afins

Afins is not a standard English term in mathematics or science. In many Francophone sources, forms derived from the French word affin meaning related or connected by affinity appear as afins; in English mathematical usage, the corresponding term is affine. When English texts refer to multiple affine mappings, the usual plural is affines. The concept, however, centers on the idea of affine relationships that preserve straight lines and certain ratios, rather than depending on distances or angles alone.

In mathematics, affine structures arise from affine transformations and affine spaces. An affine transformation is a map between vector spaces of the form f(x) = Ax + b, where A is a linear transformation and b is a translation vector. Such maps preserve collinearity and ratios of segments on parallel lines, and the composition of affine transformations remains affine. The set of all affine transformations on a space forms the affine group, which combines linear transformations with translations.

An affine space generalizes Euclidean space by focusing on points and the directions between them rather than on a fixed metric. A fundamental notion is the affine combination: a weighted sum of points with coefficients that sum to one. This leads to concepts such as affine hulls, barycentric coordinates, and affine independence, which are central to projects in geometry and computational geometry.

Common applications include computer graphics, where affine transformations map textures and shapes; robotics and computer vision, where affine invariants aid object recognition; and pure geometry, where affine concepts distinguish between properties preserved under affine maps and those preserved only under more restrictive transformations.