affiniteetti

Affiniteetti, or affine transformations, are mappings between affine spaces that preserve affine combinations of points. In Euclidean space R^n, an affine transformation T is typically written as T(x) = A x + b, where A is an n×n matrix and b is a vector in R^n. If A is invertible, T is bijective; otherwise it is a non-invertible affine map. The linear part A and the translation part b together determine how points and lines are moved.

Key properties of affine transformations include preservation of straight lines and collinearity, mapping lines to lines and planes to planes. They also preserve parallelism: parallel lines remain parallel after the transformation. Affine maps preserve affine combinations, so any relation of the form (1−t)x + t y is mapped to (1−t)T(x) + t T(y) for all t, meaning ratios of distances along the same line are preserved in the affine sense. They also take convex sets to convex sets.

Affine transformations form a group under composition when restricted to invertible maps, and can be described algebraically as the semidirect product GL(n, F) ⋉ F^n, combining linear transformations with translations. In 2D and 3D, they are widely used in computer graphics, computer vision, robotics, and geographic information systems for tasks such as image alignment, perspective correction, and coordinate changes.

Relation to other maps: affine transformations are a subset of projective transformations and form a broader class than similarity transformations, which also preserve angles and ratios of lengths. Affiniteetti thus captures linear distortions including rotation, uniform or nonuniform scaling, shearing, and translation.