生成三次B样条曲线代码

下面是 C++ 代码实现:

#include <iostream>
#include <vector>
using namespace std;

// 计算 B-Spline 基函数
double BasisFunc(int i, int k, double t, vector<double>& KnotVec)
{
    double ti = KnotVec[i];
    double ti1 = KnotVec[i+1];
    if (k == 0)
    {
        if (t >= ti && t <= ti1) return 1;
        else return 0;
    }
    double Nik1 = BasisFunc(i, k-1, t, KnotVec);
    double Ni1k1 = BasisFunc(i+1, k-1, t, KnotVec);
    double coef1 = (t-ti)/(ti+k-i-1);
    double coef2 = (ti+k-i)/double(ti+k-i);
    double coef3 = (ti1-t)/(i+k-ti1);
    double coef4 = (i+k+1-t)/double(i+k+1-ti1);
    double result = coef1*Nik1 + coef2*Ni1k1 + coef3*Nik1 + coef4*Ni1k1;
    return result;
}

// 生成 B-Spline 曲线
vector<pair<double, double>> BSpline(int k, vector<double>& ControlPoints, vector<double>& KnotVec)
{
    vector<pair<double, double>> Result;
    int n = ControlPoints.size()-1;
    int m = KnotVec.size()-1;
    for (double t=KnotVec[k-1]; t<=KnotVec[n+1]; t+=0.001) // 以 0.001 为步长
    {
        double x = 0;
        double y = 0;
        for (int i=0; i<=n; i++)
        {
            double Bi = BasisFunc(i, k, t, KnotVec);
            x += Bi * ControlPoints[i*2];
            y += Bi * ControlPoints[i*2+1];
        }
        Result.push_back(make_pair(x, y));
    }
    return Result;
}

int main()
{
    int k = 3; // 三次 B-Spline 曲线
    vector<double> ControlPoints = {1,1, 2,3, 4,-3, 7,4, 10,0}; // 控制点坐标 (x,y)
    vector<double> KnotVec = {0, 0, 0, 1, 2, 3, 3, 3}; // 结点矢量
    auto Curve = BSpline(k, ControlPoints, KnotVec);
    for (auto& p : Curve)
    {
        cout << "(" << p.first << "," << p.second << ")" << " ";
    }
    cout << endl;
    return 0;
}

程序输出的是在给定结点矢量 `KnotVec` 下生成的 B-Spline 曲线的点序列。