1. Eliseo Clementini University of L’Aquila eliseo@ing.univaq.it 2 nd International Workshop on Semantic and Conceptual Issues in GIS (SeCoGIS 2008) – 20 October 2008, Barcelona 13/04/20151

2. Presentation summary 1.Introduction 2.The geometry of the sphere 3.The 5-intersection on the plane 4.Projective relations among points on the sphere 5.Projective relations among regions on the sphere 6.Expressing cardinal directions 7.Conclusions & Future Work 1.Introduction 2.The geometry of the sphere 3.The 5-intersection on the plane 4.Projective relations among points on the sphere 5.Projective relations among regions on the sphere 6.Expressing cardinal directions 7.Conclusions & Future Work 13/04/20152

3. Introduction A flat Earth: – most spatial 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?) A flat Earth: – most spatial 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?) 13/04/20153

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 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 13/04/20154

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. 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. 13/04/20155

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? 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? 13/04/20156

7. 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. 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. The geometry of the sphere 13/04/20157

8. 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 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 The 5-intersection on the plane B C Between(B,C) Rightside(B,C) Leftside(B,C) Before(B,C) After(B,C) A  Leftside(B,C) A  Before(B,C) A  Between(B,C) A  After(B,C) A  Rightside(B,C) A  Leftside(B,C) A  Before(B,C) A  Between(B,C) A  After(B,C) A  Rightside(B,C) 13/04/20158

9. Special case of intersecting convex hulls of B and C 2-intersection Special case of intersecting convex hulls of B and C 2-intersection The 5-intersection on the plane B C Inside(B,C) Outside(B,C) A  Inside(B,C) A  Outside(B,C) 13/04/20159

10. 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 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 The 5-intersection on the plane P2 P3 P1 13/04/201510

11. 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 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 Projective relations for points on the sphere 13/04/201511

12. –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 –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 Projective relations for regions on the sphere A  Leftside(B,C) A  Before(B,C) A  Between(B,C) A  After(B,C) A  Rightside(B,C) A  Leftside(B,C) A  Before(B,C) A  Between(B,C) A  After(B,C) A  Rightside(B,C) 13/04/201512

13. –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 –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 Projective relations for regions on the sphere 13/04/2015 13 A  Inside(B,C) A  Outside(B,C)

14. –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 –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 Projective relations for regions on the sphere 13/04/201514

15. –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 –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 Projective relations for regions on the sphere 13/04/201515

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. 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. 13/04/201516

17. Expressing cardinal directions Set of relations (North, East, South, West) applied between a reference region R2 and a primary region R1. Possible mapping: –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) –… Set of relations (North, East, South, West) applied between a reference region R2 and a primary region R1. Possible mapping: –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) –…–… 13/04/201517

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 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 13/04/201518 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 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 Further work

19. Thank You Any Questions? Thanks for your Attention!!! Eliseo Clementini eliseo@ing.univaq.it Any Questions? Thanks for your Attention!!! Eliseo Clementini eliseo@ing.univaq.it 13/04/201519