In mathematics, a rigid transformation (also called Euclidean transformation or Euclidean isometry) is a geometric transformation of a Euclidean space that preserves the Euclidean distance between every pair of points. The rigid transformations include rotations, translations, reflections, or any sequence of these. Reflections are sometimes excluded from the definition of a rigid transformation by requiring that the transformation also preserve the handedness of objects in the Euclidean space. (A reflection would not preserve handedness; for instance, it would transform a left hand into a right hand.) To avoid ambiguity, a transformation that preserves handedness is known as a proper rigid transformation, or rototranslation. Any proper rigid transformation can be decomposed into a rotation followed by a translation, while any improper rigid transformation can be decomposed into an improper rotation followed by a translation, or into a sequence of reflections. Any object will keep the same shape and size after a proper rigid transformation. All rigid transformations are examples of affine transformations. The set of all (proper and improper) rigid transformations is a mathematical group called the Euclidean group, denoted for -dimensional Euclidean spaces. The set of proper rigid transformations is called special Euclidean group, denoted . In kinematics, proper rigid transformations in a 3-dimensional Euclidean space, denoted SE(3), are used to represent the linear and angular displacement of rigid bodies. According to Chasles' theorem, every rigid transformation can be expressed as a screw displacement. Formal definition A rigid transformation is formally defined as a transformation that, when acting on any vector , produces a transformed vector of the form where (i.e., is an orthogonal transformation), and is a vector giving the translation of the origin. A proper rigid transformation has, in addition, which means that R does not produce a reflection, and hence it represents a rotation (an orientation-preserving orthogonal transformation). Indeed, when an orthogonal transformation matrix produces a reflection, its determinant is −1. Distance formula A measure of distance between points, or metric, is needed in order to confirm that a transformation is rigid. The Euclidean distance formula for is the generalization of the Pythagorean theorem. The formula gives the distance squared between two points and as the sum of the squares of the distances along the coordinate axes, that is where and , and the dot denotes the scalar product. Using this distance formula, a rigid transformation has the property, Translations and linear transformations A translation of a