unity(4)3D数学-笔记11:向量,向量运算(向量相加,向量相减,向量乘除标量),三角函数,向量运算(点乘,叉乘)

逃离我推掉我的手 2023-07-03 10:33 5阅读 0赞

av28779788?p=?视频学习笔记

*  **当前为-->第四部分：3D数学**

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2lzaGVuamlhbWluZw_size_16_color_FFFFFF_t_70][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2lzaGVuamlhbWluZw_size_16_color_FFFFFF_t_70 1][]

# 向量 #

![20200205140726746.png][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2lzaGVuamlhbWluZw_size_16_color_FFFFFF_t_70 2][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2lzaGVuamlhbWluZw_size_16_color_FFFFFF_t_70 3][]

## 向量的模 ##

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2lzaGVuamlhbWluZw_size_16_color_FFFFFF_t_70 4][]

求模,在unity中随便添加1个物体,再将下例脚本拖拽其上,运行结果如图:

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2lzaGVuamlhbWluZw_size_16_color_FFFFFF_t_70 5][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2lzaGVuamlhbWluZw_size_16_color_FFFFFF_t_70 6][]

如上,白线为模长--注:为方便,**本章代码中摄像机均位于世界坐标原点(0,0,0)**

## **向量的方向** ##

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2lzaGVuamlhbWluZw_size_16_color_FFFFFF_t_70 7][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2lzaGVuamlhbWluZw_size_16_color_FFFFFF_t_70 8][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2lzaGVuamlhbWluZw_size_16_color_FFFFFF_t_70 9][]

如上,红线为单位向量-向量方向-归一化向量.

# 向量运算(1) #

![20200205181123637.png][]

## 向量相减 ##

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2lzaGVuamlhbWluZw_size_16_color_FFFFFF_t_70 10][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2lzaGVuamlhbWluZw_size_16_color_FFFFFF_t_70 11][]

为了便于观察理解,以下2个物体Y坐标皆是0,运行后感觉结果怪怪的,改成ISO正交模式后观察就容易理解了:

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2lzaGVuamlhbWluZw_size_16_color_FFFFFF_t_70 12][]  
![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2lzaGVuamlhbWluZw_size_16_color_FFFFFF_t_70 13][]  
如上,红线为T1-T2;黄线为T2-T1.以下为草图辅助理解.虚线为我们理解的T2-T1但实际并不存在.因为所有向量起点必须是世界坐标原点0,0,0.

.![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2lzaGVuamlhbWluZw_size_16_color_FFFFFF_t_70 14][]

修改1下代码,再拖放1个物体T3到场景中,将T3和T2放一起.然后t3也挂在脚本上.下例代码按空格键t3(相当于子弹)会向t1移动:

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2lzaGVuamlhbWluZw_size_16_color_FFFFFF_t_70 15][]

![20200205190703842.png][]  
![2020020519074116.png][]  
![20200205190811165.png][]  
![20200205190829684.png][]

## 向量相加 ##

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2lzaGVuamlhbWluZw_size_16_color_FFFFFF_t_70 16][]

## 向量乘除标量 ##

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2lzaGVuamlhbWluZw_size_16_color_FFFFFF_t_70 17][]

# 三角函数 #

![20200205215917841.png][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2lzaGVuamlhbWluZw_size_16_color_FFFFFF_t_70 18][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2lzaGVuamlhbWluZw_size_16_color_FFFFFF_t_70 19][]  
![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2lzaGVuamlhbWluZw_size_16_color_FFFFFF_t_70 20][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2lzaGVuamlhbWluZw_size_16_color_FFFFFF_t_70 21][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2lzaGVuamlhbWluZw_size_16_color_FFFFFF_t_70 22][]  
![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2lzaGVuamlhbWluZw_size_16_color_FFFFFF_t_70 23][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2lzaGVuamlhbWluZw_size_16_color_FFFFFF_t_70 24][]  
![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2lzaGVuamlhbWluZw_size_16_color_FFFFFF_t_70 25][]  
如上,白线会随物体运动而变化(永远指向物体自身坐标的(0,0,10)).  
不管物体如何运动,红线的终点始终在世界坐标(0,0,10).  
\-------------------------------------------------------------------

练习:计算物体右前方30度,10米远的坐标(如图是横X轴,竖Z轴,Y轴忽略不计)

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2lzaGVuamlhbWluZw_size_16_color_FFFFFF_t_70 26][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2lzaGVuamlhbWluZw_size_16_color_FFFFFF_t_70 27][]  
如下,白线终点即为所求坐标点.

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2lzaGVuamlhbWluZw_size_16_color_FFFFFF_t_70 28][]

# 向量运算(2) #

![2020020622292090.png][]

## 点乘 ##

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2lzaGVuamlhbWluZw_size_16_color_FFFFFF_t_70 29][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2lzaGVuamlhbWluZw_size_16_color_FFFFFF_t_70 30][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2lzaGVuamlhbWluZw_size_16_color_FFFFFF_t_70 31][]

**注意:根据点乘结果计算夹角范围为0-180度.**

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2lzaGVuamlhbWluZw_size_16_color_FFFFFF_t_70 32][]

结果如下图:

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2lzaGVuamlhbWluZw_size_16_color_FFFFFF_t_70 33][]  
假如我们在代码中添加判断夹角是否大于60度的逻辑,以下两种写法都可以.其中第二种方法可以避免做一次arccos运算.

当然,也可以用查表法.即在程序开始之时把各个角度的正弦余弦等提前算好存入数组中.这样在游戏中需要时不必计算(因为计算通常都是在Update中频繁运算,比较浪费计算机资源)直接读取数组相应元素即可.相当于把正弦表余弦表存入计算机,需要时读取:

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2lzaGVuamlhbWluZw_size_16_color_FFFFFF_t_70 34][]

## 叉乘 ##

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2lzaGVuamlhbWluZw_size_16_color_FFFFFF_t_70 35][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2lzaGVuamlhbWluZw_size_16_color_FFFFFF_t_70 36][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2lzaGVuamlhbWluZw_size_16_color_FFFFFF_t_70 37][]  
结合点乘与叉乘获取两向量夹角,取值范围0-360度:

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2lzaGVuamlhbWluZw_size_16_color_FFFFFF_t_70 38][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2lzaGVuamlhbWluZw_size_16_color_FFFFFF_t_70 39][]

**注意:结合点乘与叉乘,可判断夹角在0-360度**(单独用叉乘获得的夹角在0-90度,一般不用)

练习:敌人对玩家发动攻击的逻辑  
![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2lzaGVuamlhbWluZw_size_16_color_FFFFFF_t_70 40][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2lzaGVuamlhbWluZw_size_16_color_FFFFFF_t_70 41][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2lzaGVuamlhbWluZw_size_16_color_FFFFFF_t_70 42][]

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[20200205190703842.png]: https://img-blog.csdnimg.cn/20200205190703842.png
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