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Concept
Rigid transformation
Summary
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.
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 proper rigid transformation, or rototranslation. Any proper rigid transformation can be decomposed into a rotation followed by a translation, while 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 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.
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
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 does not produce a reflection, and hence it represents a rotation (an orientation-preserving orthogonal transformation).
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Ontological neighbourhood
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Screw axis
A screw axis (helical axis or twist axis) is a line that is simultaneously the axis of rotation and the line along which translation of a body occurs. Chasles' theorem shows that each Euclidean displacement in three-dimensional space has a screw axis, and the displacement can be decomposed into a rotation about and a slide along this screw axis. Plücker coordinates are used to locate a screw axis in space, and consist of a pair of three-dimensional vectors. The first vector identifies the direction of the axis, and the second locates its position.
Geometric transformation
In mathematics, a geometric transformation is any bijection of a set to itself (or to another such set) with some salient geometrical underpinning. More specifically, it is a function whose domain and range are sets of points — most often both or both — such that the function is bijective so that its inverse exists. The study of geometry may be approached by the study of these transformations. Geometric transformations can be classified by the dimension of their operand sets (thus distinguishing between, say, planar transformations and spatial transformations).
Translation (geometry)
In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure, shape or space by the same distance in a given direction. A translation can also be interpreted as the addition of a constant vector to every point, or as shifting the origin of the coordinate system. In a Euclidean space, any translation is an isometry. Displacement (geometry) If is a fixed vector, known as the translation vector, and is the initial position of some object, then the translation function will work as .
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