Rigid transformation

In mathematics, a rigid transformation (isometry) of a vector space preserves distances between every pair of points.^[1][2] Rigid transformations of the plane R², space R³, or real n-dimensional space Rⁿ are termed a Euclidean transformation because they form the basis of Euclidean geometry.^[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 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 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

$d({\mathbf {X}},{\mathbf {Y}})^{2}=(X_{1}-Y_{1})^{2}+(X_{2}-Y_{2})^{2}+\ldots +(X_{n}-Y_{n})^{2}=({\mathbf {X}}-{\mathbf {Y}})\cdot ({\mathbf {X}}-{\mathbf {Y}}).$

where X=(X₁, X₂, ..., X_n) and Y=(Y₁, Y₂, ..., Y_n), and the dot denotes the scalar product.

Using this distance formula, a rigid transformation g:Rⁿ→Rⁿ has the property,

$d(g({\mathbf {X}}),g({\mathbf {Y}}))^{2}=d({\mathbf {X}},{\mathbf {Y}})^{2}.$

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,

$d({\mathbf {v}}+{\mathbf {d}},{\mathbf {w}}+{\mathbf {d}})^{2}=({\mathbf {v}}+{\mathbf {d}}-{\mathbf {w}}-{\mathbf {d}})\cdot ({\mathbf {v}}+{\mathbf {d}}-{\mathbf {w}}-{\mathbf {d}})=({\mathbf {v}}-{\mathbf {w}})\cdot ({\mathbf {v}}-{\mathbf {w}})=d({\mathbf {v}},{\mathbf {w}})^{2}.$

A linear transformation of a vector space, L:Rⁿ→ Rⁿ, has the property that the transformation of a vector, V=av+bw, is the sum of the transformations of its components, that is,

$L({\mathbf {V}})=L(a{\mathbf {v}}+b{\mathbf {w}})=aL({\mathbf {v}})+bL({\mathbf {w}}).$

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,

$d([L]{\mathbf {v}},[L]{\mathbf {w}})^{2}=d({\mathbf {v}},{\mathbf {w}})^{2},$

that is

$d([L]{\mathbf {v}},[L]{\mathbf {w}})^{2}=([L]{\mathbf {v}}-[L]{\mathbf {w}})\cdot ([L]{\mathbf {v}}-[L]{\mathbf {w}})=([L]({\mathbf {v}}-{\mathbf {w}}))\cdot ([L]({\mathbf {v}}-{\mathbf {w}})).$

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

$d([L]{\mathbf {v}},[L]{\mathbf {w}})^{2}=({\mathbf {v}}-{\mathbf {w}})^{T}[L]^{T}[L]({\mathbf {v}}-{\mathbf {w}}).$

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

$[L]^{T}[L]=[I],$

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

$\det([L]^{T}[L])=\det[L]^{2}=\det[I]=1,$

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

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