OpenCASCADE Linear Extrusion Surface
Abstract. OpenCASCADE linear extrusion surface is a generalized cylinder. Such a surface is obtained by sweeping a curve (called the “extruded curve” or “basis”) in a given direction (referred to as the direction of extrusion and defined by a unit vector). The u parameter is along the extruded curve. The v parameter is along the direction of extrusion. The form of a surface of linear extrusion is generally a ruled surface. It can be a cylindrical surface, or a planar surface.
Key Words. OpenCASCADE, Extrusion Surface, Sweeping
1. Introduction
一般柱面（The General Cylinder）可以由一段或整个圆弧沿一个方向偏移一定的距离得到。如下图所示：
Figure 1.1 Extrusion Shapes
当将顶点拉伸时，会生成一条边；当将边拉伸时，会生成面；当将Wire拉伸时，会生成Shell，当将面拉伸时，会生成体。当将曲线沿一个方向拉伸时，会形成一个曲面，如果此方向为直线，则会生成一般柱面。如果此方向是曲线时，会生成如下图所示曲面：
Figure 1.2 Swept surface/ loft surface
本文主要介绍将曲线沿直线方向拉伸的算法，即一般柱面生成算法。并将生成的曲面在OpenSceneGraph中进行显示。
2.Cylinder Surface Definition
设W是一个单位向量，C(u)是定义在节点矢量U上，权值为wi的p次NURBS曲线。我们要得到一般柱面S(u,v)的表达式，S(u,v)是通过将C(u)沿方向W平行扫描（sweep）距离d得到的。记扫描方向的参数为v, 0<v<1，显然，S(u,v)必须满足以下两个条件：
v 对于固定的u0, S(u0, v)为由C(u0)到C(u0)+dW的直线段；
v 对于固定的v0：
所要求的柱面的表达式为：
S(u,v)定义在节点矢量U和V上，这里V＝{0，0，1，1}，U为C(u)的节点矢量。控制顶点由Pi,0=Pi和Pi,1=Pi+dW给出，权值wi,0=wi,1=wi。如下图所示为一般柱面：
Figure 2.1 A general cylinder obtained by translating C(u) a distance d along W.
其中OpenCASCADE中一般柱面的表达式如下所示：
其取值范围的代码如下所示：
// function : Bounds
// purpose :
// =======================================================================
void Geom_SurfaceOfLinearExtrusion::Bounds ( Standard_Real & U1,
Standard_Real & U2,
Standard_Real & V1,
Standard_Real & V2 ) const {
V1 = - Precision::Infinite(); V2 = Precision::Infinite();
U1 = basisCurve -> FirstParameter(); U2 = basisCurve -> LastParameter();
}
由上代码可知，参数在v方向上是趋于无穷的；在u方向上参数的范围为曲线的范围。计算柱面上点的方法代码如下所示：
// function : D0
// purpose :
// =======================================================================
void Geom_SurfaceOfLinearExtrusion::D0 ( const Standard_Real U,
const Standard_Real V,
Pnt & P) const {
XYZ Pxyz = direction.XYZ();
Pxyz.Multiply (V);
Pxyz.Add (basisCurve -> Value (U).XYZ());
P.SetXYZ(Pxyz);
}
即将柱面上点先按V方向来计算，再按U方向来计算，最后将两个方向的值相加即得到柱面上的点。
由上述代码可知，OpenCASCADE中一般柱面没有使用NURBS曲面来表示。根据这个方法，可以将任意曲线沿给定的方向来得到一个柱面，这个曲线可以是直线、圆弧、圆、椭圆等。关于柱面上更多算法，如求微分等，可以参考源程序。
3.Display the Surface
还是在OpenSceneGraph中来对一般柱面进行可视化，来验证结果。因为OpenSceneGraph的简单易用，显示曲面的程序代码如下所示：
* Copyright (c) 2013 to current year. All Rights Reserved.
*
* File : Main.cpp
* Author : eryar@163.com
* Date : 2014-11-23 10:18
* Version : OpenCASCADE6.8.0
*
* Description : Test the Linear Extrusion Surface of OpenCASCADE.
*
* Key Words : OpenCascade, Linear Extrusion Surface, General Cylinder
*
*/
// OpenCASCADE.
#define WNT
#include < Precision.hxx >
#include < gp_Circ.hxx >
#include < Geom_SurfaceOfLinearExtrusion.hxx >
#include < GC_MakeCircle.hxx >
#include < GC_MakeSegment.hxx >
#include < GC_MakeArcOfCircle.hxx >
#pragma comment(lib, " TKernel.lib " )
#pragma comment(lib, " TKMath.lib " )
#pragma comment(lib, " TKG3d.lib " )
#pragma comment(lib, " TKGeomBase.lib " )
// OpenSceneGraph.
#include < osgViewer / Viewer >
#include < osgViewer / ViewerEventHandlers >
#include < osgGA / StateSetManipulator >
#pragma comment(lib, " osgd.lib " )
#pragma comment(lib, " osgGAd.lib " )
#pragma comment(lib, " osgViewerd.lib " )
const double TOLERANCE_EDGE = 1e - 6 ;
const double APPROXIMATION_DELTA = 0.05 ;
/* *
* @brief Render 3D geometry surface.
*/
osg::Node * BuildSurface( const Handle_Geom_Surface & theSurface)
{
osg::ref_ptr < osg::Geode > aGeode = new osg::Geode();
Standard_Real aU1 = 0.0 ;
Standard_Real aV1 = 0.0 ;
Standard_Real aU2 = 0.0 ;
Standard_Real aV2 = 0.0 ;
Standard_Real aDeltaU = 0.0 ;
Standard_Real aDeltaV = 0.0 ;
theSurface -> Bounds(aU1, aU2, aV1, aV2);
// trim the parametrical space to avoid infinite space.
Precision::IsNegativeInfinite(aU1) ? aU1 = - 1.0 : aU1;
Precision::IsInfinite(aU2) ? aU2 = 1.0 : aU2;
Precision::IsNegativeInfinite(aV1) ? aV1 = - 1.0 : aV1;
Precision::IsInfinite(aV2) ? aV2 = 1.0 : aV2;
// Approximation in v direction.
aDeltaU = (aU2 - aU1) * APPROXIMATION_DELTA;
aDeltaV = (aV2 - aV1) * APPROXIMATION_DELTA;
for (Standard_Real u = aU1; (u - aU2) <= TOLERANCE_EDGE; u += aDeltaU)
{
osg::ref_ptr < osg::Geometry > aLine = new osg::Geometry();
osg::ref_ptr < osg::Vec3Array > aPoints = new osg::Vec3Array();
for (Standard_Real v = aV1; (v - aV2) <= TOLERANCE_EDGE; v += aDeltaV)
{
gp_Pnt aPoint = theSurface -> Value(u, v);
aPoints -> push_back(osg::Vec3(aPoint.X(), aPoint.Y(), aPoint.Z()));
}
// Set vertex array.
aLine -> setVertexArray(aPoints);
aLine -> addPrimitiveSet( new osg::DrawArrays(osg::PrimitiveSet::LINE_STRIP, 0 , aPoints -> size()));
aGeode -> addDrawable(aLine. get ());
}
// Approximation in u direction.
for (Standard_Real v = aV1; (v - aV2) <= TOLERANCE_EDGE; v += aDeltaV)
{
osg::ref_ptr < osg::Geometry > aLine = new osg::Geometry();
osg::ref_ptr < osg::Vec3Array > aPoints = new osg::Vec3Array();
for (Standard_Real u = aU1; (u - aU2) <= TOLERANCE_EDGE; u += aDeltaU)
{
gp_Pnt aPoint = theSurface -> Value(u, v);
aPoints -> push_back(osg::Vec3(aPoint.X(), aPoint.Y(), aPoint.Z()));
}
// Set vertex array.
aLine -> setVertexArray(aPoints);
aLine -> addPrimitiveSet( new osg::DrawArrays(osg::PrimitiveSet::LINE_STRIP, 0 , aPoints -> size()));
aGeode -> addDrawable(aLine. get ());
}
return aGeode.release();
}
/* *
* @brief Build the test scene.
*/
osg::Node * BuildScene( void )
{
osg::ref_ptr < osg::Group > aRoot = new osg::Group();
// test the linear extrusion surface.
// test linear extrusion surface of a line.
Handle_Geom_Curve aSegment = GC_MakeSegment(gp_Pnt( 3.0 , 0.0 , 0.0 ), gp_Pnt( 6.0 , 0.0 , 0.0 ));
Handle_Geom_Surface aPlane = new Geom_SurfaceOfLinearExtrusion(aSegment, gp::DZ());
aRoot -> addChild(BuildSurface(aPlane));
// test linear extrusion surface of a arc.
Handle_Geom_Curve aArc = GC_MakeArcOfCircle(gp_Circ(gp::ZOX(), 2.0 ), 0.0 , M_PI, true );
Handle_Geom_Surface aSurface = new Geom_SurfaceOfLinearExtrusion(aArc, gp::DY());
aRoot -> addChild(BuildSurface(aSurface));
// test linear extrusion surface of a circle.
Handle_Geom_Curve aCircle = GC_MakeCircle(gp::XOY(), 1.0 );
Handle_Geom_Surface aCylinder = new Geom_SurfaceOfLinearExtrusion(aCircle, gp::DZ());
aRoot -> addChild(BuildSurface(aCylinder));
return aRoot.release();
}
int main( int argc, char * argv[])
{
osgViewer::Viewer aViewer;
aViewer.setSceneData(BuildScene());
aViewer.addEventHandler( new osgGA::StateSetManipulator(
aViewer.getCamera() -> getOrCreateStateSet()));
aViewer.addEventHandler( new osgViewer::StatsHandler);
aViewer.addEventHandler( new osgViewer::WindowSizeHandler);
return aViewer.run();
return 0 ;
}
上述显示方法只是显示线框的最简单的算法，只为验证一般柱面结果，不是高效算法。显示结果如下图所示：
Figure 3.1 General Cylinder for: Circle, Arc, Line
如上图所示分别为对圆、圆弧和直线进行拉伸得到的一般柱面。根据这个原理可以将任意曲线沿给定方向进行拉伸得到一个柱面。
4.Conclusion
通过对OpenCASCADE中一般柱面的类中代码进行分析可知，OpenCASCADE的这个线性拉伸柱面Geom_SurfaceOfLinearExtrusion是根据一般柱面的定义实现的，并不是使用NURBS曲面来表示的。当需要用NURBS曲面来表示一般柱面时，需要注意控制顶点及权值的计算取值。
5. References
1. 赵罡，穆国旺，王拉柱译. 非均匀有理B样条. 清华大学出版社. 2010
2. Les Piegl, Wayne Tiller. The NURBS Book. Springer-Verlag. 1997
3. OpenCASCADE Team, OpenCASCADE BRep Format. 2014
4. Donald Hearn, M. Pauline Baker. Computer Graphics with OpenGL. Prentice Hall. 2009
5. 莫蓉，常智勇. 计算机辅助几何造型技术. 科学出版社. 2009
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