Affine
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Look up affine in Wiktionary, the free dictionary.
Affine has two main uses, one in genealogy and the other in mathematics.
Kinship and descent
Affine (see Affinity), is a relative by marriage—an "in-law" in colloquial usage. The term can also refer to groups linked through marriage, such as clans.
Mathematics
Affine is an adjective that relates to the geometry of affine spaces. Two fundamental concepts combine here, those of affine geometry and affine space.
- Affine geometry, geometry not involving any notions of origin, length or angle, but with the notion of subtraction of points giving a vector
- Affine space, an abstract structure that generalises the affine-geometric properties of Euclidean space
Also, one of the central topics of study in classical algebraic geometry is "affine algebreic variety" (see Algebraic variety), which concerns the set of common zeroes of a set of polynomials.
As an adjective, "affine" is used as a qualifier on a number of mathematical concepts which are significant in their own right:
- An affine transformation, also known as an "affine map", is a linear transformation between two vector spaces, followed by a geometric translation.
- An affine group, in an affine space, consists of all invertible affine transformations from the space into itself.
- An affine combination is a linear combination of vectors.
- An affine representation, as applied to a topological group G, is a continuous homomorphism from G to the automorphism group of an affine space, A.
- An "affine scheme" (see Spectrum of a ring), is a sheaf of commutative rings of a specific form that can subsequently be used to build general mathematical schemes.
- Mathematical schemes connect the fields of algebraic geometry, commutative algebra and number theory and are technically topological spaces, each such space considered with commutative rings for all open sets within that space.
- Affine connections are connections on the tangent bundle of a differentiable manifold. The affine connection is a part of the formal definition of the Levi-Civita connection; practical geometrical calculations on such connections are typically done using Christoffel symbology.
- In cryptography, an affine cipher is a special type of substitution cipher in that it is monoalphabetic and symmetric.fr:Affine
