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OpenCASCADE Trihedron Law

曲線的活動标架是《微分幾何》中一個很基礎的概念。有了曲線的活動标架，掃掠造型Sweep算法的實作有了一些思路。當給定一個輪廓線後，将輪廓線沿着路徑曲線掃掠可以了解為将輪廓線變換到曲線的活動标架中。 本文主要示範了Frenet活動标架的例子，讀者可以将GeomFill_TrihedronLaw其他的子類表示的其他類型活動标架自己實作，加深了解。

OpenCASCADE Trihedron Law

[email protected]

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标系，也稱為标架Frame，OpenCASCADE中叫Trihedron。這個局部坐标系随着曲線上點的運動而運動，是以也稱為活動坐标系。活動坐标系中各坐标軸的選取：

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确定後，通過叉乘即得到B。

OpenCASCADE Trihedron Law

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

定義一個活動标架有什麼作用呢？把這個問題先保留一下。本文先介紹OpenCASCADE中的标架規則Trihedron Law。

2.Trihedron Law

在OpenCASCADE中，類GeomFill_TrihedronLaw定義了曲線活動标架。其類圖如下所示：

OpenCASCADE Trihedron Law

Figure 2. Trihedron Law define Trihedron along a Curve

從基類GeomFill_TrihedronLaw派生出了各種标架，如：

l GeomFill_Fixed：固定的活動動标架，即标架沿着曲線移動時，标架的三個方向是固定的；

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

下面通過示例代碼來顯示出曲線上的Frenet标架，GeomFill_TrihedronLaw子類的用法類似。

/*
Copyright(C) 2018 Shing Liu([email protected])

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;
}

程式通過拟合幾個随機産生的點生成B樣條曲線，再将曲線按弧長等距采樣，将得到的參數計算出曲線上的點，及Frenet标架。再生成Draw腳本檔案，最後将生成的Draw腳本檔案trihedron.tcl加載到Draw Test Harness中顯示結果如下圖所示：

OpenCASCADE Trihedron Law

Figure 3. Frenet Frame

由上圖可知，局部坐标系的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

曲線的活動标架是《微分幾何》中一個很基礎的概念。有了曲線的活動标架，掃掠造型Sweep算法的實作有了一些思路。當給定一個輪廓線後，将輪廓線沿着路徑曲線掃掠可以了解為将輪廓線變換到曲線的活動标架中。

本文主要示範了Frenet活動标架的例子，讀者可以将GeomFill_TrihedronLaw其他的子類表示的其他類型活動标架自己實作，加深了解。

5. References

#include #pragma 輪廓線 局部坐标系
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