1. Projective relations in a 3D environment Roland Billen 1 & Eliseo Clementini 2 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. 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 disconnected concave square

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. Background and Motivations Projective invariants Collinearity properties e.g., three points belong to the same line RO1 RO2 PO RO1 RO2 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. Ternary projective relationships among points in R² Deriving other projective properties from collinearity beforeafter betweennonbetween rightsideleftside collinear aside U insideoutside 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 Projective relations in a 3D environment, Billen R. & Clementini E., GIScience 06, Muenster ls(A,B,C) = (1 0 0 0 0 | 0 0), bf(A,B,C) = (0 1 0 0 0 | 0 0)

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

12. Ternary projective relationships among points in R³ Almost the same that in R² Except that … beforeafter betweennonbetween rightsideleftside collinear aside U insideoutside Projective relations in a 3D environment, Billen R. & Clementini E., GIScience 06, Muenster

13. Ternary projective relationships among points in R³ The specialisation of the aside relation is not possible in R³ Set of JEPD relations (6) beforeafter betweennonbetween collinear aside U insideoutside 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. 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 beforeafter betweennonbetween collinear aside U insideoutside

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

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. Quaternary projective relationships among points in R³ Projective relations in a 3D environment, Billen R. & Clementini E., GIScience 06, Muenster internalexternal belowabove coplanar non coplanar U insideoutside

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 ????