Open CASCADE学习|Geom2d_BezierCurve 类

概述

Open CASCADE 提供了几何建模的强大工具集，其中 Geom2d_BezierCurve 类用于表示二维贝塞尔曲线。贝塞尔曲线在计算机图形学和计算机辅助设计（CAD）中具有广泛应用，本文将详细介绍 Geom2d_BezierCurve 类及其使用方法。

贝塞尔曲线简介

贝塞尔曲线是一种参数曲线，广泛应用于计算机图形学和相关领域。它由一组控制点定义，具有以下特点：

曲线的起始点和终点分别由第一个和最后一个控制点确定。
曲线的形状由中间的控制点控制。
贝塞尔曲线可以是有理的（带权重）或非有理的。

Geom2d_BezierCurve 类详解

基本概念

非有理贝塞尔曲线：由一组控制点（也称为极点）定义。
有理贝塞尔曲线：由一组带权重的控制点定义。如果所有权重相同，则曲线被视为非有理的。
次数：贝塞尔曲线的次数等于控制点的数量减去 1，并且不能超过系统定义的最大值（通常为 25）。

类的构造

#include <Geom2d_BezierCurve.hxx>

// 创建非有理贝塞尔曲线
Geom2d_BezierCurve(const TColgp_Array1OfPnt2d &CurvePoles);

// 创建有理贝塞尔曲线
Geom2d_BezierCurve(const TColgp_Array1OfPnt2d &CurvePoles, const TColStd_Array1OfReal &PoleWeights);

关键成员函数

曲线属性

Standard_Boolean IsClosed() const;
Standard_Boolean IsRational() const;
Standard_Integer Degree() const;
Standard_Integer NbPoles() const;

控制点操作

void SetPole(const Standard_Integer Index, const gp_Pnt2d &P);
void SetPole(const Standard_Integer Index, const gp_Pnt2d &P, const Standard_Real Weight);
void InsertPoleAfter(const Standard_Integer Index, const gp_Pnt2d &P, const Standard_Real Weight = 1.0);
void RemovePole(const Standard_Integer Index);

权重操作

void SetWeight(const Standard_Integer Index, const Standard_Real Weight);
Standard_Real Weight(const Standard_Integer Index) const;

曲线求导和点计算

void D0(const Standard_Real U, gp_Pnt2d &P) const;
void D1(const Standard_Real U, gp_Pnt2d &P, gp_Vec2d &V1) const;
void D2(const Standard_Real U, gp_Pnt2d &P, gp_Vec2d &V1, gp_Vec2d &V2) const;
void D3(const Standard_Real U, gp_Pnt2d &P, gp_Vec2d &V1, gp_Vec2d &V2, gp_Vec2d &V3) const;
gp_Vec2d DN(const Standard_Real U, const Standard_Integer N) const;

曲线变换

void Reverse();
Standard_Real ReversedParameter(const Standard_Real U) const;
void Transform(const gp_Trsf2d &T);

其他功能

void Segment(const Standard_Real U1, const Standard_Real U2);
void Increase(const Standard_Integer Degree);
Handle(Geom2d_Geometry) Copy() const;
void Resolution(const Standard_Real ToleranceUV, Standard_Real &UTolerance);

使用示例

完整代码示例

#include <iostream>
#include <iomanip>
#include <Geom2d_BezierCurve.hxx>
#include <TColgp_Array1OfPnt2d.hxx>
#include <gp_Pnt2d.hxx>
#include <gp_Vec2d.hxx>
#include <gp_Trsf2d.hxx>

using namespace std;

int main(int argc, char **argv) {
            
    // 设置控制精度
    const Standard_Real tolerance = 1e-7;

    // 创建一个简单的二维Bezier曲线实例
    TColgp_Array1OfPnt2d controlPoints(1, 4);
    controlPoints(1) = gp_Pnt2d(0.0, 0.0);
    controlPoints(2) = gp_Pnt2d(1.0, 2.0);
    controlPoints(3) = gp_Pnt2d(3.0, 1.0);
    controlPoints(4) = gp_Pnt2d(4.0, 3.0);

    // 创建非有理Bezier曲线
    Geom2d_BezierCurve bezierCurve(controlPoints);

    // 测试 IsClosed
    cout << "IsClosed: " << bezierCurve.IsClosed() << endl;

    // 测试 IsRational
    cout << "IsRational: " << bezierCurve.IsRational() << endl;

    // 测试 Degree
    cout << "Degree: " << bezierCurve.Degree() << endl;

    // 测试 NbPoles
    cout << "Number of Poles: " << bezierCurve.NbPoles() << endl;

    // 测试 StartPoint 和 EndPoint
    cout << "Start Point: " << bezierCurve.StartPoint().X() << ", " << bezierCurve.StartPoint().Y() << endl;
    cout << "End Point: " << bezierCurve.EndPoint().X() << ", " << bezierCurve.EndPoint().Y() << endl;

    // 测试 SetPole
    bezierCurve.SetPole(3, gp_Pnt2d(2.5, 1.5));

    // 测试 Pole
    cout << "Pole 3: " << bezierCurve.Pole(3).X() << ", " << bezierCurve.Pole(3).Y() << endl;

    // 测试 SetWeight
    TColStd_Array1OfReal weights(1, 4);
    weights(1) = 1.0;
    weights(2) = 1.0;
    weights(3) = 1.0;
    weights(4) = 1.0;

    bezierCurve.SetWeight(3, 2.0);

    // 测试 Weight
    cout << "Weight 3: " << bezierCurve.Weight(3) << endl;

    // 测试 IsRational 再次
    cout << "IsRational after setting weight: " << bezierCurve.IsRational() << endl;

    // 测试 InsertPoleAfter
    bezierCurve.InsertPoleAfter(3, gp_Pnt2d(2.5, 2.0), 1.5);

    cout << "Number of Poles after insertion: " << bezierCurve.NbPoles() << endl;

    // 测试 Increase
    bezierCurve.Increase(5);

    cout << "Degree after increasing: " << bezierCurve.Degree() << endl;

    // 测试 Reverse
    bezierCurve.Reverse();

    // 测试 ReversedParameter
    Standard_Real u = 0.5;
    Standard_Real reversedU = bezierCurve.ReversedParameter(u);
    cout << "Reversed Parameter for U=" << u << ": " << reversedU << endl;

    // 测试 D0, D1, D2, D3
    gp_Pnt2d p;
    gp_Vec2d v1, v2, v3;

    bezierCurve.D0(u, p);
    cout << "Point at U=" << u << ": " << p.X() << ", " << p.Y() << endl;

    bezierCurve.D1(u, p, v1);
    cout << "First derivative at U=" << u << ": " << v1.X() << ", " << v1.Y() << endl;

    bezierCurve.D2(u, p, v1, v2);
    cout << "Second derivative at U=" << u << ": " << v2.X() << ", " << v2.Y() << endl;

    bezierCurve.D3(u, p, v1, v2, v3);
    cout << "Third derivative at U=" << u << ": " << v3.X() << ", " << v3.Y() << endl;

    // 测试 DN
    gp_Vec2d dn = bezierCurve.DN(u, 4);
    cout << "Fourth derivative at U=" << u << ": " << dn.X() << ", " << dn.Y() << endl;

    // 测试 Segment
    bezierCurve.Segment(0.2, 0.8);

    // 测试 Transform
    gp_Trsf2d trsf;
    trsf.SetTranslation(gp_Vec2d(2.0, 3.0));
    bezierCurve.Transform(trsf);

    // 测试 Copy
    Handle(Geom2d_BezierCurve) copiedCurve = Handle(Geom2d_BezierCurve)::DownCast(bezierCurve.Copy());

    cout << "Copied curve Degree: " << copiedCurve->Degree() << endl;

    // 测试 Resolution
    Standard_Real paramTolerance;
    bezierCurve.Resolution(tolerance, paramTolerance);
    cout << "Parametric Tolerance: " << paramTolerance << endl;

    return 0;
}

输出示例

运行上述代码，预期的输出如下：

IsClosed: 0
IsRational: 0
Degree: 3
Number of Poles: 4
Start Point: 0, 0
End Point: 4, 3
Pole 3: 2.5, 1.5
Weight 3: 2
IsRational after setting weight: 1
Number of Poles after insertion: 5
Degree after increasing: 5
Reversed Parameter for U=0.5: 0.5
Point at U=0.5: 2.20833, 1.70833
First derivative at U=0.5: -2.43056, -1.09722
Second derivative at U=0.5: -4.78704, 0.101852
Third derivative at U=0.5: -39.6204, -34.7315
Fourth derivative at U=0.5: 8.7284, -102.086
Copied curve Degree: 5
Parametric Tolerance: 1.87927e-08