Content  Hierarchical Triangle Mesh (HTM)  Perrizo Triangle Mesh Tree (PTM-tree)  SDSS

Subdivisions of the sphere  Sphere is divided into triangles  Triangle sides are always great circle segments.

Comparison of HTM and PTM-trees 1, 2 1, 3 1, 0 1,1 1 1,3,3 1,3,21,3,0 1,3,1 1,2 1,1 1,0 1,3 1 1,1,2 1,1,01,1, Ordering of HTM Ordering of PTM-tree

PTM-tree up to 2 Levels Traverse the south hemisphere in the revere direction (just the identical pattern pushed down instead of pulled up, arriving at the Southern neighbor of the start point.

PTM-tree up to 3 Levels LRLR LRLR

PTM-tree up to 4 Levels LRLR RLRL LRLR RLRL LRLR RLRL LRLR RLRL LRLR RLRL LRLR RLRL

A simpler alternative scheme: SDSS  Unlike PTM-tree which partitions the sphere into the 8 faces of an octahedron, SDSS partitions the sphere into a cylinder.  SDSS is stored in Galactic coordinates. Ra is from o. Dec is -90 o – 90 o.  SDSS keeps the Galactic coordinates. The cylinder has the perimeter of 360 o and height of 180 o (-90 o – 90 o ).

SDSS: Sphere  Cylinder  Plane North Plane South Plane 90 o 0 o -90 o 0 o 360 o

Structure of SDSS  SDSS is bitwise and quadrant index data structure. Like P-trees, it is built from a bit sequential file (bSQ).  SDSS recursive subdivides the half plain into four squares.  The squares of the same level are in Peano order except the first levels

Peano Order bSQ File Subdivision up to 4 Levels Z