1. Temporal Topological Relationships of Convex Spaces in Space Syntax Theory a Hani Rezayan, b Andrew U. Frank, a Farid Karimipour, a Mahmoud R. Delavar a University of Tehran, Iran, - (rezayan, karimipr, mdelavar)@ut.ac.ir b Geo-Information, TU Vienna - frank@geoinfo.tuwien.ac.at International Symposium on Spatial-temporal Modeling Spatial Reasoning, Spatial Analysis, Data Mining and Data Fusion STM’05, August 27- 29, 2005, Beijing, China

2. Overview Temporal Topological Relationships in Convex Spaces of Cityscapes (Space Syntax Theory) Time in GIScience Computational Model Formalization Case Study: Movement of Buses in City

3. Goals Demonstrate a uniform approach to analysis of static and dynamic situations using time lifting. Show how it applies to Spatio-Temporal theories.

4. Space Syntax Theory Hillier and Hanson (1984) a spatial theory which provides means through which we could understand human settlements describes invariants in built spaces.

5. Space Syntax Theory Framework Space as container Urban grids (relations) Movement

6. Relational System of Space Generation - Step 1 of 3 Spatial decomposition of spatial configuration into elementary units of analysis. bounded spaces convex spaces axial lines

7. Relational System of Space Generation - Step 1 of 3 Example of analysis units extraction for a market (Brown, 2001)

8. Relational System of Space Generation - Step 2 of 3 Axial representation (Jiang et al., 2000)

9. Relational System of Space Generation - Step 2 of 3 Convex representation (Jiang et al., 2000)

10. Relational System of Space Generation - Step 2 of 3 Grid representation (Jiang et al., 2000)

11. GI Science and Theory : Time Changes are inevitable Time is an inherent dimension of reality deficiencies: –Lack of comprehensive ontology –Discrete or partial continuous treatments –Dominance of analytical approaches –Context-based viewpoints

12. GI Theory Development Category Theory Fundamental concepts –Category A collection of primitive element types (objects), a set of operations upon those types (morphisms), and an operator algebra which is capable of expressing the interaction between operators and elements –Morphism Homomorphism Functor

13. GI Theory Development Category Theory: Functor –a special type of mapping between categories –Let C and D be categories. A functor F from C to D is a mapping that: associates to each object X in C an object F(X) in D, associates to each morphism f : X → Y in C a morphism F(f) : F(X) → F(Y) in D –such that: Identity: F(id(X)) = id(F(X)) for every object Composition: F(g  f) = F(g)  F(f) for all morphisms f:X  Y and g:Y  Z.

14. Functional Formalization of Time Change and movement is formalized by a function from time to a position or an object property. Changing v = Time → v wherev = Any (static) type Time = Time parameter These functions are Functors!

15. Case Study: City Blocks and Moving Buses Implementation of integrated analyses for static and dynamic topological relationships Space Syntax theory –Local scale time –Moving objects –Graph

16. Points > data Point a = Point Id a a > class Points p s where >x, y :: p s → s >x (Point _ cx _) = cx >y (Point _ _ cy) = cy >xy :: s → s → p s >xy cx cy = Point (-1) cx cy >(+) :: p s → p s → p s >(-) :: p s → p s → p s

17. Instances for Static and Dynamic Points > instance Points Point a where >(+) (Point x1 y1) (Point x2 y2) = >Point (x1 + x2) (y1 + y2) > > instance Points Point (Changing a) where >(+) = lift2 (+) >

18. Research’s Critical Experiment Case Study analyseGraph 0 analyseGraph 25 analyseGraph 50 analyseGraph 75 analyseGraph 100 high integrability between 50 and 70 Activity1 Bus1 Activity2/Bus2

19. Conclusions Urgent need to integrate the theory of space of geography, planning and architecture with a formal, mathematics based theory of Geoinformation Science.

20. Conclusions Category theory is the high level abstraction that provides the environment in which a theory of space-time fields and objects is possible (as demanded by Goodchild in his keynote). Models for static analysis can be lifted to apply to dynamic situations without reprogramming.

21. Conclusions Discretization gives graphs which can be analyzed. The case study shows a the application of analytical functions to static and moving objects.