# Rigid transformation

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.^{[1]}^{[self-published source]}^{[2]}^{[3]}

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**.^{[citation needed]} 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 *R*^{T} = *R*^{−1} (i.e., *R* is an orthogonal transformation), and **t** is a vector giving the translation of the origin.

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 **R**^{n} 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

where **X**=(X_{1}, X_{2}, …, X_{n}) and **Y**=(Y_{1}, Y_{2}, …, Y_{n}), and the dot denotes the scalar product.

Using this distance formula, a rigid transformation *g*:R^{n}→R^{n} has the property,

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*: **R**^{n}→ **R**^{n}, preserves linear combinations,

A linear transformation is a rigid transformation if it satisfies the condition,

Now use the fact that the scalar product of two vectors **v**.**w** can be written as the matrix operation **v**^{T}**w**, where the T denotes the matrix transpose, we have

Thus, the linear transformation *L* is rigid if its matrix satisfies the condition

where [I] is the identity matrix. Matrices that satisfy this condition are called *orthogonal matrices.* This condition actually requires the columns of these matrices to be orthogonal unit vectors.

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

which shows that the matrix [L] can have a determinant of either +1 or −1. Orthogonal matrices with determinant −1 are reflections, and those with determinant +1 are rotations. Notice that the set of orthogonal matrices can be viewed as consisting of two manifolds in **R**^{n×n} separated by the set of singular matrices.

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.