OpenCASCADE Trihedron Law
eryar@163.com
Abstract. In differential geometry the Frenet-Serret formulas describe the kinematic properties of a particle moving along a continuous, differentiable curve in 3d space, or the geometric properties of the curve itself irrespective of any motion. More specifically, the formulas describe the derivatives of the so-called Tangent, Normal and Binormal unit vectors in terms of each other.
Key Words. Frenet-Serret Frame, TNB frame, Trihedron Law
1. Introduction
參數曲線上的局部坐標系，也稱為標架Frame，OpenCASCADE中叫Trihedron。這個局部坐標系隨著曲線上點的運動而運動，所以也稱為活動坐標系?；顒幼鴺讼抵懈髯鴺溯S的選取：
l T是參數曲線的切線方向；
l N是曲線的主法線方向，或稱主法矢；主法矢總是指向曲線凹入的方向；
l B是副法矢；當T 和N確定后，通過叉乘即得到B。
Figure 1. T, N, B frame of a curve (from wiki)
定義一個活動標架有什么作用呢？把這個問題先保留一下。本文先介紹OpenCASCADE中的標架規則Trihedron Law。
2.Trihedron Law
在OpenCASCADE中，類GeomFill_TrihedronLaw定義了曲線活動標架。其類圖如下所示：
Figure 2. Trihedron Law define Trihedron along a Curve
從基類GeomFill_TrihedronLaw派生出了各種標架，如：
l GeomFill_Fixed：固定的活動動標架，即標架沿著曲線移動時，標架的三個方向是固定的；
l GeomFill_Frenet： Frenet標架；
l GeomFill_Darboux ：Darboux標架；
l GeomFill_ConstantBiNormal：副法矢固定的標架；
3. Code Demo
下面通過示例代碼來顯示出曲線上的Frenet標架，GeomFill_TrihedronLaw子類的用法類似。
/*
Copyright(C) 2018 Shing Liu(eryar@163.com)
Permission is hereby granted, free of charge, to any person obtaining a copy
of this software and associated documentation files(the "Software"), to deal
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copies of the Software, and to permit persons to whom the Software is
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The above copyright notice and this permission notice shall be included in all
copies or substantial portions of the Software.
THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.IN NO EVENT SHALL THE
AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
SOFTWARE.
*/
#include <TColgp_Array1OfPnt.hxx>
#include <math_BullardGenerator.hxx>
#include <GCPnts_UniformAbscissa.hxx>
#include <GCPnts_UniformDeflection.hxx>
#include <GCPnts_TangentialDeflection.hxx>
#include <GCPnts_QuasiUniformDeflection.hxx>
#include <Geom_BSplineCurve.hxx>
#include <GeomAdaptor_HCurve.hxx>
#include <GeomAPI_PointsToBSpline.hxx>
#include <GeomFill_Fixed.hxx>
#include <GeomFill_Frenet.hxx>
#include <GeomFill_ConstantBiNormal.hxx>
#include <GeomFill_CorrectedFrenet.hxx>
#include <GeomFill_Darboux.hxx>
#include <GeomFill_DiscreteTrihedron.hxx>
#include <GeomFill_GuideTrihedronAC.hxx>
#include <GeomFill_GuideTrihedronPlan.hxx>
#include <BRepBuilderAPI_MakeEdge.hxx>
#include <BRepTools.hxx>
#pragma comment(lib, "TKernel.lib")
#pragma comment(lib, "TKMath.lib")
#pragma comment(lib, "TKG2d.lib")
#pragma comment(lib, "TKG3d.lib")
#pragma comment(lib, "TKGeomBase.lib")
#pragma comment(lib, "TKGeomAlgo.lib")
#pragma comment(lib, "TKBRep.lib")
#pragma comment(lib, "TKTopAlgo.lib")
void test()
{
TColgp_Array1OfPnt aPoints(1, 6);
math_BullardGenerator aBullardGenerator;
for (Standard_Integer i = aPoints.Lower(); i <= aPoints.Upper(); ++i)
{
Standard_Real aX = aBullardGenerator.NextReal() * 50.0;
Standard_Real aY = aBullardGenerator.NextReal() * 50.0;
Standard_Real aZ = aBullardGenerator.NextReal() * 50.0;
aPoints.SetValue(i, gp_Pnt(aX, aY, aZ));
}
GeomAPI_PointsToBSpline aBSplineFitter(aPoints);
if (!aBSplineFitter.IsDone())
{
return;
}
std::ofstream aTclFile("d:/tcl/trihedron.tcl");
aTclFile << std::fixed;
aTclFile << "vclear" << std::endl;
Handle(Geom_BSplineCurve) aBSplineCurve = aBSplineFitter.Curve();
Handle(GeomAdaptor_HCurve) aCurveAdaptor = new GeomAdaptor_HCurve(aBSplineCurve);
BRepBuilderAPI_MakeEdge anEdgeMaker(aBSplineCurve);
BRepTools::Write(anEdgeMaker, "d:/edge.brep");
aTclFile << "restore " << " d:/edge.brep e" << std::endl;
aTclFile << "incmesh e " << " 0.01" << std::endl;
aTclFile << "vdisplay e " << std::endl;
Handle(GeomFill_Frenet) aFrenet = new GeomFill_Frenet();
aFrenet->SetCurve(aCurveAdaptor);
GCPnts_UniformAbscissa aPointSampler(aCurveAdaptor->Curve(), 5.0);
for (Standard_Integer i = 1; i <= aPointSampler.NbPoints(); ++i)
{
Standard_Real aParam = aPointSampler.Parameter(i);
gp_Pnt aP = aCurveAdaptor->Value(aParam);
gp_Vec aT;
gp_Vec aN;
gp_Vec aB;
aFrenet->D0(aParam, aT, aN, aB);
// vtrihedron in opencascade draw 6.9.1
/*aTclFile << "vtrihedron vt" << i << " " << aP.X() << " " << aP.Y() << " " << aP.Z() << " "
<< " " << aB.X() << " " << aB.Y() << " " << aB.Z() << " "
<< " " << aT.X() << " " << aT.Y() << " " << aT.Z() << std::endl;*/
// vtrihedron in opencascade draw 7.1.0 has bug.
/*aTclFile << "vtrihedron vt" << i << " -origin " << aP.X() << " " << aP.Y() << " " << aP.Z() << " "
<< " -zaxis " << aB.X() << " " << aB.Y() << " " << aB.Z() << " "
<< " -xaxis " << aT.X() << " " << aT.Y() << " " << aT.Z() << std::endl;*/
// vtrihedron in opencascade draw 7.2.0
aTclFile << "vtrihedron vt" << i << " -origin " << aP.X() << " " << aP.Y() << " " << aP.Z() << " "
<< " -zaxis " << aB.X() << " " << aB.Y() << " " << aB.Z() << " "
<< " -xaxis " << aT.X() << " " << aT.Y() << " " << aT.Z() << std::endl;
aTclFile << "vtrihedron vt" << i << " -labels xaxis T 1" << std::endl;
aTclFile << "vtrihedron vt" << i << " -labels yaxis N 1" << std::endl;
aTclFile << "vtrihedron vt" << i << " -labels zaxis B 1" << std::endl;
aTclFile << "vsize vt" << i << " 2" << std::endl;
}
}
int main(int argc, char* argv[])
{
test();
return 0;
}
程序通過擬合幾個隨機產生的點生成B樣條曲線，再將曲線按弧長等距采樣，將得到的參數計算出曲線上的點，及Frenet標架。再生成Draw腳本文件，最后將生成的Draw腳本文件trihedron.tcl加載到Draw Test Harness中顯示結果如下圖所示：
Figure 3. Frenet Frame
由上圖可知，局部坐標系的T方向為曲線的切線方向。主法向N總是指向曲線凹側。
4. Conclusion
曲線的活動標架是《微分幾何》中一個很基礎的概念。有了曲線的活動標架，掃掠造型Sweep算法的實現有了一些思路。當給定一個輪廓線后，將輪廓線沿著路徑曲線掃掠可以理解為將輪廓線變換到曲線的活動標架中。
本文主要演示了Frenet活動標架的例子，讀者可以將GeomFill_TrihedronLaw其他的子類表示的其他類型活動標架自己實現，加深理解。
5. References
1. 趙罡, 穆國旺, 王拉柱. 非均勻有理B樣條. 清華大學出版社. 2010
2. 陳維桓. 微分幾何. 北京大學出版社. 2006
3. 朱心雄. 自由曲線曲面造型技術. 科學出版社. 2000
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