dilation

Dilation, in mathematics, refers to a transformation that enlarges or reduces objects by a scale factor with respect to a fixed point called the center. In the Euclidean plane, a dilation with center C and scale factor k ≠ 0 maps each point X to D(X) = C + k (X - C). Distances from the center are multiplied by |k|, while the center itself remains fixed. If k > 1, figures inflate; 0 < k < 1 shrinks them; k = 1 yields the identity, and k < 0 includes a rotation of 180 degrees about C.

Geometrically, a dilation is a type of similarity transformation. It preserves straight lines, angles, and the general shape of figures, while changing size. Circles map to circles with the same center but radii scaled by |k|.

Applications include computer graphics, geometric constructions, and the study of similarity in geometry. In coordinate form a dilation is equivalent to a uniform scaling plus a translation, hence it preserves ratios of lengths along lines through the center.

Other uses of the term dilation appear in biology and medicine, referring to widening or expansion of an opening or passage. Examples include pupil dilation (mydriasis), vasodilation of blood vessels, and cervical dilation during labor. In physiology, dilation describes an increase in diameter or caliber of a structure in response to stimuli.