dilataties

Dilatations, or dilations, are geometric transformations that enlarge or shrink figures with respect to a fixed point called the center, by a constant real factor k. If P is a point and C is the center, the image P' lies on the line CP and satisfies CP' = |k| · CP, with the center C remaining fixed (P' = C when P = C). The formula for a dilation with center C and scale factor k is P' = C + k(P − C).

Key properties include that lines through the center map to themselves, while lines not through the center map to lines parallel to the original. Distances from the center scale by |k|, and in two dimensions the area of any figure scales by k². If k > 0, the dilation preserves orientation; if k < 0, the transformation also includes a half-turn about the center, reversing orientation.

Dilatations are a special case of similarity transformations. They can be composed: two dilations about the same center with factors k1 and k2 yield a dilation with factor k1k2. If the centers differ, the composition is not generally a dilation but another similarity, often a spiral similarity.

In higher dimensions, dilatations are defined analogously in Euclidean space, with the same center and a real scale factor. They preserve shapes but not sizes, and they are widely used in geometry, computer graphics, design, and analysis of geometric constructions.

Etymology traces to Latin dilatatio, meaning “spreading out.” In everyday language, the term appears in mathematics, anatomy (dilatation of vessels), and physiology, reflecting a common concept of expansion or enlargement.