Introduo to Geoinformatics Topology The benefits of axiomatization

Introdução to Geoinformatics: Topology

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

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

The Axiomatization of science Newton

The Axiomatization of informatics Codd

The Axiomatization of geoinformatics Güting Frank

Building blocks: Geometry (OGC) type GEOM = {Point, Line. String, Polygon, Multi. Point, Multi. Line. String, Multi. Polygon} operations: equals, touches, disjoint, crosses, within, overlaps, contains, intersects: GEOM x GEOM → Bool

Building blocks: Time (ISO 19108) type TIME = {Instant, Period} operations: equals, before, after, begins, ends, during, contains, overlaps, meets, overlapped. By, met. By, begun. By, ended. By: TIME x TIME → Boolean

Vector geometries

Vector geometries n n Arcs and nodes Polygons

Vector geometries fonte: Universidade de Melbourne

Vector Model: Lines node vertex 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

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

Types of topology source: ESRI

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

Vector geometries n Island n Points

Topology: polygon-polyline source: ESRI

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

Topology: the OGC model source: John Elgy

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

Topology: arc-node-polygon source: ESRI

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

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

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

Geometrical operations Point in Polygon = O(n)

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

Topological relationships

Topological relationships Disjoint Point/Point Line/Line Polygon/Polygon

Topological relationships Touches Point/Line/Polygon Point/Polygon/Polygon Line/Line

Topological relationships Crosses Point/Line Point/Polygon Line/Line/Polygon

Topological relationships Overlap Point/Point Line/Line Polygon/Polygon

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

Topological relationships Equals Point/Point Line/Line Polygon/Polygon

Topological relations Interior: A◦ Exterior: ABoundary: ∂A

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

4 -intersections disjoint meet contains inside covers equal covered. By overlap

Open. GIS: 9 -intersection dimension-extended topological operations Relation disjoint 9 -intersection model meet overlap equal

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

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

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

9 -Intersection Model

DE-9 IM: dimensionally extended 9 intersection model 43

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

DE-9 IM for the example geometries I(B) B(B) E(B) I(A) 2 1 2 B(A) 1 0 1 E(A) 2 1 2 45

Region connected calculus (RCC)

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