Astronomy Application: (National Virtual Observatory data) What Ptree dimension and ordering should be used for astronomical data?, where all bodies are assumed to lie on the surface of a celestial sphere (shares its origin and equatorial plane with earth but has no specified radius) Hierarchical Triangle Mesh Tree (HTM-tree, seems to be an accepted standard) Peano Triangle Mesh Tree (PTM-tree) is a [better?] alternative - at least for data mining? (Note: RA=Recession Angle (=longitudinal angle); dec=declination (=latitudinal angle) PTM is similar to HTM used in the Sloan Digital Sky Survey project (which is a project to create a National Virtual Observatory of all [?] telescope data integrated into one repository). In both: The Celestial Sphere is divided into triangles with great circle segment sides. But PTM differs from HTM in the way in which these triangles are ordered at each level. Section 3 # 29

The difference between HTM and PTM-trees is in the ordering. 1,3,3 1,3,2 1,3,0 1,3,1 1,1,2 1,1,0 1,1,1 1.1.3 1,2 1,3 1,0 1,1 1,2 1,1 1,0 1,3 1 1 Ordering of PTM-tree Ordering of HTM Section 3 # 30 Why use a different ordering?

PTM Triangulation of the Celestial Sphere The following ordering produces a sphere-surface filling curve with good continuity characteristics, The picture at right shows the earth (blue ball at the center) and the celestial sphere out around it. Traverse southern hemisphere in the revere direction (just the identical pattern pushed down instead of pulled up, arriving at the Southern neighbor of the start point. Next, traverse the southern hemisphere in the revere direction (just the identical pattern pushed down instead of pulled up, arriving at the Southern neighbor of the start point. left Equilateral triangle (90o sector) bounded by longitudinal and equatorial line segments right right RA dec left turn Traverse the next level of triangulation, alternating again with left-turn, right-turn, left-turn, right-turn.. Section 3 # 31

PTM-triangulation - Next Level LRLR RLRL LRLR RLRL LRLR RLRL LRLR RLRL Section 3 # 32

Sphere  Cylinder  Plane Peano Celestial Coordinates Unlike PTM-trees which initially partition the sphere into the 8 faces of an octahedron, in the PCCtree scheme: The Sphere is tranformed to a cylinder, then into a rectangle, then standard Peano ordering is used on the Celestial Coordinates. Celestial Coordinates Recession Angle (RA) runs from 0 to 360o dand Declination Angle (dec) runs from -90o to 90o. South Plane 90o 0o -90o 0o 360o Sphere  Cylinder  Plane Z Z Section 3 # 33

Hilbert Ordering? In 2-dimensions, Peano ordering is 22-recursive z-ordering (raster ordering) Hilbert ordering is 44-recursive tuning fork ordering (H-trees have fanout=16)

. . . . . . . . . . . . . . . . . . down down right left up down right 0 1 2 3 4 5 6 7 8 9 A B C D E F 0 1 2 3 4 5 6 7 8 9 A B C D E F down right left up down . . . . . . . . . 0 1 2 3 4 5 6 7 8 9 A B C D E F 0 1 2 3 4 5 6 7 8 9 A B C D E F . . . 0 1 2 3 4 5 6 7 8 9 A B C D E F 0 1 2 3 4 5 6 7 8 9 A B C D E F right down up 0 1 2 3 4 5 6 7 8 9 A B C D E F 0 1 2 3 4 5 6 7 8 9 A B C D E F 0 1 2 3 4 5 6 7 8 9 A B C D E F Coordinates of a tuning-fork (upper-left) depend on ancestry. (x,y) = (ggrrbb, ggrrbb). If your parent points Down and you are the H node in your tuning-fork, your 2-bit contribution is given by: row(x) col(y) 0  00 , 00 1  00 , 01 2  01 , 01 3  01 , 00 4  10 , 00 5  11 , 00 6  11 , 01 7  10 , 01 8  10 , 10 9  11 , 10 A  11 , 11 B  10 , 11 C  01 , 11 D  01 , 10 E  00 , 10 F  00 , 11 Lookup table for Up, Left, Right Parents are similar. 1 2 3 4 5 6 7 8 9 A B C D E F