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Mathematical Principles of GIS 5th Nordic Summer School in GIScienceAugust 16 – 20, 2010 (Gävle, Sweden)
© Wolfgang Kainz 1
Mathematical Principles
of GISWolfgang Kainz
Department of Geography and Regional Research
University of Vienna, Austria
Contents
• Spatial information• History of GIS• GI-Science• Mathematical methods
© Wolfgang Kainz Department of Geography and Regional Research, University of Vienna 2
© Wolfgang Kainz 3
Space and Time
Department of Geography and Regional Research, University of Vienna
Space and Time
Creation myths start with the creation of space and time (often out of chaos).Then comes the rest…
Can we imagine something without a connection to space and time?
© Wolfgang Kainz Department of Geography and Regional Research, University of Vienna 4
We are spatiotemporal beings
• Limited to three spatial dimensions– We cannot escape from within a closed cube
(we cannot “see” higher dimensions than 3D) like
– Flatlanders cannot escape from a closed square
© Wolfgang Kainz Department of Geography and Regional Research, University of Vienna 5
Time…(Augustine)
• “…For what is time? Who can easily and briefly explain it? Who can even comprehend it in thought or put the answer into words? Yet is it not true that in conversation we refer to nothing more familiarly or knowingly than time? And surely we understand it when we speak of it; we understand it also when we hear another speak of it. …
• What, then, is time? If no one asks me, I know what it is. If I wish to explain it to him who asks me, I do not know.”
© Wolfgang Kainz Department of Geography and Regional Research, University of Vienna 6
Mathematical Principles of GIS 5th Nordic Summer School in GIScienceAugust 16 – 20, 2010 (Gävle, Sweden)
© Wolfgang Kainz 2
Importance of Space and Time
• Almost everything that happens, happens at a certain location in space and time
• The level of (geographic) detail (or scale) matters– Mapping a local event versus the global climatic change
• Time scales– 500-year flood versus property transactions
© Wolfgang Kainz Department of Geography and Regional Research, University of Vienna 7
Why is spatial special?
• We are dealing with multiple dimensions (x,y,z)• We are dealing with different levels of spatial
resolution• Representation of spatial data is more “complicated”
than of non-spatial data• We often need to transform and project data• Spatial analysis requires special methods
© Wolfgang Kainz Department of Geography and Regional Research, University of Vienna 8
© Wolfgang Kainz Department of Geography and Regional Research, University of Vienna 9
GI Science
and Systems data
information
knowledge
wisdom
acquisition
analysis
reasoning
contemplation
Disciplines Using Spatial Information
© Wolfgang Kainz Department of Geography and Regional Research, University of Vienna 10
Type of discipline Sample disciplines Development of spatial concepts
Geography, cartography, cognitive science, linguistics, psychology, philosophy
Means for capturing and processing of spatial data
Remote sensing, surveying engineering, cartography, photogrammetry
Formal and theoretical foundation
Computer science, knowledge based systems, mathematics, statistics
Applications Archaeology, architecture, forestry, geo-sciences, regional and urban planning, surveying
Support Law, economy
First, there were systems…
• Development of geographic information systems– A special type of information system dealing with
geographic information (spatial information)– Application of information technology
© Wolfgang Kainz Department of Geography and Regional Research, University of Vienna 11 © Wolfgang Kainz 12
History of GIS
Department of Geography and Regional Research, University of Vienna
Mathematical Principles of GIS 5th Nordic Summer School in GIScienceAugust 16 – 20, 2010 (Gävle, Sweden)
© Wolfgang Kainz 3
Description of the Earth
© Wolfgang Kainz 13
Land administration Geodesy & Geometry
Sumerians, Babylonians,Egyptians
Greeks
Euclid, Eratosthenes,Ptolemy
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Ga-Sur Tablet (3800 BC)
© Wolfgang Kainz 14Department of Geography and Regional Research, University of Vienna
City Map of Nippur (1500 BC)
© Wolfgang Kainz 15Department of Geography and Regional Research, University of Vienna © Wolfgang Kainz 16
Pocket-GIS
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© Wolfgang Kainz 17
The Early Years
Department of Geography and Regional Research, University of Vienna
The Early Years (1965 – 1985)
• Canada GIS• Insufficient hardware• Experimental software• “Discovery” of topology• Relational databases• Problems with acceptance and understanding
(“map data models”)• No integration of GIS and RS
© Wolfgang Kainz 18Department of Geography and Regional Research, University of Vienna
Mathematical Principles of GIS 5th Nordic Summer School in GIScienceAugust 16 – 20, 2010 (Gävle, Sweden)
© Wolfgang Kainz 4
GIS Software
Data StructuresAlgorithms
Data Input Database
Output &Visualization
Analysis
© Wolfgang Kainz 19
Vector: Arc/NodeStructure
Raster: Quadtree
Department of Geography and Regional Research, University of Vienna © Wolfgang Kainz 20
Consolidation
Department of Geography and Regional Research, University of Vienna
Consolidation (1985 – 1992)
• Introduction of PCs and minicomputers• Functional software• GIS in central and local government
organizations• GIS in private industry• GIS in education• Textbooks and journals• Functional GIS and RS software (separate)
© Wolfgang Kainz 21Department of Geography and Regional Research, University of Vienna © Wolfgang Kainz 22
1986 1987Department of Geography and Regional
Research, University of Vienna
GIS Software
Data StructuresAlgorithms
Data Input Database
Output &Visualization
Analysis
© Wolfgang Kainz 23Department of Geography and Regional Research, University of Vienna
SpatialModeling
GIS and Decision Support Systems
Management and Infrastructure
Theoretical Foundation
Data Input Database
Output &Visualization
Analysis
© Wolfgang Kainz 24
• Topology• Integration of vector- and
raster data• Separation of attribute and
geometry data
Department of Geography and Regional Research, University of Vienna
Mathematical Principles of GIS 5th Nordic Summer School in GIScienceAugust 16 – 20, 2010 (Gävle, Sweden)
© Wolfgang Kainz 5
© Wolfgang Kainz 25
Operationalization
Department of Geography and Regional Research, University of Vienna
Operationalization (1992 – 2000)
• GIS in business and science• Desire for a theoretical foundation
– Geomatics/geoinformatics– Geographic Information Science
• Spatial Modeling– Theory of spatial relations– Ontologies
• Integration of GIS and RS
© Wolfgang Kainz 26Department of Geography and Regional Research, University of Vienna
Then, came the science…
• The science behind the systems– Geoinformatics– Geomatics– Spatial information science– Spatial information theory– Geoinformation engineering– Geographic Information Science
© Wolfgang Kainz Department of Geography and Regional Research, University of Vienna 27
Space and Time in GIS
© Wolfgang Kainz 28
formal & theoretical aspects conceptual & empirical aspects
mathematics physics philosophy geography
Euclidean space
metric space
topological space
space-time absolute vs. relative
atomic vs. plenum
object vs. field
cognitive dim.
Democritus, Parmenides, Newton, LeibnizKant, Wittgenstein, LakeoffEinstein, Hawkins
Department of Geography and Regional Research, University of Vienna
SpatialModeling
GIS and Decision Support Systems
Management and Infrastructure
Theoretical Foundation
Data Input Database
Output &Visualization
Analysis
© Wolfgang Kainz 29Department of Geography and Regional Research, University of Vienna
Geographic Information Science
SpatialModeling
GIS and Decision Support Systems
Management und Infrastructure
Theoretical Foundation
Data Input Database
Output &Visualization
Analysis
© Wolfgang Kainz 30
• Theory of spatialrelations
• Integration of vectorand raster data
• Integration of attributeand geometry data(geodatabase)
Department of Geography and Regional Research, University of Vienna
Mathematical Principles of GIS 5th Nordic Summer School in GIScienceAugust 16 – 20, 2010 (Gävle, Sweden)
© Wolfgang Kainz 6
© Wolfgang Kainz 31
Spatial Modeling
Department of Geography and Regional Research, University of Vienna
Models
• Spatial modeling• Implementation of models• Application of models
© Wolfgang Kainz Department of Geography and Regional Research, University of Vienna 32
Modeling
• 2.12 “A picture is a model of reality.”• 2.14 “What constitutes a picture is that its
elements are related to one another in a determinate way.”
• 2.15 “The fact that the elements of a picture are related to one another in a determinate way represents that things are related to one another in the same way.”
© Wolfgang Kainz Department of Geography and Regional Research, University of Vienna 33
(Wittgenstein: Tractatus Logico-Philosophicus)
Spatial Modeling & Data Processing
© Wolfgang Kainz Department of Geography and Regional Research, University of Vienna 34
• Real world phenomena have a spatiotemporal extent and possess thematic characteristics (attributes).
• A (spatial) feature is a representation of a real world phenomenon.
• Spatial data are computer representations of spatial features.
• Spatial data handling extracts (spatial) informationfrom spatial data.
Real Worldphenomena
Spatial modelfeatures
GIS databasespatial data
design implementation
Data handling
Spatial Information
© Wolfgang Kainz Department of Geography and Regional Research, University of Vienna 35
Real World
Miniworld
Conceptualschema
Logicalschema
Physicalschema
Conceptual model
Logical model
Implementation
software independent
software specific
Spatial modeling
Spat
ial
mod
elin
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taba
se d
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nan
d im
plem
enta
tion
Ontology
Studies being or existence and their basic categories and relationships, to determine what entities and what types of entities exist. Ontology thus has strong implications for conceptions of reality.
© Wolfgang Kainz Department of Geography and Regional Research, University of Vienna 36
Mathematical Principles of GIS 5th Nordic Summer School in GIScienceAugust 16 – 20, 2010 (Gävle, Sweden)
© Wolfgang Kainz 7
GIScience is
• Ontology driven– What constitutes the world?
• Entities (objects, categories, concepts)• Characteristics (attributes)
– How are things related?• Relations (relationships)
© Wolfgang Kainz Department of Geography and Regional Research, University of Vienna 37
GIScience uses
• Logic (language of mathematics)• Mathematics (structures)
To make statements about the world and acquire knowledge about the world
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Mathematics
Department of Geography and Regional Research, University of Vienna © Wolfgang Kainz Department of Geography and Regional
Research, University of Vienna 40
Structures
Algebraic Order Topological
Logic
Set Theory
RelationsFunctions
Algebra OrderedSets Topology
© Wolfgang Kainz 41
Logic
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Logic
• Propositional Logic– propositions, logical operators (and, or, not, …)
• Predicate Logic– Predicates (properties or relations), quantifiers
• Logical Inference– Rules of inference
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Mathematical Principles of GIS 5th Nordic Summer School in GIScienceAugust 16 – 20, 2010 (Gävle, Sweden)
© Wolfgang Kainz 8
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Assertion and Proposition
• An assertion is a statement.– “Are you okay?”– “Give me that book.”
• A proposition is an assertion that is either true or false, but not both.– “It is raining.”– “I pass the exam.”
Department of Geography and Regional Research, University of Vienna © Wolfgang Kainz 44
Propositional Variable and Propositional Form
• A propositional variable is a proposition with an unspecified truth value denoted as P, Q, R, …, etc.
• An assertion with at least one propo-sitional variable is called a propositional form, e.g., Pand “I pass the exam.” When propositions are substituted for the variables a proposition results.
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© Wolfgang Kainz 45
Logical Operators
• Propositions and propositional variables can be combined with logical operators (or logical connectives) to form new assertions. Variables are called operands.
• Operators: not (), and (), or (), exclusive or (), implication (), equivalence ()
• "not P and Q " or "P Q "• "I study hard and I pass the exam."
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Truth Tables
• Truth tables show the truth values for all possible combinations of true and false for the operands.
• We use 0 for “false” and 1 for “true”.
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© Wolfgang Kainz 47
Negation
01
10
PP
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Conjunction(“logical and”)
111
001
010
000
QPQP
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Mathematical Principles of GIS 5th Nordic Summer School in GIScienceAugust 16 – 20, 2010 (Gävle, Sweden)
© Wolfgang Kainz 9
© Wolfgang Kainz 49
Disjunction(“logical or” or “inclusive or”)
111
101
110
000
QPQP
Department of Geography and Regional Research, University of Vienna © Wolfgang Kainz 50
Exclusive or
011
101
110
000
QPQP
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Implication
© Wolfgang Kainz Department of Geography and Regional Research, University of Vienna 51
P is premise, hypothesis, or antecedent, Q is conclusion or consequence.
– "If P then Q."– "P only if Q."– "Q if P."– "P is a sufficient condition for Q."– "Q is a necessary condition for P."
111
001
110
100
QPQP
Equivalence
© Wolfgang Kainz Department of Geography and Regional Research, University of Vienna 52
• “P is equivalent to Q ”• “P is a necessary and
sufficient condition for Q ”
• “P if and only if Q ” or “P iff Q ”
111
001
010
100
QPQP
© Wolfgang Kainz 53
Types of Propositional Forms
• A tautology is a propositional form whose truth value is true for all possible values of its propositional variables.
• A contradiction (or absurdity) is a propositional form which is always false.
• A contingency is a propositional form which is neither a tautology nor a contradiction.
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Examples
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(P Q) P is a tautology.
1111
1001
1010
1000
)( PQPQPQP
Mathematical Principles of GIS 5th Nordic Summer School in GIScienceAugust 16 – 20, 2010 (Gävle, Sweden)
© Wolfgang Kainz 10
Examples
© Wolfgang Kainz Department of Geography and Regional Research, University of Vienna 55
P P is a contradiction.
001
010
PPPP
Examples
© Wolfgang Kainz Department of Geography and Regional Research, University of Vienna 56
(P Q) Q is a contingency.
11011
01101
10010
01100
)( QQPQPQQP
© Wolfgang Kainz 57
Predicates
Predicates express a property of an object or a relationship between objects. Objects are often represented by variables.
• “x lives in y ” written as L(x,y). Here, x and y are variables, L or “lives in” is a predicate. L is said to have two arguments, x and y, or to be a 2-place predicate.
• Also, “x is equal to y” or “x = y ”, and “x is greater than y ” or “x > y ” are 2-place predicates.
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Predicates
• Values for variables must be taken from a set, the universe of discourse (or universe).
• To change a predicate into a proposition, each individual variable must be bound by either assigning a value to it, or by quantification of the variable.
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© Wolfgang Kainz 59
Universal Quantifier
Universal quantifier . It is read “for all”, “for every”, “for any”, “for arbitrary”, or “for each”.
• “For all x, P(x) “ or “xP(x) ” is interpreted as “For all values of x, the assertion P(x) is true.”
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If an assertion P(x) is true for every possible value x, then xP(x) is true; otherwise xP(x) is false.
© Wolfgang Kainz 60
Existential Quantifier
Existential quantifier . It is read as “there exists”, “for some”, or “for at least one”. A variation ! means “there exists a unique x such that …” or "there is one and only one x such that …".
“For some x, P(x) ” or “xP(x) ” is interpreted as “There exists a value of x for which the assertion P(x) is true.”
Department of Geography and Regional Research, University of Vienna
If an assertion P(x) is true for at least one value x, then xP(x) is true; otherwise xP(x) is false.
Mathematical Principles of GIS 5th Nordic Summer School in GIScienceAugust 16 – 20, 2010 (Gävle, Sweden)
© Wolfgang Kainz 11
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Logical Inference
• A theorem is a mathematical assertion which can be shown to be true.
• A proof is an argument which establishes the truth of the theorem.
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Logical Inference
© Wolfgang Kainz Department of Geography and Regional Research, University of Vienna 62
Rules of inference specify conclusions which can be drawn from assertions known or assumed to be true.
QP
PP
n
2
1
The assertions Pi are called hypotheses or premises,the assertion below the line is called conclusion. Thesymbol is read “therefore” or “it follows that” or“hence.”
© Wolfgang Kainz 63
Logical Inference
• An argument is said to be valid or correct if, whenever all the premises are true, the conclusion is true.
• An argument is correct when (P1 P2 … Pn) Q is a tautology.
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Research, University of Vienna 64
Reasoning: Rules of inference
In binary logic reasoning is based on – Deduction (modus ponens)
• Premise 1: If x is A then y is B• Premise 2: x is A• Conclusion: y is B
– Induction (moduls tollens)• Premise 1: If x is A then y is B• Premise 2: y is not B• Conclusion: x is not A
© Wolfgang Kainz Department of Geography and Regional Research, University of Vienna 65
Reasoning: Rules of inference
Example– Deduction (modus ponens)
• Premise 1: If it rains then I get wet• Premise 2: It rains• Conclusion: I get wet
– Induction (moduls tollens)• Premise 1: If it rains then I get wet• Premise 2: I do not get wet• Conclusion: It does not rain
© Wolfgang Kainz Department of Geography and Regional Research, University of Vienna 66
Generalized modus ponens
1 1
2 2
1
1
If is then is If is then is
:
If is then is : is : is
n n
x A y Bx A y B
p q
x A y Bp x Aq y B
Mathematical Principles of GIS 5th Nordic Summer School in GIScienceAugust 16 – 20, 2010 (Gävle, Sweden)
© Wolfgang Kainz 12
© Wolfgang Kainz 67
AlgebraicStructures
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Algebraic Structures
• Sets, elements, operators– Arithmetic– Map algebra– Relational algebra
• Structure preserving mappings– Homomorphism– Isomorphism
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Components of an Algebra
• A set, the carrier of the algebra,• Operations defined on the carrier, and• Distinguished elements of the carrier, the
constants of the algebra.
• Algebras are presented as tupels <carrier, operations, constants>– Example: <R,+,,0,1>
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Constants of Algebras
• Let be a binary operation on S.• An element 1S is an identity (or unit) for the
operation if 1 x = x 1 = x for every x in S.• An element 0S is a zero for the operation if
0 x = x 0 = 0 for every x in S.These constants are also called identity element and zero
element, respectively.
Examples: 1 is an identity element and 0 is a zero element for the multiplication of numbers. The number 0 is a identity element for the addition of numbers.
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© Wolfgang Kainz 71
Constants of Algebras
• Let be a binary operation on S and 1 an identity for the operation.
• If x y = 1 and y x = 1 for every y in S, then x is called a (two-sided) inverse of y with respect to the operation .
Example: In the real numbers 1/x is the inverse with respect to multiplication.
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Group
© Wolfgang Kainz Department of Geography and Regional Research, University of Vienna 72
A group is an algebra with the signature <S,, ̄,1>, with ̄ the inverse with respect to , and the following axioms:
1
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)()(
aaaaa
cbacba
If the operation is also commutative, we call the group a commutative group.
Mathematical Principles of GIS 5th Nordic Summer School in GIScienceAugust 16 – 20, 2010 (Gävle, Sweden)
© Wolfgang Kainz 13
© Wolfgang Kainz 73
Example
• <I,+,-,0> is a group where I are the integers, “+” is the addition, “-” the inverse (negative) integer, and 0 the identity for the addition.
• <R-{0},,-1,1> is a group where R are the real numbers, “” is the multiplication, “-1” the inverse, and 1 the identity for the multiplication.
Department of Geography and Regional Research, University of Vienna © Wolfgang Kainz 74
Field
A field is an algebra with the signature<F,+,, ¯,-1,0,1> and the following axioms:
1. <F,+, ¯,0> is a commutative group2. a (b c) = (a b) c3. a (b + c) = a b + a c4. (a + b) c = a c + b c5. <F-{0},,-1,1> is a commutative group
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© Wolfgang Kainz 75
Example
The real numbers <R,+,,-, -1,0,1> are a field with the addition and multiplication as binary operations, and the inverse unary operations for the addition and multiplication. The numbers 0 and 1 function as unit elements for + and , respectively.
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Boolean Algebra
A Boolean algebra is an algebra with signature <S,+,, ¯,0,1>, where + and are binary operations, and ¯ is a unary operation (complementation), with the following axioms:
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© Wolfgang Kainz 77
Boolean Algebra
complement the ofproperty 0
complement the ofproperty 1
foridentity an is 11
foridentity an is 00
law vedistributi)()()(
law vedistributi)(
law eassociativ)()(
law eassociativ)()(
law ecommutativ
law ecommutativ
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cbacbacbacba
abbaabba
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Example
<P(A),,, ¯,{},A> with ¯ as the complement relative to A, is a Boolean algebra.
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Mathematical Principles of GIS 5th Nordic Summer School in GIScienceAugust 16 – 20, 2010 (Gävle, Sweden)
© Wolfgang Kainz 14
Vector Space
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Let <V,+, ¯,0> be a commutative group, and<F,+,, ¯ ,-1,0,1> a field. V is called a vector spaceover F if for all a, b V and , F
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© Wolfgang Kainz 80
Example
The set of all vectors V with + as the vector addition is a vector space over the real numbers R where is the multiplication of a vector with a scalar.The same is true for the set of all matrices M with the matrix addition and the multiplication of a matrix with a scalar.
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Isomorphism
© Wolfgang Kainz Department of Geography and Regional Research, University of Vienna 81
Two algebras <S,,,k> and <S’,’,’,k’> are isomorphic if there exists a bijection f such that
kkfafaf
bfafbafSSf
)()4(
))(())(()3(
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:)1(
© Wolfgang Kainz 82
Isomorphism
• Two isomorphic algebras are essentially the same structure with different names.
• If we do not require f to be a bijection then we talk about a homomorphism. In general, homomorphisms generate a “smaller” image of an algebra of the same class.
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© Wolfgang Kainz 83
TopologicalStructures
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Topological Structures
• “Good behavior” of neighborhoods of points and invariants– Certain (spatial) relationships of neighborhood,
connectivity– Simple structured spaces (simplexes, cells)– Spatial relations derived from topological
invariants• Structure preserving mappings
– Topological mapping (or homeomorphism)
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Mathematical Principles of GIS 5th Nordic Summer School in GIScienceAugust 16 – 20, 2010 (Gävle, Sweden)
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Topology
• Metric space• Neighborhood• Topological space• Homeomorphism• Simplices, cells and their complexes
© Wolfgang Kainz Department of Geography and Regional Research, University of Vienna 85
What is Topology?
• Topology is the study of certain invariants of structured spaces.
• Structuring of spaces through:– a generalized notion of distance (metric space),– abstract notion of neighborhood, or– pasting together of certain well-understood
elementary objects (simplices or cells) to complexes (simplicial or cell complex).
© Wolfgang Kainz Department of Geography and Regional Research, University of Vienna 86
Spaces used in GIS
• In GIS we normally deal with objects in the Euclidean space in one, two, or three dimensions (R1, R2, or R3).
• Objects in this space are represented by nodes, arcs, polygons, and volumes.
© Wolfgang Kainz Department of Geography and Regional Research, University of Vienna 87
Properties of Space for Spatial Data
• Three-dimensional Euclidean space• Vector space• Metric space (Euclidean metric)• Topological space (topology induced by
Euclidean metric)• The topological space is structured by simple
sub-spaces (simplexes, cells, and their complexes)
© Wolfgang Kainz Department of Geography and Regional Research, University of Vienna 88
Metric Space
),(),(),(
),(),(
ifonly and if0),( and 0),(
zydyxdzxdxydyxd
yxyxdyxd
© Wolfgang Kainz 89
Let M be a set and d : M x M R a function, the metric (or distance function) on M.(M ; d) is called a metric space if the following conditions are valid for all x, y, z from M :
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Metric Space:Examples for Distance Functions
© Wolfgang Kainz Department of Geography and Regional Research, University of Vienna 90
3 21
1 33
1
( , ) ( )
( , ) max | |
( , ) | |
( , ) 0, if , and ( , ) 1, otherwise
i ii
i i i
i ii
d x y x y
d x y x y
d x y x y
d x y x y d x y
Euclideanmetric
City blockmetric
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Open Sets
© Wolfgang Kainz Department of Geography and Regional Research, University of Vienna 91
N
M
xr
K
N is an open neighborhood ofx M, if there exists an open disk K(x,r) with r > 0 andK(x,r) N.
N is an open set, if N is an open neighborhood of each of its points.
Metric Space
• An open disk of radius r around x of M is defined asK (x,r) = {y M | d (x,y) < r}
• A set N is called an (open) neighborhood of a point xof M, if there exists an open disk K (x, r) around xsuch that K (x, r) N.
• A set N M is called open set, if N is an open neighborhood of each of its points.
© Wolfgang Kainz Department of Geography and Regional Research, University of Vienna 92
© Wolfgang Kainz Department of Geography and Regional Research, University of Vienna 93
x
x
U
N
x
N
V
(N1) The point x lies in each of its neighborhoods.
(N4) Every neighborhood N of x contains a neighborhood V of x such that N is a neighborhood of every point of V.
(N3) Every superset U of a neighborhood N of x is a neighborhood of x. X is a neighborhood of x.
(N2) The intersection of two neighbor-hoods of x is itself a neighborhood.
x1N
2N
Topological Space
• Let M be a set. A topology on M is a collection O of subsets of M with the following properties:
(O1) {} O, M O(O2) A, B O A B O(O3) Ai O for all i I Ui I Ai O
• The elements of O are called open sets• (O4) N is an (open) neighborhood of x M, if
x N O.• (M,O) is called a topological space.
© Wolfgang Kainz Department of Geography and Regional Research, University of Vienna 94
Topological spacebased on concept ofneighborhood(N1 to N4):
Topological spaceBased on concept ofopen set(O1 to O3):
Definitionof open set
Definition ofneighborhood (O4)
Propertiesof open sets:O1 to O4
Propertiesof neighborhoods:N1 to N4
More theorems
Axioms
Theorems
© Wolfgang Kainz Department of Geography and Regional Research, University of Vienna 95
( ,{ ( )| })X N x x X
Topological spacebased on concept ofneighborhood(N1 to N4):
Definitionof open set
Propertiesof open sets:O1 to O4
( , )X O
Topological spaceBased on concept ofopen set(O1 to O3):
Definition ofneighborhood (O4)
Propertiesof neighborhoods:N1 to N4
Homeomorphism
• A bijective, continuous mapping with continuous inverse between two topological spaces is called a homeomorphism.
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© Wolfgang Kainz 97
Simplexes, Cells, Complexes
Simplexes, cells and their complexes are simple kinds of spaces that serve as topological equivalents of more complicated subsets of Euclidean space.
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Simplexes
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0-simplex (point)
0v1-simplex (closed line segment)
0v 1v
2-simplex (triangle)
2v
0v 1v
3-simplex (solid tetrahedron)
3v
0v 1v
2v
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Simplex
• A p-dimensional simplex Sp is a “solid” polyhedron in Rn which has internal points, is convex, and has a minimal number of vertices.– For the dimensions 1, 2, and 3, we have straight
line segments, solid triangles, and solid tetrahedrons.
• A q-dimensional face of a simplex Sp is a q-dimensional subset (simplex) of the p-dimensional simplex.
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Simplexes and Simplicial Complex
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0-simplex
1-simplex
2-simplex
3-simplex
Simplicial complex
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Simplicial Complex
A simplicial complex S is a set of simplexes in Rn
that fulfill the following conditions:– If the simplex Sp is an element of S, then each
face of Sp belongs to S.– For any two simplexes in S the intersection is
either empty or a common face.
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Cell
• A p-dimensional cell (or p-cell ) is a set which is homeomorphic to the p-dimensional unit ball.Every (open) p -simplex is a p -cell.
• Cell complexes (or CW spaces) are built starting with 0-cells (nodes) and subsequently gluing 1-cells (arcs), 2-cells (polygons), etc.
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Mathematical Principles of GIS 5th Nordic Summer School in GIScienceAugust 16 – 20, 2010 (Gävle, Sweden)
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© Wolfgang Kainz Department of Geography and Regional Research, University of Vienna 103
0-dimensional unit cell
1-dimensional unit cell
2-dimensional unit cell
3-dimensional unit cell
0-cell
1-cell
2-cell
3-cell
Cells and Cell Complex
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0-cell
1-cell
2-cell
3-cell
Cell complex
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2 2
Simplicial Complex Simplicial Complex
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Cell decompositionof a 2-dimensional space
1-dimensional skeleton 0-dimensional skeleton
Cell Decomposition
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© Wolfgang Kainz 107
Start with 0-cells
Gluing of 1-cells Gluing of 2-cells
0X
1X 2X
Generation of a cellcomplex
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Topological Mapping
© Wolfgang Kainz 108
h1M2M
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Topological Consistency Constraints
© Wolfgang Kainz Department of Geography and Regional Research, University of Vienna 109
• Every 1-cell is bounded by two 0-cells.
• For every 1-cell there are two 2-cells (left and right polygon).
• Every 2-cell is bounded by a closed cycle of 0- and 1-cells.
• Every 0-cell is surrounded by a closed cycle of 1- and 2-cells.
• 1-cells intersect only in 0-cells.
© Wolfgang Kainz 110
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Consistency Constraints: Example
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Closed Boundary Criterion
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Department of Geography and Regional Research, University of Vienna © Wolfgang Kainz 112
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Node Criterion (“Umbrella”)
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Interior, closure, boundary, and exterior
• The interior of a set A (written as A°) is the union of all open sets contained in A.
• A is closed if its complement is open. The smallest closed set that contains A is the closure of A (written as ).
• The boundary of A (written as A) is the difference of the closure and the interior.
• The exterior of A (written as A¯) is the complement of the closure.
© Wolfgang Kainz Department of Geography and Regional Research, University of Vienna 113
A
Topolocial Invariants
© Wolfgang Kainz Department of Geography and Regional Research, University of Vienna 114
A
interior
A
boundary
A
closure
A
exterior
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Topological Relationships
© Wolfgang Kainz Department of Geography and Regional Research, University of Vienna 115
Relationships between two regions can be determined based on the intersection of their boundaries and interiors (4-intersection).
A B
BABABABABAI
),(4
Spatial Relationships Between Simple Regions
© Wolfgang Kainz Department of Geography and Regional Research, University of Vienna 116
disjoint
meet
equal
inside
covered by
contains
covers
overlap
© Wolfgang Kainz 117
9-Intersection
––––
–
–
9 ),(
BABABABABABABABABA
BAI
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Order Structures
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Order Structures• Partially ordered set (poset) and lattice• Comparison of elements of a set
– Relationships of containment and inclusion• “… is contained in …”• “ … contains …”• “ … subset of …”• “ … less than or equal …”
– Representation of order diagrams as directed acyclic graph (DAG)
– Consideration of special elements (lower and upper bounds, greatest lower bound (g.l.b.), least upper bound (l.u.b.))
• Structure preserving mapping– Monotone functions
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Partially ordered set
A partially ordered set (or poset) is a set P with a relation defined for its elements x, y, and z and:(1) x x (reflexive)(2) x y and y x implies x = y (antisymetric)(3) x y and y z imply x z (transitive)
Example: numbers with ( ), sets with ()
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Order Relationships
© Wolfgang Kainz Department of Geography and Regional Research, University of Vienna 121
A
B
CD
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Order Relationships: Order Diagram
© Wolfgang Kainz Department of Geography and Regional Research, University of Vienna 122
BCD
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AA
B C
D E
Order Relationships: Order Diagram
• An order diagram is a DAG (directed acyclic graph).
• It can be represented by an adjacency matrix or list.
• Graph algorithms can be used.
© Wolfgang Kainz Department of Geography and Regional Research, University of Vienna 123
Order Structures: Sample Questions
• What regions are contained in a set of given regions?
• What is the largest region contained in a set of given ones?
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Upper and lower bounds
• Let P be a poset and S P. An elementx P is an upper bound of S if s x for alls S. A lower bound is defined by duality.
• The set of all upper or lower bounds are denoted as S* (S upper ) and S* (S lower ).
© Wolfgang Kainz Department of Geography and Regional Research, University of Vienna 125
Greatest lower bound, least upper bound
• If S* has a largest element, it is called greatest lower bound (g.l.b.), meet, or infimum. For two elements x and y we write inf{x, y} or x y (“x meet y ”).
• If S* has a least element, it is called least upper bound (l.u.b.), join, or supremum. For two elements x and y we write sup{x, y} orx y (“x join y ”).
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Example: lower bounds
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A
B C
D E
Lower bounds of B are B, Dand E.
BB
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Example: lower bounds
© Wolfgang Kainz 128
A
B C
D E
Lower bounds of C are C, Dand E.
CC
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Example: lower bounds
© Wolfgang Kainz 129
A
B C
D ELower bounds of {B,C} are Dand E.
CBB C
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Lattice
• A lattice is a poset in which a g.l.b. or l.u.b. can always be found for any two elements.
• If a g.l.b. or l.u.b. exists for every subset of the poset, we call it a complete lattice. Every finite lattice is complete.
© Wolfgang Kainz Department of Geography and Regional Research, University of Vienna 130
Order Relationships: Normal Completion
© Wolfgang Kainz Department of Geography and Regional Research, University of Vienna 131
poset
A
B C
D E
latticeA
B C
D E
X
{ }
Order Relationships: Geometric Interpretation
© Wolfgang Kainz Department of Geography and Regional Research, University of Vienna 132
A
B C
D E
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{ }
BCD
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A
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© Wolfgang Kainz 23
Monotone functions
• M and N are two posets. A functionf : MN is called a monotone function (or order preserving), if x y in M implies that f (x ) f (y ) in N.
• The function is an order-embedding when f is injective.
• If f is bijective and monotone with a monotone inverse, it is called an order isomorphism.
© Wolfgang Kainz Department of Geography and Regional Research, University of Vienna 133
Problem Solving
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Spatial problemTranslation into mathematical
problem
mathematical solution
Spatial interpretation
of solution
Solution of spatial problem
© Wolfgang Kainz Department of Geography and Regional Research, University of Vienna 135
Find the largest region contained in
given regions
Generate the posetof regions
Compute the normal completion
and g.l.b.
Spatial interpretation of
g.l.b.Solution of spatial
problem
BCD
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B C
D E
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{ }
BCD
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