affinely

Affinely relates to affine structures and transformations in geometry and linear algebra. An affine transformation is a function that preserves points, straight lines, and ratios of distances along a line. It can be expressed in the form T(x) = Ax + b, where A is a linear map and b is a fixed vector. This form is equivalent to preserving affine combinations of points.

Affine combinations are expressions of the form tx + (1 − t)y for t in the underlying field. A key property is that T(tx + (1 − t)y) = tT(x) + (1 − t)T(y). As a result, lines map to lines, parallelism is preserved, and the midpoint of a segment maps to the midpoint of its image. In short, affine mappings preserve collinearity and the ratio of segments along a line, but not necessarily distances or angles.

In general, affinely related transformations do not preserve lengths or angles unless they are special cases such as similarities or isometries. The common invariants of affine maps are geometric relations that involve straight lines and parallelism, and the preservation of affine combinations.

In geometry and its applications, the term affinely is used to describe properties or constructions that remain true under affine transformations. Affine space is an abstract setting where points and lines are defined up to translation and linear structure, without a fixed origin. In practice, affine transformations are widely used in computer graphics, engineering, and modeling to perform translations, rotations, scalings, and shears in a consistent, origin-free manner.