homothety

Homothety, also called a dilation, is a geometric transformation that maps a figure to a similar figure by expanding or contracting it about a fixed point called the center of homothety. If the center is O and the ratio is k, each point P is sent to a point P' on the line OP such that OP' equals k times OP. The center O remains fixed (OP = 0 at P = O). If k is positive, the image lies on the same side of O as P; if k is negative, the image lies on the opposite side and the orientation of the figure is reversed.

Homotheties preserve straight lines and angles; they are a type of similarity transformation. Lines through the center map to themselves, while lines not through the center map to lines parallel to the original. A circle maps to a circle, with its center along the line OC and its radius scaled by |k|. Distances between points are multiplied by |k|, and areas are multiplied by k^2. Thus shapes are preserved in form but scaled in size.

Two figures related by a homothety are similar, with corresponding points lying on rays emanating from the center. The center of homothety for two similar figures is the intersection of lines joining corresponding points. The composition of two homotheties is again a similarity; it is a homothety with ratio equal to the product of the two ratios (and center on the line through the two centers) unless the product equals 1, in which case the composition is a translation.

In practice, a common example is enlarging or reducing a geometric figure about a chosen center, such as creating a zoomed version of a diagram centered at a fixed point.