# AffineTransform

gives a TransformationFunction that represents an affine transform that maps r to m.r.

AffineTransform[{m,v}]

gives an affine transform that maps r to m.r+v.

# Details • AffineTransform gives a TransformationFunction that can be applied to vectors.
• For ordinary affine transforms in dimensions, m is an × matrix.
• AffineTransform in general supports × matrices for transformations in dimensions.

# Examples

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## Basic Examples(2)

A general affine transformation:

Transform points:

A pure rotation:

A pure translation:

## Scope(3)

Affine transform in four dimensions:

The inverse transform:

Transformation applied to a 2D shape:

Transformation applied to a 3D shape:

## Applications(5)

### Iterated Function Systems(3)

Define an iterated function system (IFS) and iterate it on point sets, by computing in each iteration:

Sierpiński carpet:

Heighway's Dragon:

Compute an iterated function system's (IFS) fixed points efficiently by randomly picking subparts of point sets:

Sierpiński carpet:

Heighway's Dragon:

Hedgehog:

Compute an iterated function system applied to graphics primitives:

Sierpiński carpet:

Hedgehog:

### Image Transformations(2)

Use an AffineTransform to rotate an image:

Affine transform of a 3D image with no translation:

## Properties & Relations(3)

Many other geometric transformations are a special case of affine transform:

In turn, an affine transformation is a special case of a linear-fractional transformation:

The composition of affine transforms is an affine transform:

## Neat Examples(1)

Nested transformations of a circle: