u goes beyond the interval [u_min, u_max], surfaces that are non-periodic with respect to the first parameter shift the parameter u to the nearest boundary u_min or u_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_min, v_max], surfaces that are non-periodic with respect to the second parameter shift the parameter v to the nearest boundary v_min or v_max, while surfaces that are periodic with respect to the second parameter add or subtract the required number of periods.

The method

void _PointOn( double u, double v, 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 _DeriveU( double u, double v, MbVector3D & s_u ),

void _DeriveV( double u, double v, MbVector3D & s_v ),

void _DeriveUU( double u, double v, MbVector3D & s_uu ),

void _DeriveUV( double u, double v, MbVector3D & s_uv ),

void _DeriveVV( double u, double v, MbVector3D & s_vv ),

void _DeriveUUU( double u, double v, MbVector3D & s_uuu ),

void _DeriveUUV( double u, double v, MbVector3D & s_uuv ),

void _DeriveUVV( double u, double v, MbVector3D & s_uvv ),

void _DeriveVVV( double u, double v, MbVector3D & s_vvv )

return the derivatives s_u, s_v, s_uu, s_uv, s_vv, s_uuu, s_uuv, s_uvv, s_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 Move( const MbVector3D & v, MbRegTransform * iReg = NULL ),

void Rotate( const MbAxis3D & axis, double angle, MbRegTransform * iReg = NULL ),

void Transform( const MbMatrix3D & m, MbRegTransform * iReg = NULL ),

methods for copying, checking for coincidence, checking for the possibility of making objects coincident, and making objects coincident,

MbSpaceItem & Duplicate( MbRegDuplicate * iReg = NULL ),

bool IsSame( const MbSpaceItem & item ),

bool IsSimilar( const MbSpaceItem & item ),

bool SetEqual( const MbSpaceItem & item ),

methods returning a type from the geometric object enumeration,

MbeSpaceType IsA(),

MbeSpaceType Type(),

MbeSpaceType Family(),

methods for retrieving and editing the internal data of an object,

MbProperty & CreateProperty( MbePrompt name ),

void GetProperties( MbProperties & properties ),

void SetProperties( MbProperties & properties ),

a method that fills the polygonal copy of a geometric object,

void CalculateWire( double sag, MbMesh & mesh ).

Most surfaces have a rectangular parameter domain. Among all surfaces, MbCurveBoundedSurface stands out as a universal surface. MbCurveBoundedSurface has curvilinear boundaries and may have arbitrary cutouts inside. MbCurveBoundedSurface is built on the basis of any surface with a rectangular parameter domain.