Unity3D学习笔记（七）：叉乘和四元素

向量的叉乘：

数学运算：a(ax,ay,az) x b(bx,by,bz) = c(aybz-azby,azbx-axby,axby-aybx)

几何意义：得到一个新的向量，同时垂直于a向量和b向量，垂直于ab向量所组成的平面，c向量是ab平面的法向量

左手螺旋定则：四指指向a，握向b，大拇指指向c

作用

1、求顺指针，逆时针关系（y>0,y<0）

2、求平面的法向量

四元数（威廉·哈密顿）：

万向节死锁(Gimbal Lock):

https://v.qq.com/x/cover/e055516g79w/e055516g79w.html

http://www.cnitblog.com/luckydmz/archive/2010/09/07/68674.html，最全

四元数和欧拉角的优缺点：

欧拉角缺点：万向节死锁 http://v.youku.com/v_show/id_XNzkyOTIyMTI=.html

四元数：xyzw四个分量，w为实部

超复数：是由一个实部 + 三个虚部组成的，如4 + 2i + 3j + 4k

复数：实部 + 虚部，如3 + 2i，5 - 3i。复数的实部为零，即为虚数；虚部为零，即为实数。

----虚数：如果一个数的平方等于负一，那么这个数就是虚数单位，如x^2 = -1，3i，10k，9j

----实数：有理数和无理数的集合

--------有理数：有限的或者无限循环的，如1/3

--------无理数：无限不循环的小数，如PI，e，根号2

四元数中存储的是轴角对儿<n(x,y,z), theta>

x = n.x * sin(theta/2)

y = n.y * sin(theta/2)

z = n.z * sin(theta/2)

w = cos(theta/2)

比如：绕着y轴旋转90度：Quaternion(0,0.707,0,0.707)

1、我们用乘法来表示四元数的旋转量的叠加

欧拉角和四元数互转

public static Quaternion Euler(Vector3 euler);

public static Quaternion Euler(float x, float y, float z);

public static Quaternion LookRotation(Vector3 forward);

public static Quaternion LookRotation(Vector3 forward, [DefaultValue("Vector3.up")] Vector3 upwards);

插值：from + (to - from) * t

线性插值：

public static Vector3 Lerp(Vector3 a, Vector3 b, float t);

a：from是起始的位置

b：to是目标位置

t：在from到to之间插值

using System.Collections;

using System.Collections.Generic;

using UnityEngine;

public class LerpTest : MonoBehaviour

{

    public Transform sphere;

    // Use this for initialization

    void Start()

    {

    }

    // Update is called once per frame

    void Update()

    {

        //API：Vector3.Lerp，线性插值，起始位置from，目标位置to，每次插剩余距离(from-to)的百分之多少t

        transform.position = Vector3.Lerp(transform.position, sphere.position, 0.05f);

        //Lerp的匀速移动

        transform.position = Vector3.Lerp(transform.position, sphere.position, /Vector3.Distance(transform.position,sphere.position) * 0.05F);

        //Lerp的匀速旋转

        //API：Mathf.Lerp();

    }

}

判断方位

using System.Collections;

using System.Collections.Generic;

using UnityEngine;

public class EnemyLookPlayer : MonoBehaviour {

    public Transform player;

    public float result1;

    public float result2;

    // Use this for initialization

    void Start () {

        Vector3 forwardVector = transform.forward;

        Vector3 enemy2Player = player.position - transform.position;

        //result1 = enemy2Player.x * forwardVector.x + enemy2Player.y * forwardVector.y + enemy2Player.z + forwardVector.z;

        result1 = Vector3.Dot(forwardVector, enemy2Player);

        Vector3 result = Vector3.Cross(forwardVector, enemy2Player);

        result2 = result.y;

    }

       // Update is called once per frame

       void Update () {

    }

    public void OnGUI()

    {

        if (result1 > )

        {

            if (result2 > )

            {

                GUILayout.Label("玩家在我的方位：右前方");

            }

            else

            {

                GUILayout.Label("玩家在我的方位：左前方");

            }

        }

        else

        {

            if (result2 > )

            {

                GUILayout.Label("玩家在我的方位：右后方");

            }

            else

            {

                GUILayout.Label("玩家在我的方位：左后方");

            }

        }

    }

}

路点移动

using System.Collections;

using System.Collections.Generic;

using UnityEngine;

public class FindPath : MonoBehaviour {

    public Transform[] paths;

    public Vector3 dir;

    public float moveSpeed;

    public float rotSpeed;

    int index = ;

    // Use this for initialization

    void Start () {

       }

       // Update is called once per frame

       void Update () {

        if (Vector3.Distance(paths[index].position, transform.position) <= 0.5f)

        {

            index++;

            index %= paths.Length;

        }

        dir = paths[index].position - transform.position;

        transform.Translate(dir *Time.deltaTime* moveSpeed,Space.World);

        Quaternion targetRotation = Quaternion.LookRotation(dir);

        transform.rotation = Quaternion.Slerp(transform.rotation, targetRotation,/ Quaternion.Angle(transform.rotation, targetRotation)* rotSpeed);

       }

}

用单例类来管理路点

using System.Collections;

using System.Collections.Generic;

using UnityEngine;

public class PathManager : MonoBehaviour {

    //用单例类来管理路点

    public static PathManager instanse = null;

    public static PathManager Instanse

    {

        get

        {

            return instanse;

        }

    }

    int index = ;

    void Awake()

    {

        instanse = this;

    }

       // Use this for initialization

       void Start () {

        //transform.GetChild(1);

    }

    // Update is called once per frame

    void Update () {

       }

    public Vector3 GetCurrentPos()

    {

        return transform.GetChild(index).position;

    }

    public bool IsReached(Vector3 targetPos)

    {

        Vector3 currPos = GetCurrentPos();

        //忽略路点的高度

        currPos.y = targetPos.y;

        //根据步径调整0.5f的值

        return Vector3.Distance(currPos, targetPos) < 0.5f;

    }

    public void MoveNext()

    {

        index++;

        //index = index % transform.childCount;

        index %= transform.childCount;

    }

}

using System.Collections;

using System.Collections.Generic;

using UnityEngine;

public class QuaternionTest : MonoBehaviour

{

    public Transform girl;

    // Use this for initialization

    void Start()

    {

        //把旋转量赋值给transform.rotation

        transform.rotation = new Quaternion(, Mathf.Sin( * Mathf.Deg2Rad), , Mathf.Cos( * Mathf.Deg2Rad));

        //用乘法表示旋转量叠加

        transform.rotation *= new Quaternion(, Mathf.Sin( * Mathf.Deg2Rad), , Mathf.Cos( * Mathf.Deg2Rad));

        //API：Quaternion.AngleAxis，输入轴角对儿，返回四元数，如绕着y轴旋转90度

        transform.rotation = Quaternion.AngleAxis(, Vector3.up);

        //用乘法表示旋转量叠加，先绕着y轴旋转45度，再绕着y轴旋转90度，结果是-225（135）

        transform.rotation = Quaternion.AngleAxis(, Vector3.up) * Quaternion.AngleAxis(, Vector3.up);

        //四元数不满足乘法交换律

        //先绕着y轴旋转45度，再绕着x轴旋转45度，结果是（45，45，0）

        transform.rotation = Quaternion.AngleAxis(, Vector3.up) * Quaternion.AngleAxis(, Vector3.right);

        //先绕着x轴旋转45度，再绕着y轴旋转45度

        transform.rotation = Quaternion.AngleAxis(, Vector3.right) * Quaternion.AngleAxis(, Vector3.up);

    }

    // Update is called once per frame

    void Update()

    {

        //API：Quaternion LookRotation(Vector3 forward);

        Vector3 dir = girl.position - transform.position;

        //transform.rotation = Quaternion.LookRotation(dir);

        //目标位置

        Quaternion targetRotation = Quaternion.LookRotation(dir);

        //Slerp球形插值，每次0.01f慢慢插

        //transform.rotation = Quaternion.Slerp(transform.rotation, targetRotation, 0.01f);

        //

        transform.rotation = Quaternion.Slerp(transform.rotation, targetRotation,  / Quaternion.Angle(transform.rotation, targetRotation));

    }

}

using System.Collections;

using System.Collections.Generic;

using UnityEngine;

public class PlayerControl : MonoBehaviour

{

    public float moveSpeed = ;

    public float rotSpeed = ;

    public float radius = ;

    // Use this for initialization

    void Start()

    {

    }

    // Update is called once per frame

    void Update()

    {

        Vector3 targetPos = PathManager.Instanse.GetCurrentPos();

        Vector3 moveDir = targetPos - transform.position;

        moveDir.y = ;

        //目标与主角y值一致

        targetPos.y = transform.position.y;

        //移动

        transform.position = Vector3.Lerp(transform.position, targetPos,  / Vector3.Distance(transform.position, targetPos) * moveSpeed);

        //旋转

        transform.rotation = Quaternion.Slerp(transform.rotation, Quaternion.LookRotation(moveDir),  / Quaternion.Angle(transform.rotation, Quaternion.LookRotation(moveDir)) * rotSpeed);

        //画线

        Debug.DrawLine(transform.position, targetPos, Color.green);

        //判断是否达到

        if (PathManager.Instanse.IsReached(transform.position))

        {

            PathManager.Instanse.MoveNext();

        }

    }

    private void OnDrawGizmos()

    {

        Gizmos.color = new Color(, , , 0.3f);

        Gizmos.DrawSphere(transform.position, radius);

    }

}

补充内容

using System.Collections;

using System.Collections.Generic;

using UnityEngine;

public class w06d3 : MonoBehaviour {

    public Transform target;

    public Transform cam;

    public Vector3 camerPosOffset;

    public float hOffset;

    public float moveSpeed = ;

    // 1、向量加法的几何意义

    // 2、向量减法的几何意义

    // 3、向量的点乘的几何意义

    // a.b = a.x * b.x + a.y * b.y + a.z * b.z

    // a.b = |a| * |b| * cos<a, b>

    // a.b = (|b| * cos<a, b>) * |a|

    // a.b = (|a| * cos<a, b>) * |b|    a向量在b向量上的投影长度 * b向量的模长

    //                          b是单位向量的话，  a向量在b向量上的投影长度

    //                        a、b 都是单位向量的话，两个向量的夹角的余弦值

    void Start () {

       if( Vector3.Angle(transform.forward, target.position - transform.position) <  )

        {

        }

       }

       void Update () {

        cam.position = transform.position + camerPosOffset;

        cam.LookAt(transform.position + Vector3.up * hOffset);

        transform.Translate((target.position - transform.position).normalized * moveSpeed * Time.deltaTime, Space.World);

       }

    private void OnTriggerEnter(Collider other)

    {

    }

    private void OnCollisionEnter(Collision collision)

    {

    }

}

using System.Collections;

using System.Collections.Generic;

using UnityEngine;

public class Pool<T> where T : UnityEngine.Object

{

    public readonly int Capacity = ;

    List<T> items = new List<T>();

    public Pool(int capacity = )

    {

        this.Capacity = capacity;

    }

    public bool Push(T item)

    {

        if (items.Count >= Capacity - ) return false;

        items.Add(item); return true;

    }

    public T Pop()

    {

        if (items.Count == ) return default(T);

        T item = items[items.Count - ];

        items.RemoveAt(items.Count - );

        return item;

    }

}

巴特西

Unity3D学习笔记（七）：叉乘和四元素

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