transformation matrix for points translations

June 19, 2013

transformation matrix for points translations

Rotation

For rotation by an angle θ clockwise about the origin, the functional form is $x' = x \cos \theta + y \sin \theta$ and $y' = - x \sin \theta + y \cos \theta$ . Written in matrix form, this becomes:

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

Similarly, for a rotation counter clockwise about the origin, the functional form is $x' = x \cos \theta - y \sin \theta$ and $y' = x \sin \theta + y \cos \theta$ and the matrix form is:

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

Scaling

For scaling (that is, enlarging or shrinking), we have $x' = s_x \cdot x$ and $y' = s_y \cdot y$ . The matrix form is:

$\begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} s_x & 0 \\ 0 & s_y \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}$

When $\ s_x s_y = 1$ , no change in size

when > 1 , size increases

when < 1 , size decreases

Shearing

For shear mapping (visually similar to slanting), there are two possibilities.

A shear parallel to the x axis has $x' = x + ky$ and $y' = y$ . Written in matrix form, this becomes:

$\begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} 1 & k \\ 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}$

A shear parallel to the y axis has $x' = x$ and $y' = y + kx$ , which has matrix form:

$\begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ k & 1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}$

reference https://en.wikipedia.org/wiki/Transformation_matrix

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