Unity贝塞尔曲线的实现方法

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一阶贝塞尔曲线

一阶贝塞尔曲线就是一条线，我们很容易根据 t 求出 t 点的位置。

P(t)=P0+(P1-P0)*t =(1-t)*P0+tP1 ； t[ 0,1] ,且其等同于线性插值。

二阶贝塞尔曲线

取平面内三个不共线的点，AB:AC=CD:CE,这个时候BD又是一条直线，可以按照一阶的贝塞尔方程来进行线性插值了。

P(B)=(1-t)*P0+tP1 ;

P(D)=(1-t)P1+tP2 ;

P(t)=(1-t)*P(B)+tP(D)

=(1-t)*((1-t)*P0+tP1)+t((1-t)P1+tP2 )

=（1-t)² *P0+2t*(1-t)*P1+t²*P2 ;t[0,1];

代码：

public LineRenderer line_b;public LineRenderer line_a;public LineRenderer line_c; public Transform start;public Transform end;public Transform c;     void Start()    {         }    void Update()    {         line_a.SetPosition(0, start.position);        line_a.SetPosition(1, c.position);        line_c.SetPosition(0, end.position);        line_c.SetPosition(1, c.position);        // float distance = Vector3.Distance(start.position, end.position);        Vector3 controlPoint = c.position;            //start.position + (start.position+ c.position).normalized * distance / 1.6f;         Vector3[] bcList = GetBeizerPathPointList(start.position, controlPoint, end.position, 50);        line_b.positionCount = bcList.Length + 1;        line_b.SetPosition(0, start.position);        for (int i = 0; i < bcList.Length; i++)        {            Vector3 v = bcList[i];            line_b.SetPosition(i + 1, v);        }      }    public static Vector3[] GetBeizerPathPointList(Vector3 startPoint, Vector3 controlPoint, Vector3 endPoint, int pointNum)    {        Vector3[] BeizerPathPointList = new Vector3[pointNum];        for (int i = 1; i <= pointNum; i++)        {            float t = i / (float)pointNum;            Vector3 point = GetBeizerPathPoint(t, startPoint,                controlPoint, endPoint);            BeizerPathPointList[i - 1] = point;        }        return BeizerPathPointList;    }     //贝塞尔曲线二次方公式    private static Vector3 GetBeizerPathPoint(float t, Vector3 p0, Vector3 p1, Vector3 p2)    {        return (1 - t) * (1 - t) * p0 + 2 * t * (1 - t) * p1 + t * t * p2;    }

三阶贝塞尔曲线

三阶贝塞尔曲线和二阶其实是同一个道理，都可以按照一阶的贝塞尔方程来进行线性插值。这里就直接上公式了。

P(t)=P0*(1-t)³ +3P1*t*(1-t)²+3P2*t²*(1-t)+P3*t³ ; t[0,1];

代码

public Transform start;public Transform end;public Transform c0;public Transform c1;     public LineRenderer line_b;    public LineRenderer line_a;    public LineRenderer line_c;    public LineRenderer line_d;    void Start()    {            }     // Update is called once per frame    void Update()    {        line_a.SetPosition(0, start.position);        line_a.SetPosition(1, c0.position);         line_c.SetPosition(0, c1.position);        line_c.SetPosition(1, c0.position);         line_d.SetPosition(0, c1.position);        line_d.SetPosition(1, end.position);           Vector3[] bcList = GetBeizerPathPointList(start.position, c0.position,c1.position, end.position, 50);        line_b.positionCount = bcList.Length + 1;        line_b.SetPosition(0, start.position);        for (int i = 0; i < bcList.Length; i++)        {            Vector3 v = bcList[i];            line_b.SetPosition(i + 1, v);        }     }     public static Vector3[] GetBeizerPathPointList(Vector3 startPoint, Vector3 controlPoint0, Vector3 controlPoint1, Vector3 endPoint, int pointNum)    {        Vector3[] BeizerPathPointList = new Vector3[pointNum];        for (int i = 1; i <= pointNum; i++)        {            float t = i / (float)pointNum;            Vector3 point = GetBeizerPathPoint(t, startPoint,                controlPoint0, controlPoint1, endPoint);            BeizerPathPointList[i - 1] = point;        }        return BeizerPathPointList;    }       //贝塞尔曲线三次方公式    private static Vector3 GetBeizerPathPoint(float t, Vector3 p0, Vector3 p1, Vector3 p2,Vector3 p3)    {        return (1 - t) * (1 - t) * (1 - t) * p0 +                3 * p1 * t * (1 - t) * (1 - t) +                3 * p2 * t * t * (1 - t) +                p3 * t * t * t;    }

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