# 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][2][3]

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 **rigid motions or **

*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.

## 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 *R*^{T} = *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.

## 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 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,

## 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 *g*(**v**):**v**→**v**+**d**. It is easy to show that this is a rigid transformation by computing,

A linear transformation of a vector space, *L*:R^{n}→ R^{n}, has the property that the transformation of a vector, **V**=a**v**+b**w**, is the sum of the transformations of its components, that is,

Each linear transformation *L* can be formulated as a matrix operation, which means *L*:**v**→[L]**v**, where [L] is an nxn matrix.

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

that is

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 nxn 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^{nxn} 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.

## References

- O. Bottema & B. Roth (1990).
*Theoretical Kinematics*. Dover Publications. reface. ISBN 0-486-66346-9. - J. M. McCarthy (2013).
*Introduction to Theoretical Kinematics*. MDA Press. reface. - Galarza, Ana Irene Ramírez; Seade, José (2007),
*Introduction to classical geometries*, Birkhauser