1. Spatial Searches in the ODM

2. slide 2 Common Spatial Questions Points in region queries 1.Find all objects in this region 2.Find all “good” objects (not in masked areas) 3.Is this point in any of the regions Region in region 4.Find regions near this region and their area 5.Find all objects with error boxes intersecting region 6.What is the common part of these regions Various statistical operations 7.Find the object counts over a given region list 8.Cross-match these two catalogs in the region

3. slide 3 Sky Coordinates of Points  Many different coordinate systems Equatorial, Galactic, Ecliptic, Supergalactic  Longitude-latitude constraints  Searches often in mix of different coordinate systems gb>40 and dec between 10 and 20 Problem: coordinate singularities, transformations  How can one describe constraints in a easy, uniform fashion?  How can one perform fast database queries in an easy fashion? Fast:Indexes Easy: simple query expressions

4. slide 4 Describing Regions Spacetime metadata for the VO (Arnold Rots)  Includes definitions of Constraint: single small or great circle Convex: intersection of constraints Region: union of convexes  Support both angles and Cartesian descriptions  Constructors for CIRCLE, RECTANGLE, POLYGON, CONVEX HULL  Boolean algebra (INTERSECTION, UNION, DIFF)  Proper language to describe the abstract regions  Similar to GIS, but much better suited for astronomy

5. slide 5 Things Can Get Complex

6. slide 6 We Do Spatial 3 Ways  Hierarchical Triangular Mesh (extension to SQL) Uses table valued functions Acts as a new “spatial access method”  Zones: fits SQL well Surprisingly simple & good  3D Constraints: a novel idea Algebra on regions, can be implemented in pure SQL

7. slide 7 PS1 Footprint  Using the projection cell definitions as centers for tessellation (T. Budavari)

8. slide 8 CrossMatch: Zone Approach  Divide space into declination zones  Objects ordered by zoneid, ra (on the sphere need wrap-around margin.)  Point search look in neighboring zones within ~ (ra ± Δ) bounding box  All inside the relational engine  Avoids “impedance mismatch”  Can “batch” comparisons  Automatically parallel  Details in Maria’s thesis r ra-zoneMax zoneMax x ra ± Δ

9. slide 9 Indexing Using Quadtrees  Cover the sky with hierarchical pixels  COBE – start with a cube  Hierarchical Triangular Mesh (HTM) uses trixels Samet, Fekete  Start with an octahedron, and split each triangle into 4 children, down to 20 levels deep  Smallest triangles are 0.3”  Each trixel has a unique htmID 2,2 2,1 2,0 2,3 2,3,0 2,3,1 2,3,22,3,3 21 23 20 22 222 223 220221

10. slide 10 Space-Filling Curve 100 103 102 101 120 1,2,1 122 121 110 113 112 111 132 133 130 131 [0.12,0.13) [0.122,0.123)[0.121,0.122)[0.120,0.121)[0.123,0.130) Triangles correspond to ranges All points inside the triangle are inside the range. [0.122,0.130) [0.120,0.121)

11. slide 11 SQL HTM Extension  Every object has a 20-deep htmID (44bits)  Clustered index on htmID  Table-valued functions for spatial joins Given a region definition, routine returns up to 10 ranges of covering triangles Spatial query is mapped to ~10 range queries  Current implementation rewritten in C#  Excellent performance, little calling overhead  Three layers General geometry library HTM kernel IO (parsing + SQL interface)

12. slide 12 Writing Spatial SQL -- region description is contained by @area DECLARE @cover TABLE (htmStart bigint,htmEnd bigint) INSERT @cover SELECT * from dbo.fHtmCover(@area) -- DECLARE @region TABLE ( convexId bigint,x float, y float, z float) INSERT @region SELECT dbo.fGetHalfSpaces(@area) -- SELECTo.ra, o.dec, 1 as flag, o.objid FROM (SELECT objID as objid, cx,cy,cz,ra,[dec] FROM Objects q JOIN @cover AS c ON q.htmID between c.HtmIdStart and c.HtmIdEnd ) AS o WHERE NOT EXISTS ( SELECT p.convexId FROM @region AS p WHERE (o.cx*p.x + o.cy*p.y + o.cz*p.z < p.c) GROUP BY p.convexId )

13. slide 13 Status  All three libraries extensively tested  Zones used for Maria’s thesis, plus various papers  New HTM code in production use since July on SDSS  Same code also used by STScI HLA, Galex  Systematic regression tests developed  Footprints computed for all major surveys  Complex mask computations done on SDSS  Loading: zones used for bulk crossmatch  Ad hoc queries: use HTM-based search functions  Excellent performance