Projective Relations on the Sphere

1. Projective Relations on the Sphere Eliseo Clementini University of L’Aquila eliseo@ing.univaq.it 2nd International Workshop on Semantic and Conceptual Issues in GIS (SeCoGIS 2008) – 20 October 2008, Barcelona

2. Presentation summary • Introduction • The geometry of the sphere • The 5-intersection on the plane • Projective relations among points on the sphere • Projective relations among regions on the sphere • Expressing cardinal directions • Conclusions & Future Work

3. Introduction • A flat Earth: • mostspatial data models are 2D • models for spatial relations are 2D • Do these models work for the sphere? • Intuitive facts on the Earth surface cannot be represented: • A is East of B, but it could also be A is West of B (Columbus teaches!) • any place is South of the North Pole (where do we go from the North Pole?)

4. Introduction • state of the art • qualitative spatial relations • 2D or 3D topological relations • 2D or 3D projective relations • topological relations on the sphere (Egenhofer 2005) • proposal • projective relations on the sphere • JEPD set of 42 relations

5. The geometry of the sphere • The Earth surface is topologically equivalent to the sphere • Straight lines equivalent to the great circles • For 2 points a unique great circle, but if the 2 points are antipodal there are infinite many great circles through them.

6. The geometry of the sphere • Two distinct great circles divide the sphere into 4 regions: each region has two sides and is called a lune. • What’s the inside of a region?

7. The geometry of the sphere • The convex hull of a region A is the intersection of all the hemispheres that contain A • The convex hull of a region can be defined if the region is entirely contained inside a hemisphere. • A convex region is always contained inside a hemisphere.

8. The 5-intersection on the plane Leftside(B,C) • It is a model for projective relations • It is based on the collinearity invariant • It describes ternary relations among a primary object A and two reference objects B and C Before(B,C) C B Between(B,C) After(B,C) Rightside(B,C)

9. The 5-intersection on the plane Outside(B,C) • Special case of intersecting convex hulls of B and C • 2-intersection C B Inside(B,C)

10. The 5-intersection on the plane P1 P1 • case of points • P1 can be between, leftside, before, rightside, after points P2 and P3 • P1 can be inside or outside points P2 and P3 if they are coincident P3 P1 P2 P1 P1

11. Projective relations for points on the sphere • case of points • P1 can be between, leftside, rightside, nonbetween points P2 and P3 • Special cases: • P2, P3 coincident • Relations inside, outside • P2, P3 antipodal • Relations in_antipodal, out_antipodal

12. Projective relations for regions on the sphere • Plain case: • External tangents exist if B and C are in the same hemisphere • Internal tangents exist if convex hulls of B and C are disjoint • Relations between, rightside, before, leftside, after

13. Projective relations for regions on the sphere • Special cases: • reference regions B, C contained in the same hemisphere, but with intersecting convex hulls (there are no internal tangents) • Relations inside and outside

14. Projective relations for regions on the sphere • Special cases: • reference regions B, C are not contained in the same hemisphere, but they lie in two opposite lunes (there are no external tangents but still the internal tangents subdivides the sphere in 4 lunes) • It is not possible to define a between region and a shortest direction between B and C • relations B_side, C_side, BC_opposite

15. Projective relations for regions on the sphere • Special cases: • If B and C’s convex hulls are not disjoint and B and C do not lie on the same hemisphere, there are no internal tangents and the convex hull of their union coincides with the sphere. • Relation entwined

16. Projective relations for regions on the sphere • The JEPD set of projective relations for three regions on the sphere is given by all possible combinations of the following basic sets: • between, rightside, before, leftside, after (31 combined relations); • inside, outside (3 combined relations); • B_side, C_side, BC_opposite (7 combined relations); • entwined (1 relation). • In summary, in the passage from the plane to the sphere, we identify 8 new basic relations. The set of JEPD relations is made up of 42 relations.

17. Expressing cardinal directions • Set of relations (North, East, South, West) applied between a reference region R2 and a primary region R1. • Possiblemapping: • North = Between(R2, North Pole). • South = Before(R2, North Pole) • East = Rightside (R2, North Pole) • West = Leftside (R2, North Pole) • undetermined dir= After(R2, North Pole) • Alternative mapping: • North = Between(R2, North Pole) – CH(R2) • …

18. Conclusions • Extension of a 2D model for projective relations to the sphere • For points, no before/after distinction • For regions, again 5 intersections plus 8 new specific relations • Mapping projective relations to cardinal directions Further work • Spatial reasoning on the sphere • Refinement of the basic geometric categorization in four directions, taking also into account user and context-dependent aspects that influence the way people reason with cardinal directions • Integration of qualitative projective relations in web tools, such as Google Earth

19. Thank You Any Questions? Thanks for your Attention!!! Eliseo Clementini eliseo@ing.univaq.it