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Concept
Point reflection
Summary
In geometry, a point reflection (also called a point inversion or central inversion) is an transformation of affine space in which every point is reflected across a specific fixed point.
A point reflection is an involution: applying it twice is the identity transformation. It is equivalent to a homothetic transformation with scale factor −1. The point of inversion is also called homothetic center.
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. A point group including a point reflection among its symmetries is called centrosymmetric.
In Euclidean space, a point reflection is an isometry (preserves distance). In the Euclidean plane, a point reflection is the same as a half-turn rotation (180° or π radians).
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 involutions, meaning 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. More narrowly, a reflection refers to a reflection in a hyperplane ( dimensional affine subspace – a point on the line, a line in the plane, a plane in 3-space), with the hyperplane being fixed, but more broadly reflection is applied to any involution of Euclidean space, and the fixed set (an affine space of dimension k, where ) is called the mirror. In dimension 1 these coincide, as a point is a hyperplane in the line.
In terms of linear algebra, assuming the origin is fixed, involutions are exactly the diagonalizable maps with all eigenvalues either 1 or −1. Reflection in a hyperplane has a single −1 eigenvalue (and multiplicity on the 1 eigenvalue), while point reflection has only the −1 eigenvalue (with multiplicity n).
The term inversion should not be confused with inversive geometry, where inversion is defined with respect to a circle.
Official source
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