Temporal Topological Relationships of Convex Spaces in Space Syntax Theory

aHani Rezayan, bAndrew U. Frank, aFarid Karimipour, aMahmoud R. Delavar

aUniversity of Tehran, Iran, - (rezayan, karimipr, mdelavar)@ut.ac.irbGeo-Information, TU Vienna - frank@geoinfo.tuwien.ac.at

International Symposium on Spatial-temporal ModelingSpatial Reasoning, Spatial Analysis, Data Mining and Data FusionSTM’05, August 27- 29, 2005, Beijing, China

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

Goals

Demonstrate a uniform approach to analysis of static and dynamic situations using time lifting.

Show how it applies to Spatio-Temporal theories.

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.

Space Syntax TheoryFramework

• Space as container

• Urban grids (relations)

• Movement

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

Relational System of Space Generation - Step 1 of 3

Example of analysis units extraction for a market (Brown, 2001)

Relational System of Space Generation - Step 2 of 3

• Axial representation

(Jiang et al.,

2000)

Relational System of Space Generation - Step 2 of 3

• Convex representation

(Jiang et al.,

2000)

Relational System of Space Generation - Step 2 of 3

• Grid representation

(Jiang et al.,

2000)

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

GI Theory DevelopmentCategory 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

GI Theory DevelopmentCategory 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:XY

and g:YZ .

Functional Formalization of Time

Change and movement is formalized by a function from time to a position or an object property.

Changing v = Time → vwhere v = Any (static) type

Time = Time parameter

These functions are Functors!

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

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

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 (+)

>

Research’s Critical Experiment Case Study

analyseGraph 0analyseGraph 25analyseGraph 50analyseGraph 75 analyseGraph 100

high integrability between50 and 70

Activity1Bus1

Activity2/Bus2

Conclusions

Urgent need to integrate the theory of space of geography, planning and architecture witha formal, mathematics based theory of Geoinformation Science.

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.

Conclusions

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