Projective relations in a 3D environment

1. Projective relations in a 3D environment Roland Billen1 & Eliseo Clementini2 1 University of Liège (Belgium) 2 University of L’Aquila (Italy)

2. TOC • Background and motivations • Ternary proj. relationships among points in R² • Ternary proj. relationships among regions in R² • Ternary proj. relationships among points in R³ • Ternary proj. relationships among bodies in R³ • Quaternary proj. relationships among points in R³ • Quaternary proj. relationships among bodies in R³ • Further research Projective relations in a 3D environment, Billen R. & Clementini E., GIScience 06, Muenster

3. disconnected concave square Background and Motivations • Qualitative Spatial Reasoning • What is projective geometry? A geometry more specific than topology and less specific than metric • E.g., topological property: • E.g., projective property: • E.g., metric property: Projective relations in a 3D environment, Billen R. & Clementini E., GIScience 06, Muenster

4. Background and Motivations • Why projective geometry? Definition of many qualitative relations • Topological: • Lakes inside Scotland • Projective: • Cities between Glasgow and Edinburgh • Lakes surrounded by mountains • Shops on the right of the road • Flags above the tree • Metric: • Edinburgh is east of Glasgow • Edinburgh is not far from Glasgow Projective relations in a 3D environment, Billen R. & Clementini E., GIScience 06, Muenster

5. RO2 RO1 PO Background and Motivations • Projective invariants • Collinearity properties • e.g., three points belong to the same line RO2 RO1 PO Projective relations in a 3D environment, Billen R. & Clementini E., GIScience 06, Muenster

6. Background and Motivations • We wished to extend our model in 3D • Could be used in • 3D GIS • Virtual Reality • Augmented Reality • Robot Navigation • Navigation in Geographic environment • … Projective relations in a 3D environment, Billen R. & Clementini E., GIScience 06, Muenster

7. U collinear aside before after rightside leftside inside outside between nonbetween Ternary projective relationships among points in R² • Deriving other projective properties from collinearity Projective relations in a 3D environment, Billen R. & Clementini E., GIScience 06, Muenster

8. Ternary projective relationships among points in R² • Partition of R² based on the two reference points • Set of JEPD relationships (7) Projective relations in a 3D environment, Billen R. & Clementini E., GIScience 06, Muenster

9. Ternary projective relationships among regions in R² • Still based on collinearity and reference objects shapes • Set of JEPD relationships (34) Projective relations in a 3D environment, Billen R. & Clementini E., GIScience 06, Muenster

10. Ternary projective relationships among regions in R² ls(A,B,C) = (1 0 0 0 0 | 0 0), bf(A,B,C) = (0 1 0 0 0 | 0 0) Projective relations in a 3D environment, Billen R. & Clementini E., GIScience 06, Muenster

11. Ternary projective relationships among regions in R² Projective relations in a 3D environment, Billen R. & Clementini E., GIScience 06, Muenster

12. U collinear aside before after rightside leftside inside outside between nonbetween Ternary projective relationships among points in R³ • Almost the same that in R² Except that … Projective relations in a 3D environment, Billen R. & Clementini E., GIScience 06, Muenster

13. U collinear aside before after inside outside between nonbetween Ternary projective relationships among points in R³ • The specialisation of the aside relation is not possible in R³ • Set of JEPD relations (6) Projective relations in a 3D environment, Billen R. & Clementini E., GIScience 06, Muenster

14. Ternary projective relationships among bodies in R³ • The relation collinear among bodies is the generalisation of the same relation among points • The partition of the space is based on tangent planes (similarity with regions in R²) Projective relations in a 3D environment, Billen R. & Clementini E., GIScience 06, Muenster

15. Ternary projective relationships among bodies in R³ • A collinearity subspace can be defined • The space is divided into a between subspace, a non-between subspace and an aside subspace Projective relations in a 3D environment, Billen R. & Clementini E., GIScience 06, Muenster

16. U collinear aside before after inside outside between nonbetween Ternary projective relationships among bodies in R³ • Same basic relationships than for points • set of JEPD relationships (18) Projective relations in a 3D environment, Billen R. & Clementini E., GIScience 06, Muenster

17. Ternary projective relationships among bodies in R³ bf(A,B,C) bt(A,B,C) bf:as(A,B,C) Projective relations in a 3D environment, Billen R. & Clementini E., GIScience 06, Muenster

18. Quaternary projective relationships among points in R³ • Three non collinear points define one an only one plane in the space  concept of coplanarity • Such a plane (called hyperplane) divides the whole space in two regions , called halfspaces • Depending on the order of the three reference points, the plane can be oriented in R³  Positive and negative halfspaces • Based on this partition, one can define projective relations between a point and three reference points  These relations are therefore quaternary • above, below, internal, external, inside and outside is a JEPD set of relations in R³ (6) Projective relations in a 3D environment, Billen R. & Clementini E., GIScience 06, Muenster

19. Quaternary projective relationships among points in R³ Projective relations in a 3D environment, Billen R. & Clementini E., GIScience 06, Muenster

20. U coplanar non coplanar below above inside outside internal external Quaternary projective relationships among points in R³ Projective relations in a 3D environment, Billen R. & Clementini E., GIScience 06, Muenster

21. Quaternary projective relationships among bodies in R³ • The concept of coplanarity between four bodies can be introduced as a generalisation of the same relation among points • We end up the same basic relationships than for points, and a set of JEPD relationships (18) Projective relations in a 3D environment, Billen R. & Clementini E., GIScience 06, Muenster

22. Quaternary projective relationships among bodies in R³ • To Build the coplanarity subspace … …We consider 8 internal and external tangent planes to the three reference bodies Projective relations in a 3D environment, Billen R. & Clementini E., GIScience 06, Muenster

23. Quaternary projective relationships among bodies in R³ Projective relations in a 3D environment, Billen R. & Clementini E., GIScience 06, Muenster

24. Quaternary projective relationships among bodies in R³ Projective relations in a 3D environment, Billen R. & Clementini E., GIScience 06, Muenster

25. Quaternary projective relationships among bodies in R³ • The all set of quaternary relations can be obtained based on the empty / non-empty intersections of the primary body A with the subspaces which satisfy the basic quaternary relations int(A,B,C,D) = (1 0 0 0 | 0 0), ext(A,B,C,D) = (0 1 0 0 | 0 0), ab(A,B,C,D) = (0 0 1 0 | 0 0), be(A,B,C,D) = (0 0 0 1 | 0 0), in(A,B,C,D) = (0 0 0 0 | 1 0), ou(A,B,C,D) = (0 0 0 0 | 0 1) Projective relations in a 3D environment, Billen R. & Clementini E., GIScience 06, Muenster

26. Quaternary projective relationships among bodies in R³ ext(A,B,C,D) int(A,B,C,D) Projective relations in a 3D environment, Billen R. & Clementini E., GIScience 06, Muenster

27. Quaternary projective relationships among bodies in R³ ab(A,B,C,D) ext:ab(A,B,C,D) Projective relations in a 3D environment, Billen R. & Clementini E., GIScience 06, Muenster

28. Further research • (at SDH 04) • Algorithms for the computation of projective relations. (done) • Reasoning system for all ternary relations, composition tables and proofs. (on going) • Extensions to n-ary relations: surrounded by, in the middle of, etc. • Extensions to other geometric types: region/line, line/line, etc. • Extensions to 3D relations. Projective relations in a 3D environment, Billen R. & Clementini E., GIScience 06, Muenster

29. Further research • (currently) • Algorithms for the computation of projective relations. (done) • Reasoning system for all ternary relations, composition tables and proofs. (almost done) • Extensions to n-ary relations: surrounded by, in the middle of, etc. (partially done) • Extensions to other geometric types: region/line, line/line, etc. • Extensions to 3D relations. (done) • Reasoning system for all quaternary relations, composition tables and proofs. • Mapping these concepts to specific environment Projective relations in a 3D environment, Billen R. & Clementini E., GIScience 06, Muenster

30. Mapping in 2D Projective relations in a 3D environment, Billen R. & Clementini E., GIScience 06, Muenster

31. Mapping in 3D Projective relations in a 3D environment, Billen R. & Clementini E., GIScience 06, Muenster

32. Thanks for attention Questions ????