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In mathematics, a rigid transformation (isometry) of a vector space preserves distances between every pair of points. Rigid transformations of the plane \(R^2\), space \(R^3\), or real n-dimensional space \(R^n\) are termed a Euclidean transformation because they form the basis of Euclidean geometry. The rigid transformations include rotations, translations, reflections, or their combination. Sometimes reflections are excluded from the definition of a rigid transformation by imposing that the transformation also preserve the handedness of figures 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, this smaller class of transformations is known as proper rigid transformations (informally, also known as roto-translations). In general, any proper rigid transformation can be decomposed as a rotation followed by a translation, while any rigid transformation can be decomposed as an improper rotation followed by a translation (or as 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 group called the Euclidean group, denoted E(n) for n-dimensional Euclidean spaces. The set of proper rigid transformation is called special Euclidean group, denoted SE(n). 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. A rigid transformation is formally defined as a transformation that, when acting on any vector v, produces a transformed vector T(v) of the form
\[T(v) = R v + t \]
where RT = R−1 (i.e., R is an orthogonal transformation), and t is a vector giving the translation of the origin.
A proper rigid transformation has, in addition,
\[ \det(R) = 1\]
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.
I hope I was helpful! \(\Huge\ddot\smile\)