Point reflection

In geometry, a point reflection or inversion in a point (or inversion through a point, or 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. In geometry, a point reflection or inversion in a point (or inversion through a point, or 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 an affine transformation. Namely, it is an isometric involutive affine transformation, which has exactly one fixed point, which is the point of inversion. It is equivalent to a homothetic transformation with scale factor equal to −1. 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 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 ( n − 1 {displaystyle n-1} 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 1 ≤ k ≤ n − 1 {displaystyle 1leq kleq n-1} ) 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 n − 1 {displaystyle n-1} 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. In two dimensions, a point reflection is the same as a rotation of 180 degrees. In three dimensions, a point reflection can be described as a 180-degree rotation composed with reflection across a plane perpendicular to the axis of rotation. In dimension n, point reflections are orientation-preserving if n is even, and orientation-reversing if n is odd. Given a vector a in the Euclidean space Rn, the 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. In Euclidean geometry, the inversion of a point X with respect to a point P is a point X* such that P is the midpoint of the line segment with endpoints X and X*. In other words, the vector from X to P is the same as the vector from P to X*.