A map obtained by composing rotations, translations, reflections, and uniform scalings (homotheties) that preserves angles and ratios of lengths between segments, producing geometrically similar figures.

Principle

A similarity is an affine composition of an isometry and a uniform scaling: f = s ∘ g where g is an isometry and s is a homothety (or vice versa). It preserves angles and multiplies all distances by a common positive factor |k| (up to orientation sign).

Demonstration

Map a triangle to a similar triangle by scaling by k=3 about its centroid and then translating and rotating to align corresponding vertices; all angle measures remain identical while side lengths are tripled.

Misapplication

Applying nonuniform (direction-dependent) scaling or shear produces an affine image that does not preserve angles or ratios of lengths; confusing conformal maps (local angle preservation) with global similarity can be misleading.

Consequence

Shapes are preserved up to scale and isometry; corresponding angles equal and ratios of corresponding side lengths are constant; similarity defines equivalence classes of figures under scaling and isometry.

Reversal

The inverse of a similarity with nonzero scale factor is a similarity with the reciprocal factor combined with the inverse isometry; negating the factor yields orientation reversal plus scaling.

Boundary

Similarity transformations form the similarity group of Euclidean space; they exclude projective transformations and general affine maps with anisotropic linear parts; conformal maps in higher dimensions may not be global similarities.

Semantic Tension

Frequently conflated with conformal transformations (which preserve angles but not necessarily global ratios) and with matrix similarity in linear algebra (an unrelated conjugation concept).

Synthesis

A similarity transformation is the class of Euclidean mappings that combine rigid motion and uniform scaling to produce figures that are geometrically the same up to a common scale.