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Formal definition. 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,
Let X be an affine space over a field k, and V be its associated vector space. An affine transformation is a bijection f from X onto itself that is an affine map; this means that a linear map g from V to V is well defined by the equation () = (); here, as usual, the subtraction of two points denotes the free vector from the second point to the first one, and "well-defined" means that ...
An affine deformation is a deformation that can be completely described by an affine transformation. Such a transformation is composed of a linear transformation (such as rotation, shear, extension and compression) and a rigid body translation. Affine deformations are also called homogeneous deformations. [3]
The Euclidean group is a subgroup of the group of affine transformations. It has as subgroups the translational group T( n ), and the orthogonal group O( n ). Any element of E( n ) is a translation followed by an orthogonal transformation (the linear part of the isometry), in a unique way: x ↦ A ( x + b ) {\displaystyle x\mapsto A(x+b)} where ...
Transformation matrix. In linear algebra, linear transformations can be represented by matrices. If is a linear transformation mapping to and is a column vector with entries, then for some matrix , called the transformation matrix of . [citation needed] Note that has rows and columns, whereas the transformation is from to .
Isomorphism. The group of fifth roots of unity under multiplication is isomorphic to the group of rotations of the regular pentagon under composition. In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if ...
The similarities group S is itself a subgroup of the affine group, so every similarity is an affine transformation. One can view the Euclidean plane as the complex plane, [b] that is, as a 2-dimensional space over the reals. The 2D similarity transformations can then be expressed in terms of complex arithmetic and are given by
Translation T is a direct isometry: a rigid motion. [1] In mathematics, an isometry (or congruence, or congruent transformation) is a distance -preserving transformation between metric spaces, usually assumed to be bijective. [a] The word isometry is derived from the Ancient Greek: ἴσος isos meaning "equal", and μέτρον metron meaning ...