Rotation

Please enable Java for an interactive construction (with Cinderella).

A Rotation of a verctor around the origin can be expressed as the multiplication of this vector with a matrix. If the vector is represented as a column vector, the matrix multiplication becomes

$\left(\begin{array}{c} x\\ y \end{array}\right) \mapsto \left(\begin{array}{cc} a &b\\ c&d \end{array}\right) \left(\begin{array}{c} x\\ y \end{array}\right) = \left(\begin{array}{c} ax+by\\ cx+dy \end{array}\right).$

The parameters

must be suitably choosen. The matrix

$\left(\begin{array}{cc} a &b\\ c&d \end{array}\right) = \left(\begin{array}{cc} \cos(\alpha)&-\sin(\alpha)\\ \sin(\alpha)&\cos(\alpha)\end{array}\right)$

results in a rotation by an angle $\alpha$ .

The following rule of thumb applies: "The columns of the matrix are the images of the unit vectors."

In the applet the vector

and the corners of the house can be moved using the mouse.

Download file:

Drehung1En.cdy

Translation $\hookleftarrow$ Contents $\hookrightarrow$ Spiral similarity