A similarity transformation with a center that maps each point to a point on the same ray from the center by multiplying distances from the center by a uniform scale factor k (the dilation factor).

Principle

Relative position vectors from the center O are scaled: for center O and factor k, P ↦ O + k·(P−O). The map commutes with collinearity and preserves angles while scaling lengths by |k|.

Demonstration

In the plane, take center O=(0,0) and factor k=2. The homothety sends P=(x,y) to P'=(2x,2y), doubling distances from O and producing a similar figure twice as large centered at O.

Misapplication

Using different scale factors for different directions (anisotropic scaling), changing the center according to point, or conflating k=0 as a nontransformation; forgetting that negative k reverses orientation and places images on the opposite ray.

Consequence

Angles are preserved and ratios of lengths along any ray from the center are multiplied by k; lines not through the center map to parallel lines, lines through the center map to themselves; shapes remain similar with predictable scale.

Reversal

If k≠0 the inverse homothety has factor 1/k and the same center; negative factors combine scaling with point reflection through the center.

Boundary

Defined for any affine or Euclidean space with a distinguished center and scalar field; the case k=1 is the identity, k=0 collapses the space to the center, and nonuniform linear scalings lie outside homotheties.

Semantic Tension

Often conflated with general uniform similarity (which may include translation and rotation); homothety is a centered uniform scaling, distinct from arbitrary similarities composed with additional isometries.

Synthesis

A homothety is the centered uniform rescaling that preserves directions and angles while multiplying all radii from the center by the same scalar factor.