Rigid motion

A transformation of a space that preserves the distance between every pair of points. In Euclidean geometry, rigid motions include translations, rotations, reflections, and their compositions.

Euclidean isometry: A transformation of Euclidean space that preserves distances between all pairs of points. Rigid motions are precisely the distance-preserving transformations of Euclidean space.
Euclidean distance: The straight-line distance between two points in Euclidean space, given by the Pythagorean formula. Preserving this distance for every point pair defines Euclidean rigid motion.
Similarity transformation: A transformation that preserves angles and multiplies all distances by the same positive scale factor. Unlike a rigid motion, it may uniformly enlarge or shrink distances.
Symmetry group: A group of transformations that preserve a specified structure or object. Rigid motions that map a figure onto itself form its symmetry group.
Translation: A transformation that moves every point by the same displacement vector. It preserves distances by shifting all points equally.
Congruence (geometry): A relation between geometric figures that can be mapped onto each other by a distance-preserving transformation. Rigid motions provide the transformations that witness congruence.
Affine transformation: A transformation that preserves points, lines, and parallelism but need not preserve lengths or angles. It preserves less geometric structure than a rigid motion.
Point group: A finite group of rigid motions that fixes at least one point and preserves a three-dimensional object or pattern. It classifies the rotational and reflectional rigid symmetries of objects and crystals.
Rotation: A transformation that turns points around a fixed center or axis by a specified angle. It preserves distances while changing directions around a center or axis.
Coordinate transformation: A rule that expresses points or coordinates in one reference system using another. Rigid motions can be represented as transformations of coordinates.

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