14.1 Class trafo

The trafo class represents a general linear transformation, which is defined for a vector $\vec{x}$ as

$\begin{displaymath} \vec{x}' = \mathsf{A}\, \vec{x} + \vec{b}\ , \end{displaymath}$

where $\mathsf{A}$ is the transformation matrix and $\vec{b}$ the translation vector. The transformation matrix must not be singular, i.e. we require $\det \mathsf{A} \ne 0$ .

Multiple trafo instances can be multiplied, corresponding to a consecutive application of the respective transformation. Note that trafo1*trafo2 means that trafo1 is applied after trafo2, i.e. the new transformation is given by $\mathsf{A} = \mathsf{A}_1 \mathsf{A}_2$ and $\vec{b} = \mathsf{A}_1 \vec{b}_2 + \vec{b}_1$ . Use the trafo methods described below, if you prefer thinking the other way round. The inverse of a transformation can be obtained via the trafo method inverse(), defined by the inverse $\mathsf{A}^{-1}$ of the transformation matrix and the translation vector $-\mathsf{A}^{-1}\vec{b}$ .

The methods of the trafo class are summarized in the following table.

=.8 trafo method =1.2function

=.8 __init__(matrix=((1,0),(0,1)), vector=(0,0)): =1.2create new trafo instance with transformation matrix and vector.

=.8 apply(x, y) =1.2apply trafo to point vector $(\mathtt{x}, \mathtt{y})$ .

=.8 inverse() =1.2returns inverse transformation of trafo.

=.8 mirrored(angle) =1.2returns trafo followed by mirroring at line through with direction angle in degrees.

=.8 rotated(angle, x=None, y=None) =1.2 returns trafo followed by rotation by angle degrees around point $(\mathtt{x}, \mathtt{y})$ , or , if not given.

=.8 scaled(sx, sy=None, x=None, y=None) =1.2 returns trafo followed by scaling with scaling factor sx in -direction, sy in -direction ( $\mathtt{sy}=\mathtt{sx}$ , if not given) with scaling center $(\mathtt{x}, \mathtt{y})$ , or , if not given.

=.8 translated(x, y) =1.2returns trafo followed by translation by vector $(\mathtt{x}, \mathtt{y})$ .

=.8 slanted(a, angle=0, x=None, y=None) =1.2returns trafo followed by XXX

=.8 `trafo` method	=1.2function
=.8 `__init__(matrix=((1,0),(0,1)), vector=(0,0)):`	=1.2create new `trafo` instance with transformation `matrix` and `vector`.
=.8 `apply(x, y)`	=1.2apply `trafo` to point vector $(\mathtt{x}, \mathtt{y})$ .
=.8 `inverse()`	=1.2returns inverse transformation of `trafo`.
=.8 `mirrored(angle)`	=1.2returns `trafo` followed by mirroring at line through with direction `angle` in degrees.
=.8 `rotated(angle, x=None, y=None)`	=1.2 returns `trafo` followed by rotation by `angle` degrees around point $(\mathtt{x}, \mathtt{y})$ , or , if not given.
=.8 `scaled(sx, sy=None, x=None, y=None)`	=1.2 returns `trafo` followed by scaling with scaling factor `sx` in -direction, `sy` in -direction ( $\mathtt{sy}=\mathtt{sx}$ , if not given) with scaling center $(\mathtt{x}, \mathtt{y})$ , or , if not given.
=.8 `translated(x, y)`	=1.2returns `trafo` followed by translation by vector $(\mathtt{x}, \mathtt{y})$ .
=.8 `slanted(a, angle=0, x=None, y=None)`	=1.2returns `trafo` followed by XXX