affinor

Affinor is a term used in some geometric and mathematical contexts to denote an affine transformation, also called an affinity. In English-language mathematics literature the term is uncommon; most texts refer to affine transformations or affine maps. An affinor acts on an n-dimensional affine space by a function of the form F(x) = A x + b, where A is a linear map represented by an n×n matrix and b is a translation vector. If b = 0, the affinor reduces to a linear transformation.

Geometrically, affinors preserve straight lines and affine combinations. They map lines to lines and preserve the ratio of segments along any line parallel to a given direction. In particular, for any points x1, x2, …, xk and coefficients αi with sum equal to 1, F(Σ αi xi) = Σ αi F(xi). They do not, in general, preserve distances or angles unless A satisfies additional constraints (such as being orthogonal with a suitable scaling).

Algebraically, affinors form the affine group of the space, consisting of all maps of the form x ↦ A x + b with A invertible and b arbitrary. The composition of affinors yields another affinor, and their inverses exist when A is invertible. Affinors are foundational in affine geometry and appear in applications ranging from computer graphics to geometric modeling, where preserving the affine structure is essential.

Note that “affinor” is not standard in modern English usage; many references simply use “affine transformation” or “affinity.” See also affine transformation, affine space, linear transformation, and the affine group.