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
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.
Contents
Formal definition
Distance formula
Translations and linear transformations
References
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,
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
where X=(X1, X2, …, Xn) and Y=(Y1, Y2, …, Yn), and the dot denotes the scalar
product.
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
g(v): v→v+d.
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,
A linear transformation L can be represented by a matrix, which means
L: v→[L]v,
where [L] is an n×n 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 vTw, 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 Rn×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.
References
1. O. Bottema & B. Roth (1990). Theoretical Kinematics (https://books.google.co
m/books?id=f8I4yGVi9ocC&printsec=frontcover&dq=kinematics&lr=&as_brr=0
&sig=YfoHn9ImufIzAEp5Kl7rEmtYBKc#PPR7,M1). Dover Publications. reface.
ISBN 0-486-66346-9.
2. J. M. McCarthy (2013). Introduction to Theoretical Kinematics (https://itunes.ap
ple.com/us/book/introduction-to-theoretical/id625965964?mt=11). MDA Press.
reface.
3. Galarza, Ana Irene Ramírez; Seade, José (2007), Introduction to classical
geometries, Birkhauser
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