二维旋转矩阵

二维旋转矩阵

在这里插入图片描述
二维旋转是绕着某一个点进行旋转。如上图所示,蓝色线绕着O点旋转 θ \theta 角度到绿色线。
假设蓝色线为 l l 绿色线为 l l'
x = r c o s ϕ x = r c o s ( ϕ + θ ) = r c o s ϕ c o s θ r s i n ϕ s i n θ = x c o s θ y s i n θ x=rcos\phi \quad x'=rcos(\phi+\theta)=rcos\phi cos\theta-rsin\phi sin\theta=xcos\theta-ysin\theta y = s i n ϕ y = r s i n ( ϕ + θ ) = r s i n ϕ c o s θ + r c o s ϕ s i n θ = x s i n θ + y c o s θ y=sin\phi \quad y'=rsin(\phi+\theta)=rsin\phi cos\theta+rcos\phi sin\theta=xsin\theta+ycos\theta

写成矩阵的形式是:
[ x y ] = [ cos θ sin θ sin θ cos θ ] [ x y ] \left[\begin{array}{l} x^{\prime} \\ y^{\prime} \end{array}\right]=\left[\begin{array}{ll} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right] *\left[\begin{array}{l} x \\ y \end{array}\right]

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转载自blog.csdn.net/qq_44864262/article/details/107620539
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