1. Introduction to Geoinformatics: Topology

2. The benefits of axiomatization
Euclid
(x + y)2 = x2 + 2xy + y2

3. The benefits of axiomatization
Euclid
(x + y)2 = x2 + 2xy + y2
Egenhofer
spatial topology

4. The Axiomatization of science
Newton

5. The Axiomatization of informatics
Codd

6. The Axiomatization of geoinformatics
Güting
Frank

7. Building blocks: Geometry (OGC)
type GEOM = {Point, LineString, Polygon, MultiPoint, MultiLineString, MultiPolygon}
operations:
equals, touches, disjoint, crosses, within, overlaps, contains, intersects: GEOM x GEOM → Bool

8. Building blocks: Time (ISO 19108)
type TIME = {Instant, Period}
operations:
equals, before, after, begins, ends, during, contains, overlaps, meets, overlappedBy, metBy, begunBy, endedBy: TIME x TIME → Boolean

9. Vector geometries

10. Vector geometries
Arcs and nodes
Polygons

11. Vector geometries
fonte: Universidade de Melbourne

12. Vector Model: Lines vertex node Lines start and end at nodes
line #1 goes from node #2 to node #1
Vertices determine shape of line
Nodes and vertices are stored as coordinate pairs

13. Vector Model: Polygons
Polygon #2 is bounded by lines 1 & 2
Line 2 has polygon 1 on left and polygon 2 on right

14. Types of topology
source: ESRI

15. Planar enforcement All the space on a map must be filled
Any point must fall in one polygon alone
Polygons must not overlap

16. Vector geometries
Island
Points

17. Topology: polygon-polyline
source: ESRI

18. Topology: polygon-polyline
Shapefile polygon spatial data model
less complex data model
polygons do not share bounding lines

19. Topology: the OGC model
source: John Elgy

20. What’s the use of a polygon?
Census tracts in São José dos Campos

21. Topology: arc-node-polygon
source: ESRI

22. Topology: arc-node-polygon
source: GIS Basics (Campbell and Chin, 2012)

23. Vectors and table Duality between entre location and atributes Lots
geoid
owner
address
cadastral ID 22
Guimarães
Caetés 768
250186 23 22 23
Bevilácqua
São João 456
110427 24
271055
Ribeiro
Caetés 790

24. Duality Location - Attributes
Praia de Boiçucanga
Praia Brava
Exemplo de Unidade Territorial Básica - UTB

25. Geometrical operations
Point in Polygon = O(n)

26. Geometrical operations
Line in Polygon = O(n•m)

27. Topological relationships

28. Topological relationships
Disjoint
Point/Point
Line/Line
Polygon/Polygon

29. Topological relationships
Touches
Point/Line
Point/Polygon
Line/Line
Line/Polygon
Polygon/Polygon

30. Topological relationships
Crosses
Point/Line
Point/Polygon
Line/Line
Line/Polygon

31. Topological relationships
Overlap
Point/Point
Line/Line
Polygon/Polygon

32. Topological relationships
Within/contains
Point/Point
Point/Line
Point/Polygon
Line/Line
Line/Polygon
Polygon/Polygon

33. Topological relationships
Equals
Point/Point
Line/Line
Polygon/Polygon

34. Topological relations
Interior: A◦
Exterior: A-
Boundary: ∂A

35. Topological Concepts Interior, boundary, exterior Green is A interior
Let A be an object in a “Universe” U.
Green is A interior
Red is boundary of A
Blue –(Green + Red) is
A exterior U A

36. 4-intersections disjoint contains inside equal
 
 
 
 
 
 
disjoint contains inside equal
 
 
 
 
 
 
 
meet covers coveredBy overlap

37. OpenGIS: 9-intersection dimension-extended topological operations
Relation
disjoint
meet
overlap
equal
9-intersection
model

38. Example Consider two polygons
A - POLYGON ((10 10, 15 0, 25 0, 30 10, 25 20, 15 20, 10 10))
B - POLYGON ((20 10, 30 0, 40 10, 30 20, 20 10))

39. 9-Intersection Matrix of example geometries
I(B)
B(B)
E(B)
I(A)
B(A)
E(A)

40. Specifying topological operations in 9-Intersection Model
Question: Can this model specify topological operation between a polygon
and a curve?

41. 9-Intersection Model

42. Examples of functions defined by SFSQL

43. DE-9IM: dimensionally extended 9 intersection model

44. 9-Intersection Matrix of example geometries
I(B)
B(B)
E(B)
I(A)
B(A)
E(A)

45. DE-9IM for the example geometries
I(B)
B(B)
E(B)
I(A) 2 1
B(A)
E(A)

46. Region connected calculus (RCC)
Basic concept: binary connection relation
C(x,y)
"x,y [C(x,y) ® C(y,x)]
"z C(z,z)
intended interpretation of C(x,y) : x & y share a point
source: Tony Cohn

47. Region connected calculus (RCC)
DC(x,y) ºdf ¬C(x,y)
x and y are disconnected
P(x,y) ºdf "z [C(x,z) ®C(y,z)]
x is a part of y
PP(x,y) ºdf P(x,y) Ù¬P(y,x)
x is a proper part of y
EQ(x,y) ºdf P(x,y) ÙP(y,x)
x and y are equal
source: Tony Cohn

48. Region connected calculus (RCC)
O(x,y) ºdf $z[P(z,x) ÙP(z,y)]
x and y overlap
DR(x,y) ºdf ¬O(x,y)
x and y are discrete
PO(x,y) ºdf O(x,y) Ù¬P(x,y) Ù ¬P(y,x)
x and y partially overlap
source: Tony Cohn

49. Region connected calculus (RCC)
EC(x,y) ºdf C(x,y) Ù¬O(x,y)
x and y externally connect
TPP(x,y) ºdf PP(x,y) Ù $z[EC(z,y) ÙEC(z,x)]
x is a tangential proper part of y
NTPP(x,y) ºdf PP(x,y) Ù ¬TPP(x,y)
x is a non tangential proper part of y
source: Tony Cohn

50. Region connected calculus (RCC)
8 provably jointly exhaustive pairwise disjoint relations
DC EC PO TPP NTPP
EQ TPPi NTPPi
source: Tony Cohn

51. Region connected calculus (RCC-8)
The eight jointly exhaustive and pairwise disjoint relations of region connection calculus (RCC8). The arrows show which relation is the next relation a configuration would transit to

52. Calculus based method (CBM)
Use 5 polymorphic binary relations between x,y:
disjoint: x Ç y = Æ
touch (a/a, l/l, l/a, p/a, p/l): x Ç y Í b(x) È b(y)
in: x Ç y Í y
overlap (a/a, l/l): dim(x)=dim(y)=dim(x Ç y) Ù x Ç y ¹ Æ Ù y ¹ x Ç y ¹ x
cross (l/l, l/a): dim(int(x))Çint(y))=max(int(x)),int(y)) Ù x Ç y ¹ Æ Ù y ¹ x Ç y ¹ x
source: Eliseo Clementini

53. Named Spatial Relationship Predicates Based on the DE-9IM