如何理解超平面？

刺骨的言语ヽ痛彻心扉 2023-07-18 14:22 4阅读 0赞

## 超平面的公式 ##

首先明确几个定义：(1) 超平面是指n维线性空间中维度为n-1的子空间。它可以把线性空间分割成不相交的两部分。比如二维空间中，一条直线是一维的，它把平面分成了两块；三维空间中，一个平面是二维的，它把空间分成了两块。(2) 法向量是指垂直于超平面的向量。

在 ![R^\{3\}][R_3] 空间中，假如有法向量**![\\large \\omega][large _omega]**，过原点的平面内任意原点出发的向量 x 必然与之满足

![\\large \\omega ^\{T\}\\ \\cdot x = 0][large _omega _T_ _cdot x _ 0]

。如果平面沿着法向量的方向上下平移了，那么这个方程就不成立了。

我们假设平移之后平面经过 ![\\large x^\{'\}(x\_\{1\}^\{'\},x\_\{2\}^\{'\},x\_\{3\}^\{'\})][large x_x_1_x_2_x_3]，平面内任意一点记为 ![\\large x(x\_\{1\},x\_\{2\},x\_\{3\})][large x_x_1_x_2_x_3 1]，法向量记为 ![\\large \\omega (\\omega \_\{1\},\\omega \_\{2\},\\omega \_\{3\})][large _omega _omega _1_omega _2_omega _3]，如下图。

![format_png][]

平面公式示意图

不难看出，![\\large x-x^\{'\}][large x-x]在平面内，当然也就和法向量垂直。于是我们有：

![\\large (x-x^\{'\})\\omega = 0][large _x-x_omega _ 0]

![\\large (x\_\{1\}-x\_\{1\}^\{'\},x\_\{2\}-x\_\{2\}^\{'\},x\_\{3\}-x\_\{3\}^\{'\})\\cdot (\\omega \_\{1\},\\omega \_\{2\},\\omega \_\{3\})) = 0][large _x_1_-x_1_x_2_-x_2_x_3_-x_3_cdot _omega _1_omega _2_omega _3_ _ 0]                                                        
化简后得：

![\\large x\_\{1\}\\omega \_\{1\} + x\_\{2\}\\omega \_\{2\} + x\_\{3\}\\omega \_\{3\} = x\_\{1\}^\{'\}\\omega \_\{1\} + x\_\{2\}^\{'\}\\omega \_\{2\} + x\_\{3\}^\{'\}\\omega \_\{3\}][large x_1_omega _1_ _ x_2_omega _2_ _ x_3_omega _3_ _ x_1_omega _1_ _ x_2_omega _2_ _ x_3_omega _3]。

即

![\\large \\omega ^\{T\}x = \\omega ^\{T\}x^\{'\}][large _omega _T_x _ _omega _T_x]

。由于其为常数项，令

![\\large b = -\\omega ^\{T\}x^\{'\}][large b _ -_omega _T_x]

，于是超平面的公式可以写成：

![\\large \\omega ^\{T\}x + b = 0][large _omega _T_x _ b _ 0]

这个结论同样适用于 ![\\large R^\{n\}][large R_n] 空间；

1.  无论超平面如何平移，系数始终是法向量 ![\\large \\omega][large _omega 1]。

## 点到超平面的距离 ##

![format_png 1][]

点到超平面距离

上图中 ![\\large x][large x] 是平面外的一点。我们要求的距离记为 ![\\large d][large d] ，也就是红色的线段。根据三角函数可以得到：

![\\large cos\\theta = \\frac\{d\}\{||x-x^\{'\}||\}][large cos_theta _ _frac_d_x-x]

（空间中一点向超平面作垂线，![\\large \\theta][large _theta] 只能是锐角，不必担心正负）。因为 ![\\large d][large d]肯定和法向量平行，所以这样来算夹角：

![\\large |(x-x^\{'\})\\omega | = ||x-x^\{'\}||\\cdot ||\\omega ||cos\\theta][large _x-x_omega _ _ _x-x_cdot _omega _cos_theta]

（因为法向量可能反向，所以给等式左边加上绝对值），联立得：

![\\large d = \\frac\{|(x-x^\{'\})\\omega |\}\{||\\omega ||\} = \\frac\{|\\omega x - \\omega x^\{'\}|\}\{||\\omega ||\}][large d _ _frac_x-x_omega _omega _ _ _frac_omega x - _omega x_omega]  
因为 ![\\large x^\{'\}][large x 1]在超平面内，![\\large \\omega x^\{'\} = -b][large _omega x_ _ -b]，于是最后得到的任意点到超平面的距离公式：

![\\large d = \\frac\{|\\omega x + b|\}\{||\\omega ||\}][large d _ _frac_omega x _ b_omega]  
  
作者：老坛酸菜君  
链接：https://www.jianshu.com/p/ba02b92baaaf  
来源：简书  
著作权归作者所有。商业转载请联系作者获得授权，非商业转载请注明出处。

[R_3]: https://private.codecogs.com/gif.latex?R%5E%7B3%7D
[large _omega]: https://private.codecogs.com/gif.latex?%5Clarge%20%5Comega
[large _omega _T_ _cdot x _ 0]: https://private.codecogs.com/gif.latex?%5Clarge%20%5Comega%20%5E%7BT%7D%5C%20%5Ccdot%20x%20%3D%200
[large x_x_1_x_2_x_3]: https://private.codecogs.com/gif.latex?%5Clarge%20x%5E%7B%27%7D%28x_%7B1%7D%5E%7B%27%7D%2Cx_%7B2%7D%5E%7B%27%7D%2Cx_%7B3%7D%5E%7B%27%7D%29
[large x_x_1_x_2_x_3 1]: https://private.codecogs.com/gif.latex?%5Clarge%20x%28x_%7B1%7D%2Cx_%7B2%7D%2Cx_%7B3%7D%29
[large _omega _omega _1_omega _2_omega _3]: https://private.codecogs.com/gif.latex?%5Clarge%20%5Comega%20%28%5Comega%20_%7B1%7D%2C%5Comega%20_%7B2%7D%2C%5Comega%20_%7B3%7D%29
[format_png]: /images/20230528/a15440ab657546a2a1192cfdfb9da392.png
[large x-x]: https://private.codecogs.com/gif.latex?%5Clarge%20x-x%5E%7B%27%7D
[large _x-x_omega _ 0]: https://private.codecogs.com/gif.latex?%5Clarge%20%28x-x%5E%7B%27%7D%29%5Comega%20%3D%200
[large _x_1_-x_1_x_2_-x_2_x_3_-x_3_cdot _omega _1_omega _2_omega _3_ _ 0]: https://private.codecogs.com/gif.latex?%5Clarge%20%28x_%7B1%7D-x_%7B1%7D%5E%7B%27%7D%2Cx_%7B2%7D-x_%7B2%7D%5E%7B%27%7D%2Cx_%7B3%7D-x_%7B3%7D%5E%7B%27%7D%29%5Ccdot%20%28%5Comega%20_%7B1%7D%2C%5Comega%20_%7B2%7D%2C%5Comega%20_%7B3%7D%29%29%20%3D%200
[large x_1_omega _1_ _ x_2_omega _2_ _ x_3_omega _3_ _ x_1_omega _1_ _ x_2_omega _2_ _ x_3_omega _3]: https://private.codecogs.com/gif.latex?%5Clarge%20x_%7B1%7D%5Comega%20_%7B1%7D%20&plus;%20x_%7B2%7D%5Comega%20_%7B2%7D%20&plus;%20x_%7B3%7D%5Comega%20_%7B3%7D%20%3D%20x_%7B1%7D%5E%7B%27%7D%5Comega%20_%7B1%7D%20&plus;%20x_%7B2%7D%5E%7B%27%7D%5Comega%20_%7B2%7D%20&plus;%20x_%7B3%7D%5E%7B%27%7D%5Comega%20_%7B3%7D
[large _omega _T_x _ _omega _T_x]: https://private.codecogs.com/gif.latex?%5Clarge%20%5Comega%20%5E%7BT%7Dx%20%3D%20%5Comega%20%5E%7BT%7Dx%5E%7B%27%7D
[large b _ -_omega _T_x]: https://private.codecogs.com/gif.latex?%5Clarge%20b%20%3D%20-%5Comega%20%5E%7BT%7Dx%5E%7B%27%7D
[large _omega _T_x _ b _ 0]: https://private.codecogs.com/gif.latex?%5Clarge%20%5Comega%20%5E%7BT%7Dx%20&plus;%20b%20%3D%200
[large R_n]: https://private.codecogs.com/gif.latex?%5Cdpi%7B120%7D%20%5Clarge%20R%5E%7Bn%7D
[large _omega 1]: https://private.codecogs.com/gif.latex?%5Cdpi%7B120%7D%20%5Clarge%20%5Comega
[format_png 1]: /images/20230528/d9c9a413e13e49a985115a3094935655.png
[large x]: https://private.codecogs.com/gif.latex?%5Cdpi%7B120%7D%20%5Clarge%20x
[large d]: https://private.codecogs.com/gif.latex?%5Cdpi%7B120%7D%20%5Clarge%20d
[large cos_theta _ _frac_d_x-x]: https://private.codecogs.com/gif.latex?%5Cdpi%7B120%7D%20%5Clarge%20cos%5Ctheta%20%3D%20%5Cfrac%7Bd%7D%7B%7C%7Cx-x%5E%7B%27%7D%7C%7C%7D
[large _theta]: https://private.codecogs.com/gif.latex?%5Cdpi%7B120%7D%20%5Clarge%20%5Ctheta
[large _x-x_omega _ _ _x-x_cdot _omega _cos_theta]: https://private.codecogs.com/gif.latex?%5Cdpi%7B120%7D%20%5Clarge%20%7C%28x-x%5E%7B%27%7D%29%5Comega%20%7C%20%3D%20%7C%7Cx-x%5E%7B%27%7D%7C%7C%5Ccdot%20%7C%7C%5Comega%20%7C%7Ccos%5Ctheta
[large d _ _frac_x-x_omega _omega _ _ _frac_omega x - _omega x_omega]: https://private.codecogs.com/gif.latex?%5Cdpi%7B120%7D%20%5Clarge%20d%20%3D%20%5Cfrac%7B%7C%28x-x%5E%7B%27%7D%29%5Comega%20%7C%7D%7B%7C%7C%5Comega%20%7C%7C%7D%20%3D%20%5Cfrac%7B%7C%5Comega%20x%20-%20%5Comega%20x%5E%7B%27%7D%7C%7D%7B%7C%7C%5Comega%20%7C%7C%7D
[large x 1]: https://private.codecogs.com/gif.latex?%5Cdpi%7B120%7D%20%5Clarge%20x%5E%7B%27%7D
[large _omega x_ _ -b]: https://private.codecogs.com/gif.latex?%5Cdpi%7B120%7D%20%5Clarge%20%5Comega%20x%5E%7B%27%7D%20%3D%20-b
[large d _ _frac_omega x _ b_omega]: https://private.codecogs.com/gif.latex?%5Cdpi%7B120%7D%20%5Clarge%20d%20%3D%20%5Cfrac%7B%7C%5Comega%20x%20&plus;%20b%7C%7D%7B%7C%7C%5Comega%20%7C%7C%7D