Shaowen Wang CyberInfrastructure and Geospatial Information Laboratory (CIGI)

1. Principles of GIS Fundamental spatial concepts Shaowen Wang CyberInfrastructure and Geospatial Information Laboratory (CIGI) Department of Geography and National Center for Supercomputing Applications (NCSA) University of Illinois at Urbana-Champaign February 21 - March, 2011

2. Things We just Learned • Databases • DBMS • Data modeling • Relational • Object-oriented • Relational databases • SQL • Extended RDBMS • Spatial data handling

3. Spatial Concepts • Euclidean geometry • Sets of geometric elements • Topology • Neighborhood • Graph • Nodes • Edges • Metric space

4. Point Object • Cartesian plane • Vector • Norm • Distance • Angle

5. Line Object • Parameterized representation • Line • Line segment • Half line

6. Polygonal Objects • Polyline • Simple closed polyline • Polygon • Convex polygon • Star-shaped polygon • Monotone • Chain • Polygon

7. Triangulation • Diagonal • Non-diagonal • TIN (Triangulated Irregular Network)

8. Sets • Organization of geometric objects • Creation of new geometric objects

9. Concepts • Element • Membership

10. Set Characteristics • Equality • Subset • Power set • Empty set • Cardinality

11. Set Operations • Intersection • Union • Difference • Complement

12. Types of Sets • Specific useful sets • Booleans • Integers • Reals • Real plane • Closed interval • Open interval • Semi-open interval

13. Relations of Sets • Product • Binary relation • Reflexive • Symmetric • Transitive • Equivalence relation

14. Functions • Domain • Codomain

15. Function Properties • Injection • Inverse function • Surjection • Bijection

16. Convexity • Visibility • Observation point • Convex hull

17. Topological Spaces • Topological properties • Topology • Point-set topology

18. Neighborhood • Neighborhoods • A collection of subsets of a given set of points S • T1: Every point in S is in some neighbor • T2: The intersection of any two neighborhoods of any point x in S contains a neighborhood of x

19. Usual Topology • Euclidean plane • Open disk • Validate T 1 and T 2

20. Travel Time Topology • Travel time relation • Symmetric • Neighborhoods • All time zones

21. Near Point • X • Subset of points in a topological space • x • An individual point in the topological space • Every neighborhood of x contains some point of X

22. Properties of A Topological Space • Open set • Closed set • Closure

23. Properties of A Topological Space • Open set • Every point of a set can be surrounded by a neighborhood that is entirely within the set • Closed set • A set contains all its near points • Closure (X -) • The union of a point set with the set of all its near points

24. Properties of A Topological Space – continued • Interior (X o) of a point set • Consists of all points that belong to the set and are not near points of the complement of the set • Boundary of a point set (∂X) • Consists of all points that are near to both the set and its complement • Connectedness • Partition into two non-empty disjoint subsets: A and B • Either A contains a point near B • Or B contains a point near A

25. Future Topics • Combinatorial topology • Network spaces • Graph • Metric spaces • Fractal geometry

26. End of This Topic