Hyperspheres in digital geometry

An analysis of volumes and surface areas of hyperspheres defined by m-neighbor distance is carried out for n-dimensional grid-point space. Closed-form expressions for volumes and surface areas are derived. Analogous to Euclidean geometry, these expressions are found to be polynomials of degree n and n−1 respectively. Like the underlying m-neighbor distance, these measures are also shown to be order-preserving. An efficient algorithm for obtaining the coefficients of volume and surface polynomials has been presented. The utility of these measures as indices of digital approximations of Euclidean space is discussed.