# Module `trafo`: Linear transformations

With the `trafo` module PyX supports linear transformations, which can then be applied to canvases, Bézier paths and other objects. It consists of the main class `trafo` representing a general linear transformation and subclasses thereof, which provide special operations like translation, rotation, scaling, and mirroring.

## Class `trafo`

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

\[\vec{x}' = \mathsf{A}\, \vec{x} + \vec{b}\ ,\]

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 \(-\mathsf{A}^{-1}\vec{b}\).

class trafo.trafo(matrix=((1, 0), (0, 1)), vector=(0, 0))

  create new `trafo` instance with transformation `matrix` and `vector`

trafo.apply(x, y)

  apply `trafo` to point vector \((\mathtt{x}, \mathtt{y})\).

trafo.inverse()

  returns inverse transformation of `trafo`.

trafo.mirrored(angle)

  returns `trafo` followed by mirroring at line through \((0,0)\) with direction `angle` in degrees.

trafo.rotated(angle, x=None, y=None)

  returns `trafo` followed by rotation by `angle` degrees around point \((\mathtt{x}, \mathtt{y})\), or \((0,0)\), if not given.

trafo.scaled(sx, sy=None, x=None, y=None)

  returns `trafo` followed by scaling with scaling factor `sx` in \(x\)-direction, `sy` in \(y\)-direction (\(\mathtt{sy}=\mathtt{sx}\), if not given) with scaling center \((\mathtt{x}, \mathtt{y})\), or \((0,0)\), if not given.

trafo.slanted(a, angle=0, x=None, y=None)

  returns `trafo` followed by slant by `angle` around point \((\mathtt{x}, \mathtt{y})\), or \((0,0)\), if not given.

trafo.translated(x, y)

  returns `trafo` followed by translation by vector \((\mathtt{x}, \mathtt{y})\).

## Subclasses of `trafo`

The `trafo` module provides a number of subclasses of the `trafo` class, each of which corresponds to one `trafo` method.

class trafo.mirror(angle)

  mirroring at line through \((0,0)\) with direction `angle` in degrees.

class trafo.rotate(angle, x=None, y=None)

  rotation by `angle` degrees around point \((\mathtt{x}, \mathtt{y})\), or \((0,0)\), if not given.

class trafo.scale(sx, sy=None, x=None, y=None)

  scaling with scaling factor `sx` in \(x\)-direction, `sy` in \(y\)-direction (\(\mathtt{sy}=\mathtt{sx}\), if not given) with scaling center \((\mathtt{x}, \mathtt{y})\), or \((0,0)\), if not given.

class trafo.slant(a, angle=0, x=None, y=None)

  slant by `angle` around point \((\mathtt{x}, \mathtt{y})\), or \((0,0)\), if not given.

class trafo.translate(x, y)

  translation by vector \((\mathtt{x}, \mathtt{y})\).