Reflection (mathematics)

In mathematics, a reflection is a mapping from a Euclidean space to itself that acts as an isometry whose fixed points form a hyperplane, called the axis of reflection in two dimensions or the plane of reflection in three dimensions, and the image of a figure under a reflection is its mirror image across that axis or plane. A reflection is an involution, meaning that applying it twice in succession returns every point to its original location, and the term is sometimes used more broadly for any non-identity isometry that is its own inverse, whose fixed points form a smaller affine subspace than a full hyperplane, such as a reflection through a single point, also called a central inversion. In a Euclidean vector space, reflection in the point at the origin is the same operation as negating every vector, and some mathematicians use the word flip as a synonym for reflection.