Point reflection

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Dual tetrahedra that are centrally symmetric to each other

In geometry, a point reflection (also called a point inversion or central inversion) is a transformation of affine space in which every point is reflected across a specific fixed point. When dealing with crystal structures and in the physical sciences the terms inversion symmetry, inversion center or centrosymmetric are more commonly used.

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).^[1] In the Euclidean plane, a point reflection is the same as a half-turn rotation (180° or π radians); a point reflection through the object's centroid is the same as a half-turn spin.