OpenCascade Ruled Surface
eryar@163.com
Abstract. A ruled surface is formed by moving a line connecting points of equal relative arc length or equal relative parametric value on two parametric curves from a start point to a terminate point on the curves. The paper focus on the ruled surface in opencascade.
Key words. OpenCascade, Ruled Surface,
1.Introduction
《解析几何》中有关于直纹面Ruled Surface的定义：一曲面S称为直纹面，如果存在一族直线使得这一族中的每一条直线全在S上。并且S上的每个点都在这一族的某一条直线上。这样一族直线称为S的一族直母线。其参数方程为：
即可以将直纹面看作是曲面对当v=0和1时得到的两个边界曲线之间进行线性插值得到的曲面。
Autodesk 3DS Max中的直纹面，图片来自：
https://knowledge.autodesk.com/support/3ds-max/learn-explore/caas/CloudHelp/cloudhelp/2015/ENU/3DSMax/files/GUID-364FE529-431B-448A-850B-DD9BBECAC90B-htm.html
直纹面是从两条曲线来构造曲面的方法，Coons曲面是由四边条界曲线来构造曲面，理解直纹面的构造原理，为进一步理解通用的放样Sweep造型打下基础。
2.Ruled Surface Parametric Equation
直纹面的参数方程也可以写成如下形式：
直接根据参数方程可以定义出相应的直纹面。在OpenCASCADE中话，可以从Geom_Surface派生新的类，并实现相应的虚函数。如实现计算对应参数u,v的值的虚函数D0()等。为了简单起见，用相应的函数计算直纹面上的点，并生成OpenCASCADE Draw Test Harness的命令脚本文件，方便在Draw中可视化。
如有名的Mobius Strip也是个直纹面：
Mobius Strip的参数方程为：
根据上述参数方程在OpenCASCADE的Draw生成Mobius Strip，代码如下所示：
const Standard_Real MOBIUS_RADIUS = 50.0;
void MobiusStrip(Standard_Real theU, Standard_Real theV, gp_Pnt& thePoint)
{
thePoint.SetX((MOBIUS_RADIUS + theU * Cos(0.5 * theV)) * Cos(theV));
thePoint.SetY((MOBIUS_RADIUS + theU * Cos(0.5 * theV)) * Sin(theV));
thePoint.SetZ(theU * Sin(0.5 * theV));
}
void TestMobiusStrip()
{
std::ofstream aTclFile("d:/mobius.tcl");
aTclFile << "pload ALL" << std::endl;
aTclFile << "vinit" << std::endl;
Standard_Real aWidth = 10.0;
Standard_Integer aN = 0;
for (Standard_Real s = -aWidth; s < aWidth; s += 1.0)
{
aTclFile << "polyline p" << ++aN ;
for (Standard_Real t = 0.0; t < M_PI * 2.0; t += 0.01)
{
gp_Pnt aPoint;
MobiusStrip(s, t, aPoint);
aTclFile << " " << aPoint.X() << " " << aPoint.Y() << " " << aPoint.Z();
}
aTclFile << "\n vdisplay p" << aN << std::endl;
}
for (Standard_Real t = 0.0; t < M_PI * 2.0; t += 0.2)
{
aTclFile << "polyline p" << ++aN;
gp_Pnt aPoint;
MobiusStrip(-aWidth, t, aPoint);
aTclFile << " " << aPoint.X() << " " << aPoint.Y() << " " << aPoint.Z();
MobiusStrip(aWidth, t, aPoint);
aTclFile << " " << aPoint.X() << " " << aPoint.Y() << " " << aPoint.Z();
aTclFile << "\n vdisplay p" << aN << std::endl;
}
}
int main(int argc, char* argv[])
{
TestMobiusStrip();
return 0;
}
在D盘生成一个mobius.tcl脚本文件，直接在Draw Test Harness中输入命令：
source d:/mobius.tcl
即可得到如下图所示的Mobius环：