geometry, a point reflection is a type of isometryof Euclidean space. It is a reflection whose mirror is a single point. An object that is invariant under a point reflection is said to possess point symmetry.
In two dimensions, a point reflection is the same as a
rotationof 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 R"n", 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.
Point reflection group
The composition of two point reflections is a translation. Specifically, point reflection at p followed by point reflection at q is translation by the vector 2(q – p).
The set consisting of all point reflections and translations is
Lie subgroupof the Euclidean group. It is a semidirect productof R"n" with a cyclic groupof order 2, the latter acting on R"n" by negation. It is precisely the subgroup of the Euclidean group that fixes the line at infinitypointwise.
In the case "n" = 1, the point reflection group is the full isometry group of the line.
Point reflections in mathematics
* Point reflection across the center of a sphere yields the
* A symmetric space is a
Riemannian manifoldwith an isometric reflection across each point. Symmetric spaces play an important role in the study of Lie groups and Riemannian geometry.
reflection (linear algebra)
Riemannian symmetric space
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