originsymmetric

Origin-symmetric is a property of a set, figure, or function indicating invariance under the negation map x → -x, or equivalently under a 180-degree rotation about the origin. A subset S of Euclidean space is origin-symmetric if x ∈ S implies -x ∈ S. In many contexts this is also called central symmetry about the origin.

For graphs of functions, origin symmetry means the graph is invariant under the point reflection (x, y) → (-x, -y). Equivalently, the function satisfies f(-x) = -f(x) for all x in the domain, i.e., the function is odd.

Examples include a circle centered at the origin (x^2 + y^2 = r^2), and any ellipse centered at the origin. Lines passing through the origin, such as y = m x, are origin-symmetric as their points come in opposite pairs. A regular polygon centered at the origin with opposite points paired is origin-symmetric. By contrast, a parabola y = x^2 is not origin-symmetric, and a circle centered at (a, 0) with a ≠ 0 is not origin-symmetric.

Notes and relationships: origin symmetry is a special case of central symmetry about the origin. It does not follow from symmetry about the x- or y-axes alone. Translating an origin-symmetric set by a nonzero vector destroys origin symmetry but may yield symmetry about other centers.

See also central symmetry, odd function, symmetry (mathematics).