# User:KumaTea/Pages/008

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## 镜像与艺术

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## 二维空间

${\displaystyle {\begin{bmatrix}x'\\y'\end{bmatrix}}={\begin{bmatrix}\cos \theta &-\sin \theta \\+\sin \theta &\cos \theta \end{bmatrix}}{\begin{bmatrix}x\\y\end{bmatrix}}}$

${\displaystyle x'=x\cos \theta -y\sin \theta \,}$
${\displaystyle y'=x\sin \theta +y\cos \theta \,}$

${\displaystyle {\begin{bmatrix}x'\\y'\end{bmatrix}}={\begin{bmatrix}\cos \theta &+\sin \theta \\-\sin \theta &\cos \theta \end{bmatrix}}{\begin{bmatrix}x\\y\end{bmatrix}}}$

${\displaystyle x'=x\cos \theta +y\sin \theta \,}$
${\displaystyle y'=-x\sin \theta +y\cos \theta \,}$

### 复平面

${\displaystyle z=x+iy\,}$

z可逆时针旋转角度θ，通过乘以${\displaystyle e^{i\theta }}$（参见欧拉公式, §2）。

 ${\displaystyle e^{i\theta }z\;}$ ${\displaystyle =(\cos \theta +i\sin \theta )(x+iy)\;}$ ${\displaystyle =(x\cos \theta +iy\cos \theta +ix\sin \theta -y\sin \theta )\;}$ ${\displaystyle =(x\cos \theta -y\sin \theta )+i(x\sin \theta +y\cos \theta )\;}$ ${\displaystyle =x'+iy'.\;}$

## 注解

1. ^ Lounesto 2001, p.30.