旋转矩阵：点旋转和坐标系旋转
旋转矩阵：点旋转和坐标系旋转
点旋转
点P在直角坐标系下的坐标为(x,y)，表示成极坐标(r,α)，则有关系式：
{ x = r ⋅ cos α y = r ⋅ sin α \left{\begin{array}{l} x=r \cdot \cos \alpha \ y=r \cdot \sin \alpha \end{array}\right.{x=r⋅cosαy=r⋅sinα
点P绕坐标系原点逆时针旋转θ角度，得点P’的坐标为(x’,y’)，极坐标表示为(r,α+θ)，则有关系式：
{ x ′ = r ⋅ cos ( α + θ ) = r ⋅ cos α ⋅ cos θ − r ⋅ sin α ⋅ sin θ y ′ = r ⋅ sin ( α + θ ) = r ⋅ sin α ⋅ cos θ + r ⋅ cos α ⋅ sin θ \left{\begin{array}{l} x^{\prime}=r \cdot \cos (\alpha+\theta)=r \cdot \cos \alpha \cdot \cos \theta-r \cdot \sin \alpha \cdot \sin \theta \ y^{\prime}=r \cdot \sin (\alpha+\theta)=r \cdot \sin \alpha \cdot \cos \theta+r \cdot \cos \alpha \cdot \sin \theta \end{array}\right.{x′=r⋅cos(α+θ)=r⋅cosα⋅cosθ−r⋅sinα⋅sinθy′=r⋅sin(α+θ)=r⋅sinα⋅cosθ+r⋅cosα⋅sinθ
化简可得：
{ x ′ = x ⋅ cos θ − y ⋅ sin θ y ′ = y ⋅ cos θ + x ⋅ sin θ \left{\begin{array}{l} x^{\prime}=x \cdot \cos \theta-y \cdot \sin \theta \ y^{\prime}=y \cdot \cos \theta+x \cdot \sin \theta \end{array}\right.{x′=x⋅cosθ−y⋅sinθy′=y⋅cosθ+x⋅sinθ
写成矩阵的形式为：
[ 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}{cc} \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 ]
则点P到点P’的旋转矩阵可表示为：
R p o i n t = [ cos θ − sin θ sin θ cos θ ] R_{point} =\left[\begin{array}{cc} \cos \theta & -\sin \theta \ \sin \theta & \cos \theta \end{array}\right]Rpoint =[cosθsinθ −sinθcosθ ]
坐标系旋转
点P在直角坐标系下的坐标为(x,y)，表示成极坐标(r,α)，则有关系式：
{ x = r ⋅ cos α y = r ⋅ sin α \left{\begin{array}{l} x=r \cdot \cos \alpha \ y=r \cdot \sin \alpha \end{array}\right.{x=r⋅cosαy=r⋅sinα
点P不动，坐标系绕原点逆时针旋转θ角度，坐标变为(x’,y’)，极坐标为(r,α-θ)，则有关系式：
{ x ′ = r ⋅ cos ( α − θ ) = r ⋅ cos α ⋅ cos θ + r ⋅ sin α ⋅ sin θ y ′ = r ⋅ sin ( α − θ ) = r ⋅ sin α ⋅ cos θ − r ⋅ cos α ⋅ sin θ \left{\begin{array}{l} x^{\prime}=r \cdot \cos (\alpha-\theta)=r \cdot \cos \alpha \cdot \cos \theta+r \cdot \sin \alpha \cdot \sin \theta \ y^{\prime}=r \cdot \sin (\alpha-\theta)=r \cdot \sin \alpha \cdot \cos \theta-r \cdot \cos \alpha \cdot \sin \theta \end{array}\right.{x′=r⋅cos(α−θ)=r⋅cosα⋅cosθ+r⋅sinα⋅sinθy′=r⋅sin(α−θ)=r⋅sinα⋅cosθ−r⋅cosα⋅sinθ
化简可得：
{ x ′ = x ⋅ cos θ + y ⋅ sin θ y ′ = y ⋅ cos θ − x ⋅ sin θ \left{\begin{array}{l} x^{\prime}=x \cdot \cos \theta+y \cdot \sin \theta \ y^{\prime}=y \cdot \cos \theta-x \cdot \sin \theta \end{array}\right.{x′=x⋅cosθ+y⋅sinθy′=y⋅cosθ−x⋅sinθ
写成矩阵的形式为：
[ 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}{cc} \cos \theta & \sin \theta \ -\sin \theta & \cos \theta \end{array}\right]\left[\begin{array}{l} x \ y \end{array}\right][x′y′ ]=[cosθ−sinθ sinθcosθ ][xy ]
则坐标系OXY旋转到坐标系O’X’Y’的旋转矩阵可表示为：
R c o o r d i n a t e = [ cos θ sin θ − sin θ cos θ ] R_{coordinate} =\left[\begin{array}{cc} \cos \theta & \sin \theta \ -\sin \theta & \cos \theta \end{array}\right]Rcoordinate =[cosθ−sinθ sinθcosθ ]
总结
- 点旋转：在坐标系OXY中，存在点P，点P经过旋转到点P’，旋转矩阵为R p o i n t R_{point}Rpoint ，则P ’ = R p o i n t ∗ P P’=R_{point}*{P}P’=Rpoint ∗P；
- 坐标系旋转：在坐标系OXY中，存在点P，坐标系OXY经过旋转变为坐标系O’X’Y’，则点P的坐标改变为P’，旋转矩阵为R c o o r d i n a t e R_{coordinate}Rcoordinate ，则P ’ = R c o o r d i n a t e ∗ P P’=R_{coordinate}*{P}P’=Rcoordinate ∗P；
当点旋转和坐标系旋转的旋转角度和旋转方向一样时，可以发现R p o i n t R_{point}Rpoint 与R c o o r d i n a t e R_{coordinate}Rcoordinate 是互逆关系，即R p o i n t = R c o o r d i n a t e t R_{point}=R^{t}_{coordinate}Rpoint =Rcoordinatet 。