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 θ角度到绿色线。 假设蓝色线为 l l l,绿色线为 l ′ 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 x=rcosϕx′=rcos(ϕ+θ)=rcosϕcosθ−rsinϕsinθ=xcosθ−ysinθ 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 y=sinϕy′=rsin(ϕ+θ)=rsinϕcosθ+rcosϕsinθ=xsinθ+ycosθ
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] [x′y′]=[cosθsinθ−sinθcosθ]∗[xy]