L1-Spatial Concepts

L1 - Spatial Concepts

NGEN06 & TEK230:

Algorithms in Geographical Information Systems

by: Irene Rangel, updated 2015-11-02 by Sadegh Jamali 1

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)?2

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. Spatial queries

2. Set-based queries

3. Topological queries

4. Graph-based queries

5. Euclidean queries

6. 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.

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But how storing explicitly?

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:UnionIntersectionComplement

Logical Operators:ORANDNOT

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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 Etc.

Uganda Kampala

Egypt Cairo

Nigeria Lagos

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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 polygonTwo lines intersect

Not a topological relationship: an object is close to another (spatial relationship) 15

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.

AA

A

Connected Not Connected18

L1-Spatial Concepts

4-intersection model

Boundary ( ) 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

A

Definitions of boundary and interiors of connected objects (A) in R2

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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 Ø A meets B

¬ Ø ¬ Ø Ø Ø A equals B

Ø ¬ Ø ¬ Ø Ø A inside B

¬ Ø ¬ Ø ¬ Ø Ø A coveredBy B

Ø ¬Ø Ø ¬ Ø B inside A

¬ Ø ¬ Ø Ø ¬ Ø A covers B

¬ Ø ¬ Ø ¬ Ø ¬ Ø 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 equals B

A disjoint B A contains B

A inside B

A meets B A covers B

A coveredBy 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

Link-node structure:

a data structure that consists of a set of data records (nodes) linked together and

organized by references (links).

Data structures to store topological data explicitly

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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:

2) Manhattan distance: d(p,q)= |xp- xq| + |yp- yq|

) -( ) -( ),( 22qpqp yyxxqpd

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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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