In geometrya point reflection or inversion in a point or inversion through a pointor central inversion is a type of isometry of Euclidean space. An object that is invariant under a point reflection is said to possess point symmetry ; if it is invariant under point reflection through its center, it is said to possess central symmetry or to be centrally symmetric.
Point reflection can be classified as No Sense - Center - Point* - Reflection affine transformation. Namely, it is an isometric involutive affine transformation, which has exactly one fixed pointwhich is the point of inversion. The point of inversion is also called homothetic center.
The term reflection is loose, and considered by some an abuse of language, with inversion preferred; however, point reflection is widely used. Such maps are involutionsmeaning that they have order 2 — they are their own inverse: applying them twice yields the identity map — which is also true of other maps called reflections.
In dimension 1 these coincide, as a point is a hyperplane in the line. The term inversion should not be confused with inversive geometrywhere inversion is defined with respect to a circle. In two dimensions, a point reflection is the same as a rotation of degrees. In three dimensions, a point reflection can be described as a degree rotation composed with reflection across a plane perpendicular to the axis of rotation.
In dimension npoint reflections are orientation -preserving if n is even, and orientation-reversing if n is odd. Given a vector a in the Euclidean space R nthe formula for the reflection of a across the point p is.
In the case where p is the origin, point reflection is simply the negation of the vector a. This mapping is an isometric involutive affine transformation which has exactly one fixed pointwhich is P.
This is an example of linear transformation. This is an example of non-linear affine transformation. The composition of two point reflections is a translation.
The set consisting of Tori Amos - Tori The Fox point reflections and translations is Lie subgroup of the Euclidean group.
It is a semidirect product of R n with a cyclic group of order 2, the latter acting on R n by negation. It is precisely the subgroup of the Euclidean group that fixes the line at infinity pointwise. These rotations are mutually commutative. Therefore, inversion in a point in even-dimensional space is an orientation-preserving isometry or direct isometry.
Therefore, it reverses rather than preserves orientationit is an indirect isometry. The following point groups in three dimensions contain inversion:. Closely related to inverse in a point is No Sense - Center - Point* - Reflection in respect to a planewhich can be thought of as a "inversion in a plane".
The operation commutes with every other linear transformationbut not with translation : it is in the center of the general linear group. In mathematics, reflection through the origin refers to the point reflection of Euclidean space R n across the origin of the Cartesian coordinate system.
It is a product of n orthogonal reflections reflection through the axes of any orthogonal basis ; note that orthogonal reflections commute. Analogously, it No Sense - Center - Point* - Reflection a longest element of the orthogonal group, with respect to the generating set of reflections: elements of the orthogonal group all have length at most n with respect to the generating set No Sense - Center - Point* - Reflection reflections, [note 2] and reflection through the origin has length n, though it is not unique in this: other maximal combinations of rotations and possibly reflections also have maximal length.
In SO 2 rreflection through the origin is the farthest point from the identity element with respect to the usual metric. Reflection through the identity extends to an automorphism of a Clifford algebracalled the Playing With Lifes - Agathocles - If This Is Cruel Whats Vivisection Then? involution or grade involution.
Reflection through the identity lifts to a pseudoscalar. From Wikipedia, the free encyclopedia. Redirected from Central symmetry. Geometric symmetry operation. This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. Not to be confused with inversive geometryin which inversion is through a circle instead of a point.
Further information: Clifford algebra. Further information: Spin group. Categories : Euclidean symmetries Functions and mappings Clifford algebras Quadratic forms.
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