affine transformation
English
Noun
affine transformation (plural affine transformations)
- (geometry, linear algebra) A geometric transformation that preserves lines and parallelism, but in general not lengths or angles; (more formally) an automorphism of an affine space: a mapping of an affine space onto itself that preserves both the dimension of any affine subspace and the ratio of the lengths of any pair of parallel line segments.
- An affine transformation does not in general preserve angles between lines or distances between points, but it does preserve ratios of distances between points lying on a straight line.
- Given an affine space, every affine transformation on can be represented as the composition of a linear transformation on and a translation of .
- Examples of affine transformations include translation, scaling, homothety, similarity, reflection, rotation, shear and compositions of them in any combination and sequence.
- 1965, Michael B. P. Slater (translator), P. S. Modenov, A. S. Parkhomenko, Geometric Transformations, Volume 1: Euclidean and Affine Transformations, Academic Press, page 145,
- Just as for plane transformations, we may show that the set of all affine transformations of space form a group.
- Under an affine transformation of space, the image of a line is a line, and the image of a plane is a plane.
- 1982, George E. Martin, Transformation Geometry, Springer, page 169,
- Theorem 15.2 A transformation such that the images of every three collinear points are themselves collinear is an affine transformation.
- Are the affine transformations the same as those transformations for which the images of any three noncollinear points are themselves noncollinear? We shall see the answer is "Yes."
- 2004, Solomon Khmelnik, Computer Arithmetic of Geometrical Figures: Algorithms and Hardware Design, Mathematics in Computer Comp., page 8,
- Most striking and well-known examples of affine transformation applications are computer tomography (see for instance [1]) and information compression for telecommunication systems (see [2]).
- This book describes affine transformations (displacements, turns, scaling, shifts) of -dimensional figures, where .
Synonyms
- (geometric transformation that preserves lines and parallelism): affinity
Hypernyms
- affine map, affine mapping, affine homomorphism (mappings from one space to a (generally) different one)
Hyponyms
- linear transformation
Holonyms
- affine group
Derived terms
- affine transformation matrix
Translations
geometric transformation that preserves lines and parallelism
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Further reading
Geometric transformation on Wikipedia.Wikipedia
Affine geometry on Wikipedia.Wikipedia
Affine group on Wikipedia.Wikipedia
Affine space on Wikipedia.Wikipedia
Affine variety on Wikipedia.Wikipedia
Complex affine space on Wikipedia.Wikipedia
Anamorphosis on Wikipedia.Wikipedia
- Affine transformation on Encyclopedia of Mathematics