Mathematical transformation that preserves distances
This article may be confusing or unclear to readers. In particular, the lead refers correctly to transformations of Euclidean spaces, while the sections describe only the case of Euclidean vector spaces or of spaces of coordinate vectors. The "formal definition" section does not specify which kind of objects are represented by the variables, call them vaguely as "vectors", suggests implicitly that a basis and a dot product are defined for every kind of vectors. Please help clarify the article. There might be a discussion about this on the talk page.(August 2021) (Learn how and when to remove this message)
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 rigid motion, a Euclidean motion, or a proper rigid transformation.
In dimension two, a rigid motion is either a translation or a rotation. In dimension three, every rigid motion can be decomposed as the composition of a rotation and a translation, and is thus sometimes called a rototranslation. In dimension three, all rigid motions are also screw motions (this is Chasles' theorem)
In dimension at most three, 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 rigid motions is called the special Euclidean group, and denoted SE(n).
In kinematics, rigid motions in a 3-dimensional Euclidean space are used to represent displacements of rigid bodies. According to Chasles' theorem, every rigid transformation can be expressed as a screw motion.
^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
and 25 Related for: Rigid transformation information
In mathematics, a rigidtransformation (also called Euclidean transformation or Euclidean isometry) is a geometric transformation of a Euclidean space...
In physics, a rigid body, also known as a rigid object, is a solid body in which deformation is zero or negligible. The distance between any two given...
instance, active transformations are useful to describe successive positions of a rigid body. On the other hand, passive transformations may be useful in...
functions) Rigid body, in physics, a simplification of the concept of an object to allow for modelling Rigidtransformation, in mathematics, a rigid transformation...
every rigidtransformation that is not a rigid motion is the product of r and a rigid motion. A glide reflection is an example of a rigidtransformation that...
engine, a robotic arm or the human skeleton. Geometric transformations, also called rigidtransformations, are used to describe the movement of components in...
affine transformations: those where the determinant of A {\displaystyle A} is positive. In the last case this is in 3D the group of rigidtransformations (proper...
obtained using a rigidtransformation [Z] to characterize the relative movement allowed at each joint and separate rigidtransformation [X] to define the...
realization, if it exists, is unique up to rigidtransformations, i.e. distance-preserving transformations of Euclidean space (rotations, reflections...
infinitesimal transformation is a limiting form of small transformation. For example one may talk about an infinitesimal rotation of a rigid body, in three-dimensional...
to represent rigidtransformations in three dimensions. Since the space of dual quaternions is 8-dimensional and a rigidtransformation has six real degrees...
sequence of rigidtransformations along links and around joints in a mechanical system. The principle that the sequence of transformations around a loop...
an orthogonal transformation (an origin-preserving rigidtransformation) with a uniform scaling (dilation). All similarity transformations (which globally...
A real unitary matrix is an orthogonal matrix, which describes a rigidtransformation (an isometry of Euclidean space R k {\displaystyle \mathbb {R} ^{k}}...
dimension. This means: applying a rigidtransformation, followed by a translation and then the inverse rigidtransformation, has the same effect as a single...
dimensions is defined similarly. A rotation of axes is a linear map and a rigidtransformation. Coordinate systems are essential for studying the equations of curves...
mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective...
a combination of translations, rotations (together also called rigidtransformations), and uniform scalings. In other words, the shape of a set of points...
automorphisms are often called transformations, for example rigidtransformations, affine transformations, projective transformations. Category theory, which...
dimensions is defined similarly. A rotation of axes is a linear map and a rigidtransformation. 2D computer graphics#Rotation Cartan–Dieudonné theorem Clockwise...
rotations together form the rigid motions or rigid displacements. This set forms a group under composition, the group of rigid motions, a subgroup of the...
obtained using rigidtransformations [Z] to characterize the relative movement allowed at each joint and separate rigidtransformations [X] to define the...
aligning three dimensional models given an initial guess of the rigidtransformation required. The ICP algorithm was first introduced by Chen and Medioni...