Rigid transformation

A rigid transformation, also called a Euclidean transformation or Euclidean isometry, is a geometric transformation of Euclidean space that preserves the distance between every pair of points. The principal rigid transformations are rotations, translations and reflections; when handedness, or orientation, is also preserved, so reflections are excluded, a rigid transformation is instead called a rigid motion or a proper rigid transformation. Formally, a rigid transformation can be written as T(v) = Rv + t, where R is an orthogonal matrix representing a rotation or reflection component and t is a translation vector. Because rigid transformations preserve distance between every pair of points by definition, the terms Euclidean transformation and Euclidean isometry are used interchangeably with rigid transformation.