Introdução to
Geoinformatics: Geometries
Vector Model
Lines: fundamental spatial data model
vertex
node
node
vertex
vertex 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 : fundamental spatial data model
• Polygon #2 is bounded by lines 1 & 2
• Line 2 has polygon 1 on left and polygon 2 on right
Vector Model
Shapefile polygon spatial data model
• less complex data model
• polygons do not share bounding lines
Vector geometries
Vector geometries


Arcs and nodes
Polygons
Vector geometries

Island

Points
Vector geometries
fonte: Universidade de Melbourne
Vector geometries: the OGC model
fonte: John Elgy
Para que serve um polígono?
Setores censitários em São José dos Campos
Vectors and table

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
Vector and raster geometries
Vector
Raster
fonte: Mohamed Yagoub
Raster geometry
Extent
célula
Resolution
source: Mohamed Yagoub
Raster geometries (cell spaces)
Regular space partitions
Many attributes per
cell
Cell space
Cellular Data Base Resolution
2500 m
2.500 m e 500 m
Rasters or vectors?
source: Mohamed Yagoub
Raster geometry
fonte: Mohamed Yagoub
The mixed cell problem
fonte: Mohamed Yagoub
Cells or vectors?
Cells or vector?
Cells or vectors? (RADAM x SRTM)
Cells or vectors? (RADAM x LANDSAT)
Raster or vectors?

“Boundaries drawn in thematic maps (such as soil,
vegetation, and geology) are rarely accurate. Drawing
them as thin lines often does not adequately represent
their character. We should not worry so much about the
exact locations and elegant graphical representations.”
(P. A. Burrough)
2,5 D geometries
isolines
TIN
2,5 D geometries
Grey-coloured relief
Shaded relief
2,5D geometries
Regular grid
2,5 D geometries
TIN (triangular irregular networks)
Conversion btw geometries
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
Line/Polygon
Point/Polygon
Polygon/Polygon
Line/Line
Topological relationships
Crosses
Point/Line
Point/Polygon
Line/Line
Line/Polygon
Topological relationships
Overlap
Point/Point
Line/Line
Polygon/Polygon
Topological relationships
Within/contains
Point/Point
Line/Line
Point/Line
Line/Polygon
Point/Polygon
Polygon/Polygon
Topological relationships
Equals
Point/Point
Line/Line
Polygon/Polygon
Topological relations
Interior: A◦
Exterior: A-
Boundary: ∂A
Topological Concepts

Interior, boundary, exterior

Let A be an object in a “Universe” U.
U
Green is A interior ( Ao )
Red is boundary of A (A)
A
Blue –(Green + Red) is
A exterior ( A )
4-intersections
 
 
disjoint
 
 
meet
 
 
 
 
contains
inside
 
 
covers
 
 
equal
 
 
coveredBy
overlap
 
 
OpenGIS: 9-intersection dimension-extended
topological operations
 ( Ao  B o ) ( Ao  B) ( Ao  B  ) 


o

 (A  B ) (A  B) (A  B ) 


 ( A  B o ) ( A  B) ( A  B  ) 


Relation
disjoint
9-intersection
 0 0 1
model


 0 0 1
1 1 1 


meet
overlap
equal
 0 0 1


 0 11 
1 1 1 


 1 1 1


 1 1 1
 1 1 1


1 0 0 


0 1 0
 0 0 1


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))
44
9-Intersection Matrix of example geometries
I(B)
B(B)
E(B)
I(A)
B(A)
E(A)
45
Specifying topological operations in 9Intersection Model
Question: Can this model specify topological operation between a polygon
and a curve?
9-Intersection Model
DE-9IM: dimensionally extended 9 intersection
model
49
9-Intersection Matrix of example geometries
I(B)
B(B)
E(B)
I(A)
B(A)
E(A)
50
DE-9IM 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
51