Similarity transformation

A geometric transformation that preserves distances up to a common scale factor, and therefore preserves angles and shape while allowing changes in size, position, or orientation.

Euclidean transformation: A transformation of Euclidean space that preserves distances between all pairs of points. A similarity is a Euclidean transformation when its scale factor is one.
Similarity matrix: A matrix representing a linear transformation that scales all lengths by the same factor while preserving angles. Its columns encode the rotation or reflection and uniform scaling of a planar similarity.
Isometry: A distance-preserving transformation between metric spaces. Isometries preserve every length, whereas similarities may multiply all lengths by one factor.
Similar triangles: Triangles with equal corresponding angles and proportional corresponding side lengths. Their proportional sides are the central triangle-level consequence of similarity.
Dilation (geometry): A transformation that scales distances from a fixed center by a constant factor. Dilation supplies the size change in a similarity transformation.
Affine transformation: A transformation that preserves points, lines, and parallelism, but not necessarily lengths or angles. Every similarity is affine, while general affine maps can distort shape.
Shear mapping: A linear transformation that shifts points parallel to a fixed direction by amounts proportional to another coordinate. A shear can change angles and shape, unlike a similarity.
Thales' theorem: A geometric theorem relating parallel lines to proportional segments in triangles. It uses similarity to infer inaccessible lengths from proportional segments.
Rotation (geometry): A rigid transformation that turns points around a fixed center by a specified angle. Rotations can change orientation without changing a figure’s size or shape.
Complex plane: A representation of complex numbers as points in a plane with real and imaginary axes. Multiplication by a nonzero complex number expresses planar scaling and rotation compactly.

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