This article may be confusing or unclear to readers. In particular, the lead refers correctly to transformations of Euclidean spaces, while the sections describe only the case of Euclidean vector spaces or of spaces of coordinate vectors. The "formal definition" section does not specify which kind of objects are represented by the variables, call them vaguely as "vectors", suggests implicitly that a basis and a dot product are defined for every kind of vectors. (August 2021)
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.[self-published source]
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 E(n) for n-dimensional Euclidean spaces. The set of proper rigid transformations 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
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,
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.
A measure of distance between points, or metric, is needed in order to confirm that a transformation is rigid. The Euclidean distance formula for Rn is the generalization of the Pythagorean theorem. The formula gives the distance squared between two points X and Y as the sum of the squares of the distances along the coordinate axes, that is
Using this distance formula, a rigid transformation g : Rn → Rn has the property,
Translations and linear transformations
A translation of a vector space adds a vector d to every vector in the space, which means it is the transformation
It is easy to show that this is a rigid transformation by showing that the distance between translated vectors equal the distance between the original vectors:
A linear transformation of a vector space, L : Rn → Rn, preserves linear combinations,
where [L] is an n×n matrix.
A linear transformation is a rigid transformation if it satisfies the condition,
Matrices that satisfy this condition form a mathematical group under the operation of matrix multiplication called the orthogonal group of n×n matrices and denoted O(n).
Compute the determinant of the condition for an orthogonal matrix to obtain
The set of rotation matrices is called the special orthogonal group, and denoted SO(n). It is an example of a Lie group because it has the structure of a manifold.