Karl Jones at 12:48, 24 September 2016

2016-09-24T12:48:58Z

Karl Jones: Created page with "File:Fractal_fern_diagram.png|thumb|An image of a fern-like fractal that exhibits affine self-similarity. Each of the leaves of the fern is related to each other leaf by an..."

2016-09-24T12:48:26Z

Created page with "File:Fractal_fern_diagram.png|thumb|An image of a fern-like fractal that exhibits affine self-similarity. Each of the leaves of the fern is related to each other leaf by an..."

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[[File:Fractal_fern_diagram.png|thumb|An image of a fern-like fractal that exhibits affine self-similarity. Each of the leaves of the fern is related to each other leaf by an affine transformation. For instance, the red leaf can be transformed into both the small dark blue leaf and the large light blue leaf by a combination of reflection, rotation, scaling, and translation.]]In [[geometry]], an '''affine transformation''', '''affine map''', or an '''affinity''' (from the Latin ''affinis'', "connected with") is a [[Function (mathematics)|function]] between [[Affine space|affine spaces]] which preserves points, straight lines and planes.

== Description ==

Also, sets of parallel lines remain parallel after an affine transformation. An affine transformation does not necessarily preserve angles between lines or distances between points, though it does preserve ratios of distances between points lying on a straight line.

Examples of affine transformations include translation, scaling, homothety, similarity transformation, reflection, rotation, shear mapping, and compositions of them in any combination and sequence.

Unlike a purely linear transformation, an affine map need not preserve the zero point in a linear space. Thus, every linear transformation is affine, but not every affine transformation is linear.

All [[Euclidean space|Euclidean spaces]] are affine, but there are affine spaces that are non-Euclidean.

In affine coordinates, which include Cartesian coordinates in Euclidean spaces, each output coordinate of an affine map is a linear function (in the sense of calculus) of all input coordinates.

Another way to deal with affine transformations systematically is to select a point as the origin; then, any affine transformation is equivalent to a linear transformation (of position vectors) followed by a translation.

== See also ==

* [[3D projection]]
* [[Affine geometry]]
* [[Affine space]]
* [[Bent function]]
* [[Flat (geometry)]]
* [[Homography]]
* [[Transformation matrix # affine transformation]]

== External links ==

* [https://en.wikipedia.org/wiki/Affine_transformation Affine transformation] @ Wikipedia

[[Category:Functions]]
[[Category:Geometry]]
[[Category:Mathematics]]

← Older revision		Revision as of 12:48, 24 September 2016
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−	[[File:Fractal_fern_diagram.png\|thumb\|An image of a fern-like fractal that exhibits affine self-similarity. Each of the leaves of the fern is related to each other leaf by an affine transformation. For instance, the red leaf can be transformed into both the small dark blue leaf and the large light blue leaf by a combination of reflection, rotation, scaling, and translation.]]In [[geometry]], an '''affine transformation''', '''affine map''', or an '''affinity''' (from the Latin ''affinis'', "connected with") is a [[Function (mathematics)\|function]] between [[Affine space\|affine spaces]] which preserves points, straight lines and planes.	+	[[File:Fractal_fern_diagram.png\|thumb\|An image of a fern-like [[fractal]] that exhibits affine [[self-similarity]]. Each of the leaves of the fern is related to each other leaf by an affine transformation. For instance, the red leaf can be transformed into both the small dark blue leaf and the large light blue leaf by a combination of reflection, rotation, scaling, and translation.]]In [[geometry]], an '''affine transformation''', '''affine map''', or an '''affinity''' (from the Latin ''affinis'', "connected with") is a [[Function (mathematics)\|function]] between [[Affine space\|affine spaces]] which preserves points, straight lines and planes.

	== Description ==		== Description ==

Affine transformation - Revision history

Karl Jones at 12:48, 24 September 2016

Karl Jones: Created page with "File:Fractal_fern_diagram.png|thumb|An image of a fern-like fractal that exhibits affine self-similarity. Each of the leaves of the fern is related to each other leaf by an..."