circularring

Circularring is an informal term sometimes used to describe an annular region in the plane—a ring-shaped area bounded by two concentric circles. In formal mathematics, the standard term for this object is an annulus, and circularring appears in informal discussions, textbooks, or software documentation as a shorthand for that shape.

Formal definition: An annulus A is the set of points in the plane satisfying r ≤ |x| ≤ R, where 0 ≤ r < R and |x| denotes the distance from the origin. If r = 0, the region reduces to a disk, whereas for r > 0 the region has a hole in the center. The boundary of an annulus consists of two concentric circles: the inner circle of radius r and the outer circle of radius R.

Key properties: An annulus has exactly two boundary components, is connected, and is compact when considered as a subset of the plane. In polar coordinates, it is described by the radial bounds r ≤ ρ ≤ R. Topologically, it is homeomorphic to a cylinder S^1 × [0,1], and its fundamental group is isomorphic to the integers Z.

Applications: Annular regions appear in various areas of mathematics and applied fields. In complex analysis, they serve as domains of convergence for Laurent series. In potential theory and conformal mapping, annuli model rings and circular symmetry. In imaging and computer graphics, ring-shaped masks and filters resemble circularrings.

See also: annulus, ring, topology, polar coordinates, Laurent series. Note that circularring is a nonstandard label; the formal term is annulus.