aspheric

Aspheric is a term used across disciplines to describe non-spherical forms or spaces with specific topological properties. In optics it refers to a surface whose profile deviates from a sphere to correct aberrations; in mathematics and geometry it refers to spaces whose higher homotopy groups vanish.

In optics, aspheric lenses are designed to reduce spherical aberration and improve optical performance, especially for wide fields of view. They can be based on conic-section profiles or incorporate higher-order polynomial terms. Aspheric elements are common in eyeglasses, camera lenses, and laser systems, enabling thinner, lighter lenses and sharper images. The surface sag describes how the height varies with radius and is often expressed by a conic-based formula with additional aspheric terms. Manufacturing methods include precision grinding and polishing, computer-controlled shaping, and molding, with quality control by interferometric testing.

In mathematics and topology, an aspherical space X is defined by the vanishing of all higher homotopy groups: π_n(X)=0 for every n≥2. Equivalently, the universal covering space of X is contractible. This property implies that the fundamental group π_1(X) captures the essential topological information. Examples include the torus and surfaces of genus greater than zero, as well as many manifolds with nonpositive curvature or hyperbolic geometry. Asphericity is a central notion in geometric topology, linking geometry, algebraic topology, and group theory.

See also: spherical aberration, conic section, hyperbolic geometry, contractible, fundamental group.