Affine Transformation
An Affine Transformation is a function between affine spaces which preserves points, straight lines and planes.
- See: Translation (Geometry), Geometry, Affine Spaces, Parallel (Geometry), Scaling (Geometry), Homothetic Transformation, Similarity Transformation (Geometry), Reflection (Mathematics), Rotation, Shear Mapping, Function Composition, Affine Space.
References
2015
- (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/Affine_transformation Retrieved:2015-6-9.
- In geometry, an affine transformation, affine map[1] or an affinity (from the Latin, affinis, "connected with") is a function between affine spaces which preserves points, straight lines and planes. 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.
If [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] are affine spaces, then every affine transformation [math]\displaystyle{ f : X \to Y }[/math] is of the form [math]\displaystyle{ x \mapsto Mx + b }[/math] , where [math]\displaystyle{ M }[/math] is a linear transformation on [math]\displaystyle{ X }[/math] and [math]\displaystyle{ b }[/math] is a vector in [math]\displaystyle{ Y }[/math] . 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.
For many purposes an affine space can be thought of as Euclidean space, though the concept of affine space is far more general (i.e., all 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.
- In geometry, an affine transformation, affine map[1] or an affinity (from the Latin, affinis, "connected with") is a function between affine spaces which preserves points, straight lines and planes. 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.
- ↑ Berger, Marcel (1987), p. 38.
2006
- (Everitt, 2006) ⇒ Brian S. Everitt. (2006). “The Cambridge Dictionary of Statistics. 3rd Edition.” Cambridge University Press. ISBN:0521690277
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- Affine invariance: A term applied to statistical procedures which give identical results after the data has been subjected to an affine transformation. An example is Hotelling’s T 2 test. [Canadian Journal of Statistics, 2003, 31, 437–55.]
- Affine transformation: The transformation, Y= AX+b where A is a nonsingular matrix and b is any vector of real numbers. Important in many areas of statistics particularly multivariate analysis.
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