Unity【Lerp & Slerp】- 线性插值与球形插值的区别

在Unity的向量Vector和四元数Quaternion类中，均包含线性插值Lerp和球形插值Slerp的函数，那么两者之间有何区别，通过下面的例子进行观察：

图一中黄色线与红色线相交的点是从点A到点B进行线性插值得出的结果，图二则是球形插值得出的结果，或许称之为弧形插值更容易理解。二者的区别从图中可以明显看出，从四元数的角度来看，线性插值每帧得出的旋转结果是不均匀的，从代数的角度思考，如果两个单位四元数之间进行插值，如图一中的线性插值，得到的四元数并不是单位四元数，因此球形插值更为合理，因为它是不改变长度的。

测试代码如下：

using UnityEngine;
using UnityEditor;

/// 
/// 线性插值
/// 
public class LerpExample : MonoBehaviour
{
    [SerializeField] private Transform a; //点A
    [SerializeField] private Transform b; //点B

    private void OnDrawGizmos()
    {
        Handles.Label(transform.position, "O");
        Handles.Label(a.position, "A");
        Handles.Label(b.position, "B");

        Handles.color = Color.cyan;
        //以transform.position作为点O
        //绘制OA线段
        Handles.DrawLine(transform.position, a.position);
        //绘制OB线段
        Handles.DrawLine(transform.position, b.position);

        for (int i = 1; i < 10; i++)
        {
            //插值点
            Vector3 l = Vector3.Lerp(a.position, b.position, i * .1f);
            Handles.color = Color.red;
            //绘制点O到插值点的线段
            Handles.DrawLine(transform.position, l);
            Handles.color = Color.yellow;
            //绘制插值点之间的线段
            Handles.DrawLine(l, Vector3.Lerp(a.position, b.position, (i - 1) * .1f));
            Handles.Label(l, $"线性插值{i}");
        }
        Handles.DrawLine(b.position, Vector3.Lerp(a.position, b.position, .9f));
    }
}

using UnityEngine;
using UnityEditor;

/// 
/// 球形插值
/// 
public class SlerpExample : MonoBehaviour
{
    [SerializeField] private Transform a; //点A
    [SerializeField] private Transform b; //点B

    private void OnDrawGizmos()
    {
        Handles.Label(transform.position, "O");
        Handles.Label(a.position, "A");
        Handles.Label(b.position, "B");

        Handles.color = Color.cyan;
        //以transform.position作为点O
        //绘制OA线段
        Handles.DrawLine(transform.position, a.position);
        //绘制OB线段
        Handles.DrawLine(transform.position, b.position);

        for (int i = 1; i < 10; i++)
        {
            //插值点
            Vector3 l = Vector3.Slerp(a.position, b.position, i * .1f);
            Handles.color = Color.red;
            //绘制点O到插值点的线段
            Handles.DrawLine(transform.position, l);
            Handles.color = Color.yellow;
            //绘制插值点之间的线段
            Handles.DrawLine(l, Vector3.Slerp(a.position, b.position, (i - 1) * .1f));
            Handles.Label(l, $"球形插值{i}");
        }
        Handles.DrawLine(b.position, Vector3.Slerp(a.position, b.position, .9f));
    }
}

在对Transform组件中的Position坐标和Rotation旋转进行插值运算时， 通常用Vector3中的插值函数去处理Position，用Quaternion中的插值函数去处理Rotation。

如果我们使用Vector3中的插值函数去处理Rotation，则会出现如下这种情况：

代码如下：

using UnityEngine;
using System.Collections;

public class Example : MonoBehaviour
{
    [SerializeField] private Transform point1;
    [SerializeField] private Transform point2;

    private IEnumerator ExampleCoroutine(Transform target)
    {
        float beginTime = Time.time;
        Vector3 beginPosition = transform.position;
        Vector3 beginRotation = transform.eulerAngles;
        for (var duration = 2f; (Time.time - beginTime) < duration;)
        {
            float t = (Time.time - beginTime) / duration;
            transform.position = Vector3.Lerp(beginPosition, target.position, t);
            transform.eulerAngles = Vector3.Lerp(beginRotation, target.eulerAngles, t);
            yield return null;
        }
        transform.position = target.position;
        transform.eulerAngles = target.rotation.eulerAngles;
    }
    private void OnGUI()
    {
        if (GUILayout.Button("POINT1", GUILayout.Width(200f), GUILayout.Height(50f)))
        {
            StartCoroutine(ExampleCoroutine(point1));
        }
        if (GUILayout.Button("POINT2", GUILayout.Width(200f), GUILayout.Height(50f)))
        {
            StartCoroutine(ExampleCoroutine(point2));
        }
    }
}

下面是使用Quaternion.Lerp得出的结果：

代码如下：

using UnityEngine;
using System.Collections;

public class Example : MonoBehaviour
{
    [SerializeField] private Transform point1;
    [SerializeField] private Transform point2;

    private IEnumerator ExampleCoroutine(Transform target)
    {
        float beginTime = Time.time;
        Vector3 beginPosition = transform.position;
        Quaternion beginRotation = transform.rotation;
        for (var duration = 2f; (Time.time - beginTime) < duration;)
        {
            float t = (Time.time - beginTime) / duration;
            transform.position = Vector3.Lerp(beginPosition, target.position, t);
            transform.rotation = Quaternion.Lerp(beginRotation, target.rotation, t);
            yield return null;
        }
        transform.position = target.position;
        transform.rotation = target.rotation;
    }
    private void OnGUI()
    {
        if (GUILayout.Button("POINT1", GUILayout.Width(200f), GUILayout.Height(50f)))
        {
            StartCoroutine(ExampleCoroutine(point1));
        }
        if (GUILayout.Button("POINT2", GUILayout.Width(200f), GUILayout.Height(50f)))
        {
            StartCoroutine(ExampleCoroutine(point2));
        }
    }
}