What does affine transformation mean?
Definitions for affine transformation
affine trans·for·ma·tion
This dictionary definitions page includes all the possible meanings, example usage and translations of the word affine transformation.
Princeton's WordNet
affine transformationnoun
(mathematics) a transformation that is a combination of single transformations such as translation or rotation or reflection on an axis
Wiktionary
affine transformationnoun
a linear transformation between vector spaces followed by a translation.
Wikipedia
Affine transformation
In Euclidean geometry, an affine transformation or affinity (from the Latin, affinis, "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles. More generally, an affine transformation is an automorphism of an affine space (Euclidean spaces are specific affine spaces), that is, a function which maps an affine space onto itself while preserving both the dimension of any affine subspaces (meaning that it sends points to points, lines to lines, planes to planes, and so on) and the ratios of the lengths of parallel line segments. Consequently, sets of parallel affine subspaces 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. If X is the point set of an affine space, then every affine transformation on X can be represented as the composition of a linear transformation on X and a translation of X. Unlike a purely linear transformation, an affine transformation need not preserve the origin of the affine space. Thus, every linear transformation is affine, but not every affine transformation is linear. Examples of affine transformations include translation, scaling, homothety, similarity, reflection, rotation, shear mapping, and compositions of them in any combination and sequence. Viewing an affine space as the complement of a hyperplane at infinity of a projective space, the affine transformations are the projective transformations of that projective space that leave the hyperplane at infinity invariant, restricted to the complement of that hyperplane. A generalization of an affine transformation is an affine map (or affine homomorphism or affine mapping) between two (potentially different) affine spaces over the same field k. Let (X, V, k) and (Z, W, k) be two affine spaces with X and Z the point sets and V and W the respective associated vector spaces over the field k. A map f: X → Z is an affine map if there exists a linear map mf : V → W such that mf (x − y) = f (x) − f (y) for all x, y in X.
ChatGPT
affine transformation
An affine transformation, also known as an affine map, is a function between affine spaces that preserves points, straight lines, and planes. It also preserves ratios of lengths of segments on a straight line. The collection of all such transformations form a mathematical group. Importantly, unlike purely linear transformations, affine transformations include translations, scaling, rotation, and shear transformations.
Wikidata
Affine transformation
In geometry, an affine transformation or affine map or an affinity is a transformation which preserves straight lines and ratios of distances between points lying on a straight line. It does not necessarily preserve angles or lengths, but does have the property that sets of parallel lines will remain parallel to each other after an affine transformation. Examples of affine transformations include translation, geometric contraction, expansion, homothety, reflection, rotation, shear mapping, similarity transformation, and spiral similarities and compositions of them. An affine transformation is equivalent to a linear transformation followed by a translation.
Matched Categories
Numerology
Chaldean Numerology
The numerical value of affine transformation in Chaldean Numerology is: 1
Pythagorean Numerology
The numerical value of affine transformation in Pythagorean Numerology is: 8
Translations for affine transformation
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"affine transformation." Definitions.net. STANDS4 LLC, 2024. Web. 19 Apr. 2024. <https://www.definitions.net/definition/affine+transformation>.
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