A Solution to the Isolatitude, Equi-area, Hierarchical Pixel-Coordinate System

Gorski et al (1999b) have earlier presented the outline of a pixelisation-to-spherical-coordinate transformation scheme which simultaneously satisfies three properties which are especially useful for rapid analyses of maps on a sphere: (i) equal spacing of pixels along lines of constant latitude, (ii) equal pixel `areas' (solid angles) and (iii) hierarchical scaling with increasing numbers of pixels. Their outline is based on the division of the sphere into twelve regions covering equal solid angles, which are hierarchically subdivided in a way compatible with these three criteria. In this paper, a complete derivation of this scheme is presented, including, in particular, (1) the angle theta^* defining the limit between polar and equatorial regions, and (2) the transformations from the unit interval [0,1] \wedge [0,1] to spherical coordinates in a polar region.