dilatations

A dilatation, or dilation, is a geometric transformation that enlarges or shrinks a figure with respect to a fixed point called the center of dilation and by a positive scale factor k. Each point P in the plane is carried to a point P′ on the line CP such that CP′ = k·CP. If k > 1 the image is an enlargement; if 0 < k < 1 it is a reduction. If k < 0 the transformation reverses orientation and corresponds to a reflection through the center combined with scaling by |k|.

Dilatations are a type of similarity transformation: they preserve straight lines and angles, and they map figures to similar copies. A line through the center remains the same line, while a line not passing through the center is mapped to a line parallel to the original. Circles map to circles, with radii scaled by |k| and centers mapped accordingly. In particular, a circle centered at the center of dilation remains centered at that point with radius scaled by k.

Algebraically, in a vector space with center C, the image of a point X is X′ = C + k(X − C). In the complex plane, if z denotes a point and zC the center, then z′ = zC + k(z − zC). Dilations with a fixed center form a one-parameter family within the group of similarities, and a dilation about the origin reduces to simple scalar multiplication.

Outside geometry, the term dilatation can also describe the widening of a hollow organ or vessel in biology and medicine; in mathematics, however, the focus is on the scaling transformation described above.