三次B样条曲线拟合C++

B样条曲线的方程：P=${\sum }_{i=0}^n{P}_i{F}_{i,k}(t)$
其中$F_{i,k}(t)$为基函数，三次B样条的基函数为：
$F_{0,3}(t)=\frac{1}{6}{(1-t)}^3$

$F_{1,3}(t)=\frac{1}{6}(3t^3-6t^2+4)$

$F_{2,3}(t)=\frac{1}{6}(-3t^3+3t^2+3t+1)$

$F_{3,3}(t)=\frac{1}{6}{t}^3$

所以，三次B样条的方程式为:
$P=P_0{F}_{0,3}(t)+P_1{F}_{1,3}(t)+P_2{F}_{2,3}(t)+P_3{F}_{3,3}(t)$
把基函数代入可以简化为：
$P=w_0+w_{1}t+w_2{t}^2+w_3{t}^3$ (0≤t<≤1)
其中，$w_0=\frac{1}{6}(P_0+4P_1+P_2)$

$w_1=-\frac{1}{2}(P_0-P_2)$

$w_2=\frac{1}{2}(P_0-2P_1+P_2)$

$w_3=-\frac{1}{6}(P_0-3P_1+3P_2-P_3)$

因此，每四个离散点就可拟合一段曲线，比如$P_0,P_1,P_2,P_3$可以拟合一段光滑曲线，$P_1,P_2,P_3,P_4$可以拟合下一段，相邻两段曲线是平滑过渡的，以此类推N个点可以拟合出N-3段平滑相接的曲线。
以上是非闭合曲线的拟合，闭合曲线只需离散点集首尾相连，也就是说，还需用$P_{N-1},P_0,P_1,P_2$拟合一段曲线。
c++代码如下：

/*B样条曲线拟合
@return 返回拟合得到的曲线
@discretePoints 输入的离散点，至少4个点
@closed 是否拟合闭合曲线，true表示闭合，false不闭合
@stride 拟合精度
*/
vector<Point2f> BSplineFit(vector<Point2f> discretePoints, bool closed, double stride = 0.01) {
	vector<Point2f> fittingPoints;
	for (int i = 0; i < (closed ? discretePoints.size() : discretePoints.size() - 1); i++) {
		Point2f xy[4];
		xy[0] = (discretePoints[i] + 4 * discretePoints[(i + 1) % discretePoints.size()] + discretePoints[(i + 2) % discretePoints.size()]) / 6;
		xy[1] = -(discretePoints[i] - discretePoints[(i + 2) % discretePoints.size()]) / 2;
		xy[2] = (discretePoints[i] - 2 * discretePoints[(i + 1) % discretePoints.size()] + discretePoints[(i + 2) % discretePoints.size()]) / 2;
		xy[3] = -(discretePoints[i] - 3 * discretePoints[(i + 1) % discretePoints.size()] + 3 * discretePoints[(i + 2) % discretePoints.size()] - discretePoints[(i + 3) % discretePoints.size()]) / 6;
		for (double t = 0; t <= 1; t += stride) {
			Point2f totalPoints = Point2f(0, 0);
			for (int j = 0; j < 4; j++) {
				totalPoints += xy[j] * pow(t, j);
			}
			fittingPoints.push_back(totalPoints);
		}
	}
	return fittingPoints;
}

非闭合拟合效果：

闭合曲线拟合效果：