homothetia

A homothety, or dilation, is a similarity transformation of Euclidean space with a fixed center C and a real ratio k ≠ 0 that sends every point P to a point P' on the line CP such that CP' = k·CP. Equivalently, P' can be written as P' = C + k(P − C).

The center C is fixed by the transformation. Lines through C are mapped to themselves; lines not through C are mapped to lines parallel to the original. Distances are multiplied by the absolute value |k|, and angles are preserved. If k > 0 the orientation is preserved; if k < 0 the orientation is reversed.

Special cases include k = 1, which yields the identity transformation, and k = −1, which is a half-turn or point reflection about C.

In two or three dimensions, a homothety scales all figures about the center by the factor |k|, producing similar figures. It is a basic example of a centric similarity. If two figures are related by a homothety, the centers of the corresponding dilation are given by the lines joining corresponding points; for two similar figures there are typically two centers: an internal and an external center of homothety, corresponding to the two possible orientations of the mapping.

The concept extends to spaces of any dimension and underpins constructions and proofs involving similarity and scaling in geometry.