Affine Transformation
An affine transformation is any transformation that preserves collinearity (i.e., all points lying
on a line initially still lie on a line
after transformation) and ratios of distances
(e.g., the midpoint of a line segment remains the midpoint after transformation).
In this sense, affine indicates a special class of projective transformations that
do not move any objects from the affine space to the plane
at infinity or conversely. An affine transformation is also called an affinity.
Geometric contraction, expansion, dilation, reflection, rotation, shear, similarity transformations, spiral similarities, and translation are all affine transformations, as are their combinations. In general, an affine transformation is a composition of rotations, translations, dilations, and shears.
While an affine transformation preserves proportions on lines, it does not necessarily preserve angles or lengths. Any triangle can be transformed into any other by an affine transformation, so all triangles are affine and, in this sense, affine is a generalization of congruent and similar.
A particular example combining rotation and expansion is the rotation-enlargement transformation
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(1)
|
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(2)
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Separating the equations,
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(3)
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(4)
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This can be also written as
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(5)
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(6)
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where
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(7)
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(8)
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The scale factor is then defined by
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(9)
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and the rotation angle by
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(10)
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An affine transformation of is a map
of the form
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(11)
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for all , where
is a linear transformation
of
. If
, the
transformation is orientation-preserving;
if
, it is orientation-reversing.