centerreflection

Center reflection is a type of symmetry transformation found in Euclidean geometry. The transformation maps every point P in a plane or space to a point P′ such that the center point C lies exactly midway between P and P′. Algebraically, a center reflection with respect to a center C can be described by the mapping x ↦ 2C – x, where x and C are position vectors. This operation is also called point reflection, central symmetry, or inversion about a point. A center reflection is an isometry; it preserves distances between points but reverses orientation, making it an improper isometry.

Because of its nature, a center reflection has several useful properties. First, applying a center reflection twice returns every point to its original position, so the transformation is an involution. Additionally, a center reflection followed by a translation can be combined to give a new center reflection whose center is shifted accordingly, leading to the group structure of symmetries in Euclidean space. In three dimensions, center reflection is the same as negation of all coordinates about a chosen origin, and it can be represented by the matrix −I in Cartesian coordinates.

Center reflections frequently occur in crystallography, where they describe point symmetry in crystal lattices, and in physics, where elementary particles that are their own antiparticles are said to be symmetric under such transformation. The concept also appears in computer graphics as a simple way to mirror an object around a point.