Rigid transformation
Encyclopedia
In mathematics, a rigid transformation or a Euclidean transformation is a transformation
from a Euclidean space
to itself that preserves distances between every pair of points (isometry
). In particular, any object will have the same shape
and size
before and after a rigid transformation.
Rigid transformations include rotations
, translations, or their combination (sometimes called roto-translations). 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. To avoid ambiguity, this smaller class of transformations is known as proper rigid transformations. In general, any proper rigid transformation can be decomposed as a rotation followed by a translation. Any rigid transformation can be decomposed as an improper rotation followed by a translation (or as a sequence of reflections).
The set of all proper rigid transformations is a group
called the Euclidean group
.
All rigid transformations are affine transformations. Rigid transformations which involve a translation are not linear transformation
s. Not all transformations are rigid transformations. An example is a shear
, which changes two axes in different ways, or a similarity transformation, which preserves angles but not lengths.
In mechanics
, (proper) rigid transformations are used to represent the linear
and angular displacement
of rigid bodies
.
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 is a rotation
(an orientation-preserving orthogonal transformation).
Transformation
-Mathematics:* Transformation * Transformation * Integral transform* Data transformation * Transformation matrix-Natural science:* Phase transformation, a physical transition from one medium to another...
from a Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
to itself that preserves distances between every pair of points (isometry
Isometry
In mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...
). In particular, any object will have the same shape
Shape
The shape of an object located in some space is a geometrical description of the part of that space occupied by the object, as determined by its external boundary – abstracting from location and orientation in space, size, and other properties such as colour, content, and material...
and size
Size
The word size may refer to how big something is. In particular:* Measurement, the process or the result of determining the magnitude of a quantity, such as length or mass, relative to a unit of measurement, such as a meter or a kilogram...
before and after a rigid transformation.
Rigid transformations include rotations
Rotation (mathematics)
In geometry and linear algebra, a rotation is a transformation in a plane or in space that describes the motion of a rigid body around a fixed point. A rotation is different from a translation, which has no fixed points, and from a reflection, which "flips" the bodies it is transforming...
, translations, or their combination (sometimes called roto-translations). Sometimes reflections
Reflection (mathematics)
In mathematics, a reflection is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as set of fixed points; this set is called the axis or plane of reflection. The image of a figure by a reflection is its mirror image in the axis or plane of reflection...
are excluded from the definition of a rigid transformation by imposing that the transformation also preserve the handedness
Orientation (mathematics)
In mathematics, orientation is a notion that in two dimensions allows one to say when a cycle goes around clockwise or counterclockwise, and in three dimensions when a figure is left-handed or right-handed. In linear algebra, the notion of orientation makes sense in arbitrary dimensions...
of figures in the Euclidean space. To avoid ambiguity, this smaller class of transformations is known as proper rigid transformations. In general, any proper rigid transformation can be decomposed as a rotation followed by a translation. Any rigid transformation can be decomposed as an improper rotation followed by a translation (or as a sequence of reflections).
The set of all proper rigid transformations is a group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
called the Euclidean group
Euclidean group
In mathematics, the Euclidean group E, sometimes called ISO or similar, is the symmetry group of n-dimensional Euclidean space...
.
All rigid transformations are affine transformations. Rigid transformations which involve a translation are not linear transformation
Linear transformation
In mathematics, a linear map, linear mapping, linear transformation, or linear operator is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. As a result, it always maps straight lines to straight lines or 0...
s. Not all transformations are rigid transformations. An example is a shear
Shear (mathematics)
In mathematics, shear mapping or transvection is a particular kind of linear mapping. Linear mapping is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication...
, which changes two axes in different ways, or a similarity transformation, which preserves angles but not lengths.
In mechanics
Mechanics
Mechanics is the branch of physics concerned with the behavior of physical bodies when subjected to forces or displacements, and the subsequent effects of the bodies on their environment....
, (proper) rigid transformations are used to represent the linear
Displacement (vector)
A displacement is the shortest distance from the initial to the final position of a point P. Thus, it is the length of an imaginary straight path, typically distinct from the path actually travelled by P...
and angular displacement
Angular displacement
Angular displacement of a body is the angle in radians through which a point or line has been rotated in a specified sense about a specified axis....
of rigid bodies
Rigid body
In physics, a rigid body is an idealization of a solid body of finite size in which deformation is neglected. In other words, the distance between any two given points of a rigid body remains constant in time regardless of external forces exerted on it...
.
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,
- detDeterminantIn linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...
(R) = 1
which means that R is a rotation
Rotation (mathematics)
In geometry and linear algebra, a rotation is a transformation in a plane or in space that describes the motion of a rigid body around a fixed point. A rotation is different from a translation, which has no fixed points, and from a reflection, which "flips" the bodies it is transforming...
(an orientation-preserving orthogonal transformation).