L1-Spatial Concepts
L1 - Spatial Concepts
NGEN06 & TEK230:
Algorithms in Geographical Information Systems
by: Irene Rangel, updated 2015-11-02 by Sadegh Jamali
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L1-Spatial Concepts
Concept of Space
Object model –> set of objects
(vector)
Field-based model -> set of locations
with properties
(raster or grid)
How do we store geographic data (geometry)?
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L1-Spatial Concepts
Aim
 Understand the relationship between spatial queries
and mathematical concepts.
 Know how topological relationships are defined in GIS.
 Get knowledge about the relationship between type of
queries and suitable methods of storing geographic
data.
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L1-Spatial Concepts
Content
1.
2.
3.
4.
5.
6.
Spatial queries
Set-based queries
Topological queries
Graph-based queries
Euclidean queries
Storing relationships or deriving in real-time?
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L1-Spatial Concepts
Spatial Queries
Set-based query:
Is Uganda a country in Africa?
Africa
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L1-Spatial Concepts
Spatial Queries
Topological queries
Which countries are neighbours to Uganda?
Africa
Topological
Relationships
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L1-Spatial Concepts
Spatial Queries
Graph-based queries
How long is the traveling distance from Uganda to
Egypt?
Africa
Relationships
between
elements
distance between elements
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L1-Spatial Concepts
Spatial Queries
Euclidean queries
What is the area of Uganda?
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L1-Spatial Concepts
Do we always need coordinates to answer
spatial queries?
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L1-Spatial Concepts
We are not always relying on coordinates (or the
Euclidean space) in GIS; by storing set-based,
topological and graph-based data explicitly (without
using coordinates) we can answer many spatial queries
without considering coordinates.
But how storing explicitly?
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L1-Spatial Concepts
Set-based queries
Countries_in_africa ={Egypt, Uganda, ...}
Z={..., -2, -1, 0 1 2, ...}
R= the real numbers
R2 = R x R
E= {x=(x1,x2) | x  R2 , 0<x1<100, 0< x2<100 }
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L1-Spatial Concepts
Set algebra
Set Operations:
Union
Intersection
Complement
Logical Operators:
OR
AND
NOT
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L1-Spatial Concepts
Data structure to store set based data explicitely
Is Uganda a country in Africa?
Table: Countries_in_Africa
Country
Capital
Uganda
Kampala
Egypt
Cairo
Nigeria
Lagos
Etc.
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L1-Spatial Concepts
Topological queries
Topology is derived from the Greek and means
the science of position.
Topological Space:
A set and a number of subsets (which follow
certain rules)
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L1-Spatial Concepts
Topological Relationships?
Using a rubber sheet (where all points, lines
and areas are drawn), topological relationships
are the properties that remain between the
points, lines and areas for all possible kinds of
deformation of the rubber sheet (except tearing).
<- Rubber sheet transformation
Examples:
Point is inside a polygon
Two lines intersect
Not a topological relationship: an object is close to another
(spatial relationship)
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L1-Spatial Concepts
Topological transformation
1) There should be one-to-one correspondence between the
elements in the original and transformed set (bijection).
2) Two points that are ”connected” in the original set should also
be ”connected” in the transformed set.
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L1-Spatial Concepts
Topological Relationships
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L1-Spatial Concepts
4-intersection model
It is defined using the boundary and the interior of
objects.
This terminology is defined for cells (2 dimensional,
connected sets without holes - closed) in R2.
A
A
A
Connected
Not Connected
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L1-Spatial Concepts
4-intersection model
Definitions of boundary and interiors of connected objects (A) in R2
Boundary ( A )
Interior (Ao)
Point
The empty set
Point
Line
The end points
The line apart from the
end points
Area
The line(s) that
constitute the
border of the area
The area inside the
border lines
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L1-Spatial Concepts
Definitions of topological relationships (using
the 4-intersection model)
∂A ∩ ∂B
A0 ∩ B0
∂A ∩ B0
A0 ∩ ∂ B
Topological relationship
Ø
Ø
Ø
Ø
A disjoint B
¯
Ø
Ø
Ø
A meets B
¯
¯
Ø
Ø
A equals B
Ø
¯
¯
Ø
A inside B
¯
¯
¯
Ø
A coveredBy B
Ø
¯
Ø
¯
B inside A
¯
¯
Ø
¯
A covers B
¯
¯
¯
¯
A overlaps B
Ø = empty set
¬ Ø = not empty set
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L1-Spatial Concepts
Definitions of topological relationships (using
the 4-intersection model)
A disjoint B
A inside B
A contains B
A meets B
A equals B
A coveredBy B
A covers B
A overlaps B
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L1-Spatial Concepts
Other models of topological relationships
• 9-intersection model (DE-9IM)
The 4-intersection model is actually not that suitable for
expressing topological relationships between line and/or
point objects.
DE-9IM was proposed to be an international standard by
the International Standard Organization –ISO19125-1.
Defines topological relationships using interior, exterior
and boundary of objects.
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L1-Spatial Concepts
Data structures to store topological data
explicitly
Link-node structure:
a data structure that consists of a set of data
records (nodes) linked together and
organized by references (links).
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L1-Spatial Concepts
Graph-based queries
• Also called network queries
• They consider distances
• Shortest (fastest) route is a typical example.
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L1-Spatial Concepts
Graph-based queries
Traveling time between airports.
The points (A, B, ... , H) are airports (i.e. elements in the set airports).
The edges denote that there are flight routes between the airports.
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L1-Spatial Concepts
Graph-based and metric queries
• In a graph-based query there is no restriction on the
distances between the elements in the set.
• Metric query is a sub-set of a graph-based query that set
constraints on the distances-> they must obey the rules
of a metric.
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L1-Spatial Concepts
Metric query
A metric (d) is a distance measure between two
elements in a set.
The metric has to obey the 3 following rules (where p,q,
and r are elements in the set, d=metric):
1. d(p,q)>=0, d(p,q)=0  p=q
2. d(p,q)=d(q,p) (symmetry)
3. d(p,q)<=d(p,r)+d(r,q) (triangle inequality)
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L1-Spatial Concepts
Metric query
There are an infinite number of metrics.
Two of the most common metrics (in R2) in GIS are:
1) Euclidean distance: d ( p, q)  ( x p - xq ) 2  ( y p - y q ) 2
2) Manhattan distance: d(p,q)= |xp- xq| + |yp- yq|
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L1-Spatial Concepts
Data structures to store graph-based data
explicitly
• Graphs can be stored as matrixes.
• Sparse graphs are normally stored in
adjacency list (Sedgewick, 2002 ).
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L1-Spatial Concepts
Euclidean queries
What is the distance between a building and a
road?
-> Require coordinates to be stored.
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L1-Spatial Concepts
Storing relationships or deriving in real time?
• In some cases the relationships can be derived
from stored coordinate data.
– Traveling distance (but not traveling time)
Storing relationships explicitly:
– Advantages:
• It saves processing time
• Could enhance the quality of the answer
– Disadvantages:
• It takes more space in memory
• It entails redundancy (storing same information twice)
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L1-Spatial Concepts
Storing relationships or deriving in real time?
A few things you should consider before you decide
what relationships should be stored explicitly:
–
–
–
–
What type of queries will the database serve?
How will the database be maintained?
Will the database be connected to other databases?
…
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