

Geometric and Algebraic Properties of HEALPixHEALPix is a genuinely curvilinear partition of the sphere into exactly equal area quadrilaterals of varying shape. The baseresolution 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 baseresolution pixel that is needed to reach a desired highresolution 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 isolatitude rings located between the upper and lower corners of the equatorial baseresolution 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 nongeodesic and take the very simple forms in the equatorial zone, and , or , 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.
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 isolatitude 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 baseresolution 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 baseresolution elements (curvilinear quadrilaterals) into a [0,1] x [0,1] square exists. This tree structure allows one to implement efficiently all applications involving nearestneighbour searches [Wandelt, Hivon & Górski (1998)], and also allows for an immediate construction of the fast Haar wavelet transform on HEALPix. Eric Hivon 20100618 



