Geometric and Algebraic Properties of HEALPix

HEALPix is a genuinely curvilinear partition of the sphere into exactly equal area quadrilaterals of varying shape. The base-resolution comprises twelve pixels in three rings around the poles and equator.

The resolution of the grid is expressed by the parameter N_side which defines the number of divisions along the side of a base-resolution pixel that is needed to reach a desired high-resolution partition.

All pixel centers are placed on 4 x N_side-1 rings of constant latitude, and are equidistant in azimuth (on each ring). All iso-latitude rings located between the upper and lower corners of the equatorial base-resolution pixels, the equatorial zone, are divided into the same number of pixels: N_eq= 4 x N_side. The remaining rings are located within the polar cap regions and contain a varying number of pixels, increasing from ring to ring with increasing distance from the poles by one pixel within each quadrant.

Pixel boundaries are non-geodesic and take the very simple forms $\cos \theta = a \pm b \cdot \phi$ in the equatorial zone, and $\cos \theta = a + b / \phi^2$ , or $\cos \theta = a + b / (\pi/2 - \phi) ^2$ , in the polar caps. This allows one to explicitly check by simple analytical integration the exact area equality among pixels, and to compute efficiently more complex objects, e.g. the Fourier transforms of individual pixels.

**Figure 3:** Cylindrical projection of the **HEALPix** division of a sphere and two natural pixel numbering schemes (RING and NESTED) allowed by **HEALPix**. Both numbering schemes map the two dimensional distribution of discrete area elements on a sphere into the one dimensional, integer pixel number array, which is essential for computations involving data sets with very large total pixel numbers. From top to bottom: Panel one (resolution parameter N_side = 2) and panel two (N_side = 4) show the RING scheme for pixel numbering, with the pixel number winding down from north to south pole through the consecutive isolatitude rings. Panel three (resolution parameter N_side = 2) and panel four (N_side = 4) show the NESTED scheme for pixel numbering within which the pixel number grows with consecutive hierarchical subdivisions on a tree structure seeded by the twelve base-resolution pixels.

Specific geometrical properties allow HEALPix to support two different numbering schemes for the pixels, as illustrated in Figure 3.

First, in the RING scheme, one can simply count the pixels moving down from the north to the south pole along each iso-latitude ring. It is in the RING scheme that Fourier transforms with spherical harmonics are easy to implement.

Second, in the NESTED scheme, one can arrange the pixel indices in twelve tree structures, corresponding to base-resolution pixels. Each of those is organised as shown in Fig. 1. This can easily be implemented since, due to the simple description of pixel boundaries, the analytical mapping of the HEALPix base-resolution elements (curvilinear quadrilaterals) into a [0,1] x [0,1] square exists. This tree structure allows one to implement efficiently all applications involving nearest-neighbour searches [Wandelt, Hivon & Górski (1998)], and also allows for an immediate construction of the fast Haar wavelet transform on HEALPix.