參考：
原理
公式
應用

一.原理

貝塞爾曲線在起始點和終止點鎖定的情況下做均勻移動，
這意味著：

1. 在平面內選3個不同線的點并且依次用線段連接。如下所示

2. 在AB和BC線段上找出點D和點E，使得 AD/AB = BE/BC

3. 連接DE，在DE上尋找點F，F點需要滿足：DF/DE = AD/AB = BE/BC

4. 當D點從A到B、E點從B到C、F點從D到E時，F點走過的就是貝塞爾曲線

5. 以上是二階貝塞爾曲線，更高的也是同樣的原理

線性

三次

四次

二. 公式和運用

公式：

代碼：

using UnityEditor;
using UnityEngine;
public class BezierCurve : MonoBehaviour
{
    public Vector3[] points;

    public void Reset()
    {
        points = new Vector3[] {new Vector3(1f, 0f, 0f),
                new Vector3(2f, 0f, 0f),
                new Vector3(3f, 0f, 0f),
                new Vector3(4f, 0f, 0f)
            };
    }

    public Vector3 GetPoint_TwoPower(float t)
    {
        //父節點位置會變，所以在世界坐標下計算
        return transform.TransformPoint(BezierHelper.GetPoint_TwoPower(points[0], points[1], points[2], t));
    }

    public Vector3 GetPoint_ThreePower(float t)
    {
        //父節點位置會變，所以在世界坐標下計算
        return transform.TransformPoint(BezierHelper.GetPoint_ThreePower(points[0], points[1], points[2], points[3], t));
    }

    public Vector3 GetVelocity_TwoPower(float t)
    {
        //return transform.TransformPoint(Bezier.GetFirstDerivative(points[0], points[1], points[2], t)) - transform.position;
        return transform.TransformDirection(BezierHelper.GetVelocity_TwoPower(points[0], points[1], points[2], t));
    }

    public Vector3 GetVelocity_ThreePower(float t)
    {
        //return transform.TransformPoint(Bezier.GetFirstDerivative(points[0], points[1], points[2], t)) - transform.position;
        return transform.TransformDirection(BezierHelper.GetVelocity_ThreePower(points[0], points[1], points[2], points[3], t));
    }

    public Vector3 GetDirection(float t)
    {
        return GetVelocity_ThreePower(t).normalized;    //長度變小點
    }

    /// <summary>
    /// 貝塞爾曲線
    /// </summary>
    public static class BezierHelper
    {
        /// <summary>
        /// 二次方公式
        /// </summary>
        /// <param name="p0"></param>
        /// <param name="p1"></param>
        /// <param name="p2"></param>
        /// <param name="t"></param>
        /// <returns></returns>
        public static Vector3 GetPoint_TwoPower(Vector3 p0, Vector3 p1, Vector3 p2, float t)
        {
            return (1 - t) * (1 - t) * p0 + 2 * t * (1 - t) * p1 + t * t * p2;         //公式代替Lerp
                                                                                       //return Vector3.Lerp(Vector3.Lerp(p0, p1, t), Vector3.Lerp(p1, p2, t), t); //這個也可以
        }

        /// <summary>
        /// 獲取二次貝塞爾曲線的切線。也就是速度、也就是對公式求導。
        /// </summary>
        /// <param name="p0"></param>
        /// <param name="p1"></param>
        /// <param name="p2"></param>
        /// <param name="t"></param>
        /// <returns></returns>
        public static Vector3 GetVelocity_TwoPower(Vector3 p0, Vector3 p1, Vector3 p2, float t)
        {
            return
                2f * (1f - t) * (p1 - p0) +
                2f * t * (p2 - p1);
        }

        /// <summary>
        /// 三次方公式
        /// </summary>
        /// <param name="p0"></param>
        /// <param name="p1"></param>
        /// <param name="p2"></param>
        /// <param name="t"></param>
        /// <returns></returns>
        public static Vector3 GetPoint_ThreePower(Vector3 p0, Vector3 p1, Vector3 p2, Vector3 p3, float t)
        {
            t = Mathf.Clamp01(t);
            float oneMinusT = 1f - t;
            return
                oneMinusT * oneMinusT * oneMinusT * p0 +
                3f * oneMinusT * oneMinusT * t * p1 +
                3f * oneMinusT * t * t * p2 +
                t * t * t * p3;
        }

        /// <summary>
        /// 獲取三次貝塞爾曲線的切線。也就是速度、也就是對公式求導。
        /// </summary>
        /// <param name="p0"></param>
        /// <param name="p1"></param>
        /// <param name="p2"></param>
        /// <param name="t"></param>
        /// <returns></returns>
        public static Vector3 GetVelocity_ThreePower(Vector3 p0, Vector3 p1, Vector3 p2, Vector3 p3, float t)
        {
            t = Mathf.Clamp01(t);
            float oneMinusT = 1f - t;
            return
                3f * oneMinusT * oneMinusT * (p1 - p0) +
                6f * oneMinusT * t * (p2 - p1) +
                3f * t * t * (p3 - p2);
        }
    }
}

[CustomEditor(typeof(BezierCurve))]
public class BerzierCurveEditor : Editor
{
    BezierCurve curve;
    Transform handleTransform;
    Quaternion handleRotation;

    private const int lineSteps = 20;

    private void OnSceneGUI()
    {
        curve = target as BezierCurve;

        handleTransform = curve.transform;
        handleRotation = Tools.pivotRotation == PivotRotation.Local ? handleTransform.rotation : Quaternion.identity;   //處理Local和Global的切換時

        //處理Scene上的點
        Vector3 p0 = ShowPoint(0);
        Vector3 p1 = ShowPoint(1);
        Vector3 p2 = ShowPoint(2);
        Vector3 p3 = ShowPoint(3);

        //畫第一個點
        Vector3 lineStart = curve.GetPoint_TwoPower(0f);
        Handles.color = Color.gray;
        Handles.DrawLine(lineStart, lineStart + curve.GetDirection(0f));

        //lineSteps越大，曲線越平滑
        for (int i = 1; i <= lineSteps; i++)
        {
            //畫點
            Vector3 lineEnd = curve.GetPoint_ThreePower(i / (float)lineSteps);
            Handles.color = Color.red;
            Handles.DrawLine(lineStart, lineEnd);

            //畫切線
            Handles.color = Color.gray;
            Handles.DrawLine(lineEnd, lineEnd + curve.GetDirection(i / (float)lineSteps));

            lineStart = lineEnd;
        }
    }

    private Vector3 ShowPoint(int index)
    {
        Vector3 point = handleTransform.TransformPoint(curve.points[index]);    //世界坐標下的點
        EditorGUI.BeginChangeCheck();                           //檢查在代碼塊內更改了的任何控件。
        point = Handles.PositionHandle(point, handleRotation);  //處理點的移動

        if (EditorGUI.EndChangeCheck()) //如果有改動
        {
            Undo.RecordObject(curve, "Move Point");//在撤消菜單中可見
                                                   //EditorUtility.SetDirty(curve);
            curve.points[index] = handleTransform.InverseTransformPoint(point); //本地坐標的點
        }
        return point;
    }
}