affine transformation

English

Noun

affine transformation (plural affine transformations)

1. (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 $X$, every affine transformation on $X$ can be represented as the composition of a linear transformation on $X$ and a translation of $X$.

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

   • 1965, P. S. Modenov, A. S. Parkhomenko, translated by Michael B. P. Slater, 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 $n$-dimensional figures, where $n=1,2,3,4$.

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

• Chinese:

  Mandarin: 仿射變換／仿射变换 (fǎngshè biànhuàn)

• Estonian: afiinne teisendus
• Finnish: affiinikuvaus, affiini kuvaus
• French: transformation affine f
• German: affine Abbildung f
• Hungarian: affin transzformáció (hu)
• Icelandic: vildarvörpun f, vildarmótun f
• Italian: trasformazione affine f
• Japanese: アフィン変換 (afinhenkan )
• Russian: аффи́нное преобразова́ние n (affínnoje preobrazovánije)
• Spanish: transformación afín f
• Swedish: affin transformation c

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