image.png

对数学公式一直不感冒，今天看了个图发现这玩意原来挺简单的
https://blog.csdn.net/xiexian1204/article/details/49592765

暴力法求三阶贝塞尔曲线：

    public static Vector3 BezierCurve(Vector3 p0, Vector3 p1, Vector3 p2, Vector3 p3, float t)
    {
        //向量，三线
        Vector3 P01 = p1 - p0;  //向量0-1
        Vector3 P12 = p2 - p1;  //向量1-2
        Vector3 P23 = p3 - p2;  //向量2-3
        //根据t拆分向量，加上原始点坐标，为终点【坐标】
        Vector3 P01T = P01 * t + p0;
        Vector3 P12T = P12 * t + p1;
        Vector3 P23T = P23 * t + p2;

        //向量，三线变两线
        Vector3 P012 = P12T - P01T;
        Vector3 P123 = P23T - P12T;
        //根据t拆分向量，加上原始点坐标，为终点【坐标】
        Vector3 P012T = P012 * t + P01T;
        Vector3 P123T = P123 * t + P12T;

        //向量，两线变一线
        Vector3 P0123 = P123T - P012T;
        //根据t拆分向量，加上原始点坐标，为终点【坐标】
        Vector3 P0123T = P0123 * t + P012T;

        return P0123T;
    }

下面有温柔的数学公式：
https://blog.csdn.net/begonia__z/article/details/51199454

    public static Vector3 BezierCurve(Vector3 P0, Vector3 P1, Vector3 P2, Vector3 P3, float t)
    {
        Vector3 B = Vector3.zero;
        float t1 = (1 - t) * (1 - t) * (1 - t);
        float t2 = (1 - t) * (1 - t) * t;
        float t3 = t * t * (1 - t);
        float t4 = t * t * t;
        B = P0 * t1 + 3 * t2 * P1 + 3 * t3 * P2 + P3 * t4;
        //B.y = P0.y*t1 + 3*t2*P1.y + 3*t3*P2.y + P3.y*t4;
        //B.z = P0.z*t1 + 3*t2*P1.z + 3*t3*P2.z + P3.z*t4;
        return B;
    }

其实上面那个化简一下应该和下面的一样，不过下面的效率高还是用下面的吧。

二阶贝塞尔曲线

// p：当前点 t：当前进度(0-1) p0：起点 p1：中间点 p2：终点
p = (1-t)^2 * p0 + 2(1-t)tp1 + t^2p2

    /// <summary>
    /// 二阶贝塞尔
    /// </summary>
    /// <param name="t">0-1</param>
    /// <param name="p0">起点</param>
    /// <param name="p1">中间点</param>
    /// <param name="p2">终点</param>
    /// <returns></returns>
    public Vector3 GetQuadraticCurvePoint(float t, Vector3 p0, Vector3 p1, Vector3 p2)
    {
        float u = 1 - t;    //剩余进度
        float tt = t * t;   //进度*进度
        float uu = u * u;   //剩余进度*剩余进度
        return (uu * p0) + (2 * u * t * p1) + (tt * p2);
    }