最近项目中用到了贝塞尔曲线，学习完成后记录一下自己的学习结果吧。

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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