Editing Rotation matrix (section)

=== Decomposition into shears ===

For the 2D case, a rotation matrix can be decomposed into three [[shear matrix|shear matrices]] ({{Harvnb|Paeth|1986}}):

:<math>\begin{align}
 R(\theta)
 &{}= 
  \begin{bmatrix}
    1 & -\tan \frac{\theta}{2}\\
    0 & 1
  \end{bmatrix}
  \begin{bmatrix}
    1 & 0\\
    \sin \theta & 1  
  \end{bmatrix}
  \begin{bmatrix}
    1 & -\tan \frac{\theta}{2}\\
    0 & 1
  \end{bmatrix}
\end{align}
</math>

This is useful, for instance, in computer graphics, since shears can be implemented with fewer multiplication instructions than rotating a bitmap directly. On modern computers, this may not matter, but it can be relevant for very old or low-end microprocessors.

A rotation can also be written as two shears and a [[squeeze mapping]] (an area preserving [[scaling (geometry)|scaling]]) ({{Harvnb|Daubechies|Sweldens|1998}}):

:<math>\begin{align}
 R(\theta)
 &{}= 
  \begin{bmatrix}
    1 & 0\\
    \tan\theta & 1
  \end{bmatrix}
  \begin{bmatrix}
    1 & -\sin\theta\cos\theta\\
    0 & 1
  \end{bmatrix}
  \begin{bmatrix}
    \cos\theta & 0\\
    0 & \frac{1}{\cos\theta}
  \end{bmatrix}
\end{align}
</math>