1. Peter BogaertGeos 2005 A Qualitative Trajectory Calculus and the Composition of its Relations Authors: Nico Van de Weghe, Bart Kuijpers, Peter Bogaert and Philippe De Maeyer Department of Geography, Ghent University, Belgium {nico.vandeweghe, peter.bogaert, philippe.demaeyer}@ugent.be Theoretical Computer Science, Hasselt University, Belgium bart.kuijpers@uhasselt.be

2. Peter BogaertGeos 2005 Overview  Problem statement  The Qualitative Trajectory Calculus – Double Cross (QTC C )  Composition Table  Composition Rules Table  Concluding Remarks

3. Peter BogaertGeos 2005 Problem Statement Qualitative Reasoning: Region Connection Calculus (RCC) Database: 9-Intersection Model Randell, D., Cui, Z, and Cohn, A.G., 1992. A Spatial Logic Based on Regions and Connection, In: Nebel, B., Swartout, W., and Rich, C. (Eds.), Proc. of the 3rd Int. Conf. on Knowledge Representation and Reasoning (KR), Morgan Kaufmann, San Mateo, USA, 165 ‑ 176. Egenhofer, M. and Franzosa, R., 1991. Point ‑ Set Topological Spatial Relations, International Journal of Geographical Information Systems, 5 (2), 161 ‑ 174. a lot of work has been done in stating the dyadic topological relations between two polygons Two approaches

4. Peter BogaertGeos 2005 Problem Statement DC(k,l)EC(k,l)PO(k,l) TPP(k,l)NTPP(k,l) TPPI(k,l)NTPPI(k,l) EQ(k,l) 8 possible topologic relations Constraints upon the ways these relations change

5. Peter BogaertGeos 2005 Problem Statement Developing a calculus for representing and reasoning about movements of objects in a qualitative framework. continuously moving objects: often only “disconnected from” “How do we handle changes in movement between moving objects, if there is no change in their topological relationship?” DC(k,l)

6. Peter BogaertGeos 2005 QTC (Van de Weghe, N. (2004) Representing and Reasoning about Moving Objects: A Qualitative Approach, PhD thesis, Belgium, Ghent University, Faculty of Sciences, Department of Geography, 268 pp.) QTC B 1D 2D Distance, speed QTC C 2D Double Cross Concept A calculus for representing and reasoning about movements of two disconnected point-like objects in a qualitative framework.

7. Peter BogaertGeos 2005 QTC – Simplification

8. Peter BogaertGeos 2005  Freksa, Ch., 1992. Using Orientation Information for Qualitative Spatial reasoning, In: Frank, A.U., Campari, I., and Formentini, U. (Eds.), Proc. of the Int. Conf. on Theories and Methods of Spatio ‑ Temporal Reasoning in Geographic Space, Pisa, Italy, Lecture Notes in Computer Science, Springer ‑ Verlag, (639), 162 ‑ 178. Double-Cross Concept

9. Peter BogaertGeos 2005 QTC C Double-Cross Calculus (Freksa) QTC – DOUBLE-CROSS (QTC C ) l k

10. Peter BogaertGeos 2005 Double-Cross Calculus (Freksa) QTC C l k l k QTC – DOUBLE-CROSS (QTC C )

11. Peter BogaertGeos 2005 Double-Cross Calculus (Freksa) QTC C l k l k l k QTC – DOUBLE-CROSS (QTC C )

12. Peter BogaertGeos 2005 QTC – DOUBLE-CROSS (QTC C ) l k

13. Peter BogaertGeos 2005 0 – + – QTC – DOUBLE-CROSS (QTC C ) l k

14. Peter BogaertGeos 2005 – + –– 0 QTC – DOUBLE-CROSS (QTC C ) l k

15. Peter BogaertGeos 2005 0 – ––– + QTC – DOUBLE-CROSS (QTC C ) k l

16. Peter BogaertGeos 2005 0 – + –––– – + QTC – DOUBLE-CROSS (QTC C ) l k

17. Peter BogaertGeos 2005 Qualitative Trajectory Calculus (QTC) QTC B2D QTC C

18. Peter BogaertGeos 2005 Qualitative Trajectory Calculus (QTC) QTC B2D QTC C

19. Peter BogaertGeos 2005 Qualitative Trajectory Calculus (QTC) QTC B2D QTC C

20. Peter BogaertGeos 2005 R 1 (k,l)  R 2 (l,m) = R 3 (k,m)  originate from temporal reasoning (Allen 1983)  encodes all possible compositions of relations  simple table look ‑ up  useful from a computational point of view Composition Table  Idea: to compose a finite set of new facts and rules from existing ones

21. Peter BogaertGeos 2005 Composition Table

22. Peter BogaertGeos 2005 Composition Table  QTC C defines 81 different relations  Difficult visual presentation  A composition table would have 81*81 (6561) entries

23. Peter BogaertGeos 2005  Useful for visual presentation  Instrument to help generate the composition table Composition Rules Table  Reduction of the number of entrances

24. Peter BogaertGeos 2005 Composition Rules Table  Based on 2 rules  Which rotation do we need, such that l of R2 matches l of R1?  How is k moving with respect to l in R1, and how is m moving with respect to l in R2?

25. Peter BogaertGeos 2005  Which rotation do we need, such that l of R2 matches l of R1? Composition Rules Table (k,l)(– + + –) C (l,m)(– – – +) C k l l m (k,m)

26. Peter BogaertGeos 2005 (case 1 of n) (k,l)(– + + –) C (l,m)(– – – +) C k l l m l (k,m) k m Composition Rules Table

27. Peter BogaertGeos 2005 (k,l)(– + + –) C (l,m)(– – – +) C (case 1 of n) (k,m) k m k l l m Composition Rules Table

28. Peter BogaertGeos 2005 m k – (k,m)(– ) C (k,l)(– + + –) C (l,m)(– – – +) C (case 1 of n) (k,m) k l l m Composition Rules Table

29. Peter BogaertGeos 2005 k l l m m k + (k,m)(– + ) C (k,m)(– ) C (k,l)(– + + –) C (l,m)(– – – +) C (case 1 of n) (k,m) Composition Rules Table

30. Peter BogaertGeos 2005 m k – (k,m)(– + ) C (k,m)(– ) C (k,m)(– + – ) C (k,l)(– + + –) C (l,m)(– – – +) C (case 1 of n) (k,m) k l l m Composition Rules Table

31. Peter BogaertGeos 2005 k l l m m k + (k,m)(– + ) C (k,m)(– ) C (k,m)(– + – ) C (k,m)(– + – + ) C (k,l)(– + + –) C (l,m)(– – – +) C (case 1 of n) (k,m) Composition Rules Table

32. Peter BogaertGeos 2005 k l l m m k l (k,l)(– + + –) C (l,m)(– – – +) C (case 2 of n) (k,m) Composition Rules Table

33. Peter BogaertGeos 2005 k l l m k (k,l)(– + + –) C (l,m)(– – – +) C (case 2 of n) (k,m) m Composition Rules Table

34. Peter BogaertGeos 2005 k l l m k (k,l)(– + + –) C (l,m)(– – – +) C (case 2 of n) (k,m) m (k,m)(– – + +) C Composition Rules Table

35. Peter BogaertGeos 2005 (k,l)(– + + –) C (l,m)(– – – +) C > 180° k l m l (k,m) l (n of n cases) Composition Rules Table

36. Peter BogaertGeos 2005 (k,l)(– + + –) C (l,m)(– – – +) C > 180° < 360° k l m l l k l m (n of n cases) (k,m) Composition Rules Table

37. Peter BogaertGeos 2005 > 180° and < 360° (n of n cases)(k,l)(– + + –) C (l,m)(– – – +) C (k,m) Composition Rules Table

38. Peter BogaertGeos 2005 Which rotation do we need, such that l of R 2 matches l of R 1 > 180° and < 360° (n of n cases)(k,l)(– + + –) C (l,m)(– – – +) C (k,m) Composition Rules Table

39. Peter BogaertGeos 2005 How is k moving with respect to l in R 1, and how is m moving with respect to l in R 2 (n of n cases)(k,l)(– + + –) C (l,m)(– – – +) C (k,m) Composition Rules Table

40. Peter BogaertGeos 2005 How is k moving with respect to l in R 1, and how is m moving with respect to l in R 2 (n of n cases)(k,l)(– + + –) C (l,m)(– – – +) C (k,m) Composition Rules Table

41. Peter BogaertGeos 2005 Composition Rules Table

42. Peter BogaertGeos 2005  independent  goal: movement of object k with respect to object n (R 3 )  incomplete data  incomplete answer  two scientific teams 'Puzzling the Past' (k,n)?

43. Peter BogaertGeos 2005 (k,l)(– + + –) C (l,n)(– – – +) C Team I Team II (k,m)(– + – +) C (m,n)(– + – +) C k l l m m n n k (k,n)(k,n) (k,n)(k,n) 'Puzzling the Past'

44. Peter BogaertGeos 2005 Team IIITeam IV Team I Team II   Team III Team IV (k,n)(k,n) 'Puzzling the Past'

45. Peter BogaertGeos 2005 Team V (k,n)(– + – +) C Team IIITeam IV 'Puzzling the Past'

46. Peter BogaertGeos 2005 Complex researches (huge number of anchor points, teams, measurements per team, updates, etc.) Cuncluding Remarks  IMPLEMENTATION OF QTC C The composition-rules table forms a basis for reasoning about incomplete spatio-temporal knowledge in information systems. Can be used in a variety of research domains: geomorphology, geology, archaeology, and biology.

47. Peter BogaertGeos 2005 A Qualitative Trajectory Calculus and the Composition of its Relations Authors: Nico Van de Weghe, Bart Kuijpers, Peter Bogaert and Philippe De Maeyer Department of Geography, Ghent University, Belgium {nico.vandeweghe, peter.bogaert, philippe.demaeyer}@ugent.be Theoretical Computer Science, Hasselt University, Belgium bart.kuijpers@uhasselt.be