Practical Properties of the Spiral Points

Robert Raskin

Jet Propulsion Lab, 300-320
Pasadena, CA 91109

Email: rob.raskin@jpl.nasa.gov


A uniform hexagonal grid over the sphere can be closely approximated by a set known as the spiral points (Rakhmanov et al., 1994; Saff and Kuijlaars, 1997). These points are generated along a spiral from pole to pole, with successive points separated in latitude by a fixed increment of sin(latitude) and in longitude by a fixed arc length along a parallel. Rakhmanov et al. found that this family of grids possesses excellent asymptotic properties with respect to uniformity, although the reasons for its fine performance and close approximation to a hexagonal grid are not known. A significant advantage of this set is that it can be generated for any number of grid points n. Some practical properties of this grid are presented, including possible data structures, methods of finding adjacent points, and uniformity of point distribution, based upon empirical tests performed for a wide range of values of n.

Rakhmanov, E.A., E.B. Saff, and Y. M. Zhou, Minimal discrete energy on the sphere, Mathematical Research Letters, 1, 647, 1994. Saff, E. B. and A.B.J. Kuijlaars, Distributing many points on the sphere, Mathematical Intelligencer, 19, 5, 1997.