c# - 在 C# 中给定开始、结束和沿贝塞尔曲线的 1 个点时，查找 QuadraticBezierSegment 的控制点 - QuadraticBezier 3-point Interpolation

Question

这类似于我之前询问的关于三次贝塞尔曲线的问题。我有一个起点、一个终点和一个沿着二次贝塞尔曲线的点。鉴于这三个点，我希望能够在 WPF 中绘制一个 QuadraticBezierSegment，但我需要单个 ControlPoint 值（在 QuadraticBezierSegment 中它是 Point1）才能绘制它。

是否有计算或方法可以确定该值并因此绘制我的 QuadraticBezier？

谢谢！

Answer 1

最佳二次拟合比最佳三次拟合更简单。这是一些代码：

static class DrawingUtility
{
    static void bez3pts1(double x0, double y0, double x3, double y3, double x2, double y2, out double x1, out double y1)
    {
        // find chord lengths
        double c1 = Math.Sqrt((x3 - x0) * (x3 - x0) + (y3 - y0) * (y3 - y0));
        double c2 = Math.Sqrt((x3 - x2) * (x3 - x2) + (y3 - y2) * (y3 - y2));
        // guess "best" t
        double t = c1 / (c1 + c2);
        // quadratic Bezier is B(t) = (1-t)^2*P0 + 2*t*(1-t)*P1 + t^2*P2
        // solving gives P1 = [B(t) - (1-t)^2*P0 - t^2*P2] / [2*t*(1-t)] where P3 is B(t)
        x1 = (x3 - (1 - t) * (1 - t) * x0 - t * t * x2) / (2 * t * (1 - t));
        y1 = (y3 - (1 - t) * (1 - t) * y0 - t * t * y2) / (2 * t * (1 - t));
    }

    // pass in a PathFigure and it will append a QuadraticBezierSegment connecting the previous point to int1 and endPt
    static public void QuadraticBezierFromIntersection(PathFigure path, Point startPt, Point int1, Point endPt)
    {
        double x1, y1;
        bez3pts1(startPt.X, startPt.Y, int1.X, int1.Y, endPt.X, endPt.Y, out x1, out y1);
        path.Segments.Add(new QuadraticBezierSegment { Point1 = new Point(x1, y1), Point2 = endPt } );
    }
}