计算弧形段长度的公式

Question

考虑 class：

public class Point3D
{

    public double X;
    public double Y;
    public double Z;

    public Point3D(double x, double y, double z)
    {
        X = x;
        Y = y;
        Z = z;
    }

    public double DistanceTo(Point3D to)
    {
        double dX = Math.Abs(to.X - X);
        double dY = Math.Abs(to.Y - Y);
        double dZ = Math.Abs(to.Z - Z);
        return Math.Sqrt(dX * dX + dY * dY + dZ * dZ);
    }

}

和class：

public class Segment
{

    public Point3D From;
    public Point3D To;
    public double? Radius;

    public Segment(Point3D from, Point3D to, double? radius)
    {
        From = from;
        To = to;
        Radius = radius;
    }

    public double Length
    {
        get
        {
            double straightLength = From.DistanceTo(To);
            if (Radius == null)
                return straightLength;
            if (Radius < straightLength/ 2d)
                throw new Exception();
            // Compute the arcuate segment length
        }
    }

}

我想计算通过 From 和 To 3D 点的弧长（Radius）。

欢迎提供帮助！

Answer 1

根据http://mathworld.wolfram.com/IsoscelesTriangle.html

straightLength / 2 = Radius * sin( 1/2 * angle)

因此：

angle = 2 * arcsin( straightLength / 2 / Radius)

和

arcLength = Radius * angle;

Answer 2

多一个参考： from math.stackexchange

S=弧长=r*theta, 其中 r= 半径和 Cos(theta)=（2 个向量的点积）/（这些向量的模数的乘积）

计算弧形段长度的公式

Formula to compute an arcuate segment length

c#

computational-geometry