OpenCASCADE Trihedron Law-卡核

OpenCASCADE Trihedron Law

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

参数曲线上的局部坐标系，也称为标架Frame，OpenCASCADE中叫Trihedron。这个局部坐标系随着曲线上点的运动而运动，所以也称为活动坐标系。活动坐标系中各坐标轴的选取：

l T是参数曲线的切线方向；

l N是曲线的主法线方向，或称主法矢；主法矢总是指向曲线凹入的方向；

l B是副法矢；当T 和N确定后，通过叉乘即得到B。

Figure 1. T, N, B frame of a curve (from wiki)

定义一个活动标架有什么作用呢？把这个问题先保留一下。本文先介绍OpenCASCADE中的标架规则Trihedron Law。

2.Trihedron Law

在OpenCASCADE中，类GeomFill_TrihedronLaw定义了曲线活动标架。其类图如下所示：

Figure 2. Trihedron Law define Trihedron along a Curve

从基类GeomFill_TrihedronLaw派生出了各种标架，如：

l GeomFill_Fixed：固定的活动动标架，即标架沿着曲线移动时，标架的三个方向是固定的；

l GeomFill_Frenet： Frenet标架；

l GeomFill_Darboux ：Darboux标架；

l GeomFill_ConstantBiNormal：副法矢固定的标架；

3. Code Demo

下面通过示例代码来显示出曲线上的Frenet标架，GeomFill_TrihedronLaw子类的用法类似。

/*
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
in the Software without restriction, including without limitation the rights
to use, copy, modify, merge, publish, distribute, sublicense, and / or sell
copies of the Software, and to permit persons to whom the Software is
furnished to do so, subject to the following conditions :

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;
}