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Theory & Methods of GIScience Version 1.3 [03 October 2010]

© Wolfgang Kainz 1

© Wolfgang Kainz 1

Theory and Methods ofGeographicInformationScience

© Wolfgang Kainz 2

Spatial Information

Space, Time, and us…GISystemsData, information,…

© Wolfgang Kainz 3

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?

Limited to three spatial dimensionsWe cannot escape from within a closed cube (we cannot “see” higher dimensions than 3D) likeFlatlanders cannot escape from a closed square

© Wolfgang Kainz 4

We are spatiotemporal beings

© Wolfgang Kainz 5

Enuma Elish(Babylonian Creation Myth)“When on high the heaven had not been named,Firm ground below had not been called by name,When primordial Apsu, their begetter,And Mummu-Tiamat, she who bore them all,Their waters mingled as a single body,No reed hut had sprung forth, no marshland had appeared,None of the gods had been brought into being,And none bore a name, and no destinies determined ̶ Then it was that the gods were formed in the midst of heaven.”

© Wolfgang Kainz 6

Metamorphoses(P. Ovidius Naso)

“…Before there was earth or sea or the sky that covers everything, Nature appeared the same throughout the whole world: what we call chaos: a raw confused mass, nothing but inert matter, badly combined discordant atoms of things, confused in the one place……This conflict was ended by a god and a greater order of nature, since he split off the earth from the sky, and the sea from the land…”


© Wolfgang Kainz 2

© Wolfgang Kainz 7

Genesis(The Bible)

“In the beginning God created the heaven and the earth…”

© Wolfgang Kainz 8

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 9

Importance of Space and Time

Almost everything that happens, happens at a certain location in space and timeThe level of (geographic) detail (or scale) matters

Mapping a local event versus the global spread of bird flu

Time scales500-year flood versus property transactions

© Wolfgang Kainz 10

Geographic vs. Spatial

Geographic refers to the surface of the Earth or the space near to itSpatial refers to any space (other planets, the human body,…)


Why is spatial special?

We are dealing with multiple dimensions (x,y,z)We are dealing with different levels of spatial resolutionRepresentation of spatial data is more “complicated” than of non-spatial dataWe often need to project dataSpatial analysis requires special methods


GI Science

Systems data

information

knowledge

wisdom

acquisition

analysis

reasoning

contemplation


© Wolfgang Kainz 3


Disciplines Using Spatial Information

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)


GIS Functional Modules

Input

Data Management

Manipulation and Analysis Output


History of GISFirst GIS (CGIS)

Commercial GIS Software

Geo-Information Technology

1960s

1980s

1990s

topology, data structures, DBMS

operationalization

Geographic Information Science

theory & widespread use

2000s


Then, came the science…

The science behind the systemsGeoinformaticsGeomaticsSpatial information scienceSpatial information theoryGeoinformation engineeringGeographic Information Science


Models

Spatial modelingImplementation of modelsApplication of models


© Wolfgang Kainz 4


Cultural Background

Spatial Modeling

GIS and Spatial Decision Support Systems (SDSS)

Management and Infrastructure

Theoretical Foundation

Data Input Database

Output andVisualization

Analysis


Modeling(Wittgenstein: Tractatus Logico-Philosophicus)

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.”


Real Worldphenomena

Spatial modelfeatures

GIS databasespatial data

design implementation

Spatial Modeling & Data Processing

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.

Data handling

Spatial Information


Real World

Miniworld

Conceptualschema

Logicalschema

Physicalschema

Conceptual model

Logical model

Implementation

software independent

software specific

Spatial modeling

Spat

ial

mod

elin

gD

atab

ase

desi

gnan

d im

plem

enta

tion


Geography, Cartography, andGIScience


Geography

Science of the surface of the Earth and its spatial differentiation, its physical characteristics as well as the space and place of human lives and actions.Geography also deals with humans in their physical and social environment.Established as a scientific discipline at universities since the mid 19th century


© Wolfgang Kainz 5


Geography

GEOGRAPHY

Thematic

PhysicalGeography

Natural Science

HumanGeography

Social Science

Regional

RegionalResearch

Regional and Environmental

Planning


Cartography

Cartography is a discipline dealing with the collection, processing, storage, and analysis of spatial information; and in particular with its visualization by cartographic representations.


Cartography

Cartography

GeneralCartography

Theories andMethods

CartographicModeling

CartographicTechniques

AppliedCartography

TopographicCartography

ThematicCartography


Cartography as a Science: subject of research

Cartographic expressions and their graphic elements (Arnberger)Epistemological aspect: object relationships in space, timeCommunication of spatial information


Content of GIScience (Goodchild 1992)

Data collection and measurementNature of spatial variationDiscretization, generalization, abstraction, approximation

Data captureFrom secondary data sources to primary data sources

Spatial statisticsSpatial data uncertaintySpatial data quality

Data modeling and theories of spatial dataObject vs. field

Data structures, algorithms and processesDisplayAnalytical tools

Integration of GIS and spatial analysisInstitutional, managerial and ethical issues


GIS Research (Goodchild 1992)

Research on the generic issues that surround the use of GIS technology…Both “research about GIS” and “research with GIS”


© Wolfgang Kainz 6


GIScience (Goodchild 2004)Geographic data display two general properties

Spatial dependence• Tobler’s First Law of Geography (“All things are related,

but nearby things are more related than distant things.”)Spatial heterogeneity

• Expectations vary across the Earth’s surface (results of an analysis depend on the place)

Form vs. ProcessForm: the world as it looks (static)Process: dynamics


What makes a Science?

Science [from Latin scientia = knowledge] isa system of acquiring knowledgean organized body of knowledge

Science adheres to the scientific methoda body of techniques for the investigation of phenomena and the acquisition of new knowledge based on observation and reasoning


Science is

Pure science (basic science)Scientific theories without consideration of applications

Applied scienceScientific theories for the solution of practical problems


Approaches to science

EmpiricismAll knowledge of non-analytic truths is justified by experience• An analytic statement is true by definition

(without any knowledge of the world); it isa priori true. The contrary is a synthetic statement whose truth depends on (some) knowledge of the world

Scientific realismdefines science in terms of ontology


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.


GIScience is

Ontology drivenWhat constitutes the world?• Entities (objects, categories, concepts)• Characteristics (attributes)

How are things related?• Relations (relationships)


© Wolfgang Kainz 7


GIScience uses

LogicMathematics

To make statements about the world and acquire knowledge about the world


Structures

Algebraic Order Topological

Logic

Set Theory

RelationsFunctions

AlgebraOrdered

SetsTopology


Logic

Propositional logicPredicate logicLogical inference


Logic

Propositional Logicassertion, proposition, propositional variable, logical operators, propositional form, truth table, tautology, contradiction, contingency

Predicate Logicpredicates, quantifiers

Logical Inference


Assertion and Proposition

An assertion is a statement.“Are you okay?”“Give me the book.”

A proposition is an assertion that is either true or false, but not both.

“It is raining.”“I pass the exam.”


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., P and “I pass the exam.” When propositions are substituted for the variables a proposition results.


© Wolfgang Kainz 8


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."


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”.


Negation

01

10

PP


Conjunction(“logical and”)

111

001

010

000

QPQP


Disjunction(“logical or” or “inclusive or”)

111

101

110

000

QPQP


Exclusive or

011

101

110

000

QPQP


© Wolfgang Kainz 9


Implication

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


Implication

The converse of P Q is the proposition Q P.The contrapositive of P Q is the proposition Q P.


Equivalence

“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


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.


Examples

(P Q) P is a tautology.

1111

1001

1010

1000

)( PQPQPQP


Examples

P P is a contradiction.

001

010

PPPP




Examples

(P Q) Q is a contingency.

11011

01101

10010

01100

)( QQPQPQQP


Logical Identities

1 ( ) idempotence of

2 idempotence of

3 ( ) ( ) commutativity of

4 ( ) ( ) commutativity of

5 [( ) ] [ ( )] associativity of

6 [( ) ] [ ( )] associativity of

7 ( ) ( ) D

P P P

P (P P)

P Q Q P

P Q Q P

P Q R P Q R

P Q R P Q R

P Q P Q

e Morgan's Law

8 ( ) ( ) De Morgan's Law

9 [ ( )] [( ) ( )] distributivity of over

P Q P Q

P Q R P Q P R


Logical Identities

10 [ ( )] [( ) ( )] distributivity of over

11 ( 1) 1

12 ( 1)

13 ( 0)

14 ( 0) 0

15 ( ) 1

16 ( ) 0

17 ( ) double negation

18 ( ) ( ) implication

P Q R P Q P R

P

P P

P P

P

P P

P P

P P

P Q P Q


Logical Identities

19 ( ) [( ) ( )] equivalence

20 [( ) ] [ ( )] exportation

21 [( ) ( )] absurdity

22 ( ) ( ) contrapositive

P Q P Q Q P

P Q R P Q R

P Q P Q P

P Q Q P


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 yare 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.


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.




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.”


Universal Quantifier

If an assertion P(x) is true for every possible value x, then xP(x) is true; otherwise xP(x) is false.


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.”


Existential Quantifier

If an assertion P(x) is true for at least one value x, then xP(x) is true; otherwise xP(x) is false.


Logical Relationships Involving Quantifiers

1 ( ) ( ), where is an arbitrary element of the universe

2 ( ) ( ), where is an arbitrary element of the universe

3 ( ) ( )

4 ( ) ( )

5 ( ) ( )

6 [ ( ) ] [ ( ) ]

7 [ ( ) ]

xP x P c c

P c xP x c

x P x xP x

xP x xP x

x P x xP x

xP x Q x P x Q

xP x Q

[ ( ) ]

8 [ ( ) ( )] [ ( ) ( )]

9 [ ( ) ( )] [ ( ) ( )]

x P x Q

xP x xQ x x P x Q x

xP x xQ x x P x Q x


Logical Relationships Involving Quantifiers

10 [ ( ) ] [ ( ) ]

11 [ ( ) ] [ ( ) ]

12 [ ( ) ( )] [ ( ) ( )]

13 [ ( ) ( )] [ ( ) ( )]

xP x Q x P x Q

xP x Q x P x Q

x P x Q x xP x xQ x

xP x xQ x x P x Q x




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.


Logical Inference

Rules of inference specify conclusions which can be drawn from assertions known or assumed to be true.

Q

P

P

P

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.”


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.


Rules of Inference Related to the Language of Propositions

Rule of Inference Tautological Form Name

( ) addition

( ) simplification

[ ( )] modus ponens

[ ( )] modus tollens

[( ) ] disjunctive syllogism

PP P Q

P Q

P QP Q P

P

P

P Q P P Q Q

Q

Q

P Q Q P Q P

P

P Q

P P Q P Q

Q



Rule of Inference Tautological Form Name

[( ) ( )] [ ] hypotetical syllogism

conjunction

P Q

Q R P Q Q R P R

P R

P

Q

P Q



Rule of Inference Tautological Form

( ) ( )

[( ) ( ) ( )] [ ]

P Q R S

Q S P Q R S Q S P R

P R

constructive dilemma

destructive dilemma

Rule of Inference Tautological Form

( ) ( )

[( ) ( ) ( )] [ ]

P Q R S

P R P Q R S P R Q S

Q S




Rules of Inference Involving Predicates and Quantifiers

( )universal instatiation

( )

( )universal generalization

( )

( )existential instantiation

( )

( )existential generalization

( )

xP x

P c

P x

xP x

xP x

P c

P c

xP x


Set Theory, Relations, and Functions


Set Theory

A set is a collection of well-distinguished objects. Any object of the collection is called an element, or member of the set.An element x of the set S is written asx S. If x is not an element of S we write x S.


Set Theory: Specification

Listing of elements{1,2,3,4,5}

Implicit description by means of a predicate and a free variable

{x | x I x > 10}

Graphic representation (Venn diagram)

A


Relations Between Sets:Subset

If each element of a set A is an element of a set B then A is subset of B, written as A B. B is called superset of A, written as B A.

We call A a proper subset of B when A Band A B.If U is the universe of discourse thenA U.


Relations Between Sets:Equality

Two sets A and B are equal if and only if A B and B A.




Empty Set

A set with no members is called emptyset, null set, or void set. It is written as {} or .The empty set is subset of every set.


ABAB

Operations on Sets:Union

The union of two sets A and B, written as A B, is the set

A B = {x | x A x B}


Operations on Sets:Intersection

The intersection of two sets A and B, written as A B, is the set

A B = {x | x A x B}

If A B = {} then A and B are disjoint.


BAA

BAB

Operations on Sets:Intersection


Operations on Sets:Difference

The difference of two sets A and B, written as A - B, is the set

A - B = {x | x A x B}

AB

A-BB


Operations on Sets:Complement

The complement of a set A, written as Āis the set

Ā = U - A = {x | x A}

U

AAA




Operations on SetsPower Set

The power set of a set A, written as P(A), is the set of all subsets of A.

When the set has n elements, the power set has 2n elements.


Rules for Set Operations

)()( then , and

)()( then , and

{}{}

{}

DBCADCBA

DBCADCBA

ABA

A

AA

AAA

AAA



BAABA

ABA

AA

ABABA

BBABA

ABA

BAA

)(

{})(

{}

then , If

then , If



BABA

BABA

AA

AA

UAA

CABACBA

CABACBA

{}

)()()(

)()()(


Binary Relations

The Cartesian product (or cross product) of two sets A and B, denoted as AxB, is the set of all pairs{<a,b> | a A b B}.A binary relation R over AxB is a subset of AxB. The set A is the domain of R; Bis the codomain. We write <a,b> Ras aRb.


Special Properties of Relations

Let R be a binary relation on A. ThenR is reflexive if xRx for every x in A.R is irreflexive if xRx for no x in A.R is symmetric if xRy implies yRx for every x and y in A.R is antisymmetric if xRy and yRxtogether imply x = y for every x, y in A.R is transitive if xRy and yRz together imply xRz for every x,y, and z in A.




Graphic Representaion of Binary Relations

a bDirected graph

a

c d

b

A = {a, b, c, d}R = {<a,a>, <a,c>, <b,c>}


Special Relations

A reflexive, symmetric and transitive relation is called equivalence relation.A reflexive, antisymmetric and transitive relation is called order relation.


Composition of Relations

Let R1 be a relation from A to B and R2 a relation from B to C. The composite relation from A to C, written as R1R2 is defined asR1R2 = {<a,c> | a A c C b[b B <a,b>R1 <b,c>R2]}

The composition is not commutative, but associative.


Functions

A function (or map, or transformation) ffrom A to B, written as f: AB, is a relation from A to B such that for every aA, there exists a unique bB such that <a,b>f. We write f (a) = b. A is called the domain, and B is called codomain off ; a is the argument and b the value of the function for the argument a.


Composition of Functions

Let g : AB and f : BC be functions. The composite function fg is a function from A to C, and(f g)(x ) = f (g (x )) for all x in A.The composition of functions is associative, but not commutative.


Classes of Functions

A function f from A to B is surjective (a surjection, or onto) if f (A ) = B.A function f from A to B is injective (an injection, or one-to-one) if a1 a2implies f (a1) f (a2).A function f from A to B is bijective (a bijection, or one-to-one and onto) if f is both surjective and injective.




Classes of Functions

If every horizontal line intersects the graph of a function at least once, then the function is surjective.If no horizontal line intersects the graph more than once, then the function is injective.If every horizontal line intersects the graph of a function exactly once, then the function is bijective.


Example: surjective function (not injective)


Example: injective function (not surjective)


Example: bijective function


Example: not injective, not surjective


Algebraic Structures




Structures

Algebraic StructuresTopological StructuresOrder Structures


Algebraic Structures (Algebras)

Structure and OperationsSpecial elementsVarieties

GroupFieldBoolean AlgebraVector Space

Homomorphism


Components of an Algebra

A set, the carrier of the algebra,Operations defined on the carrier, andDistinguished elements of the carrier, the constants of the algebra.

Algebras are presented as tupels <carrier, operations, constants>

Example: <R,+,,0,1>© Wolfgang Kainz 106

Signature and Variety

Two algebras have the same signature(or are of the same species) if they have corresponding operations and corresponding constants.A set of axioms for the elements of the carrier, together with a signature specifies a class of algebras called variety.


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.


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) inverseof y with respect to the operation .




Group

A group is an algebra with the signature <S,, ̄,1>, with ̄ the inverse with respect to , and the following axioms:

1

11

)()(

aa

aaa

cbacba


Group

If the operation is also commutative, we call the group a commutative group.


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.


Field

A field is an algebra with the signature <F,+,, ¯,-1,0,1> and the following axioms:

<F,+, ¯,0> is a commutative groupa (b c) = (a b) ca (b + c) = a b + a c(a + b) c = a c + b c<F-{0},,-1,1> is a commutative group


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.


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:




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

aa

aa

aa

aa

cabacba

cabacba

cbacba

cbacba

abba

abba


Example

<P(A),,, ¯,{},A> with ¯ as the complement relative to A, is a Boolean algebra.


Vector Space

Let <V,+, ¯,0> be a commutative group, and <F,+,, ¯ ,-1,0,1> a field. V is called a vector space over F if for all a, b V and , F

aa

aa

aaa

baba

1

)()(

)(

)(


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 matricesM with the matrix addition and the multiplication of a matrix with a scalar.


Isomorphism

Two algebras <S,,,k> and <S’,’,’,k’> are isomorphic if there exists a bijection f such that

kkf

afaf

bfafbaf

SSf

)()4(

))(())(()3(

)()()()2(

:)1(


Isomorphism

Two isomorphic algebras are essentially the same structure with different names.If we do not require f to be a bijectionthen we talk about a homomorphism. In general, homomorphisms generate a “smaller” image of an algebra of the same class.




Topology


Topologic Structures

Metric spaceNeighborhoodTopological spaceHomeomorphismSimplices, cells and their complexes


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, orpasting together of certain well-understood elementary objects (simplices or cells) to complexes (simplicial or cell complex).


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.


Metric Space

Let M be a set and d : M x M R a function, the metric on M. (M ; d) is called a metric spaceif the following conditions are fulfilled for all x, y, z from M :

),(),(),(

),(),(

ifonly and if 0),( and 0),(

zydyxdzxd

xydyxd

yxyxdyxd


Examples

The following functions define metrics on the real numbers R3:

otherwise ,1),( and , if ,0),(

||),(

||max),(

)(),(

3

1

31

3

1

2

yxdyxyxd

yxyxd

yxyxd

yxyxd

i ii

iii

i ii

Euclideanmetric

City blockmetric




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 x of M, if there exists an open diskK (x, r) around x such that K (x, r) N. A set N M is called open set, if N is an open neighborhood of each of its points.


Open Sets

N

Mx

r

K

N is an open neighborhood ofx M, if there exists an open disk K(x,r) with r > 0 and K(x,r) N.

N is an open set, if N is an open neighborhood of each of its points.


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 neighborhoods of x is itself a

neighborhood.

x1N

2N


Homeomorphism

A bijective, continuous mapping with continuous inverse between two topological spaces is called a homeomorphism.


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.


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.




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.


Simplexes0-simplex (point)

0v1-simplex (closed line segment)

0v 1v

2-simplex (triangle)

2v

0v 1v

3-simplex (solid tetrahedron)

3v

0v 1v

2v


0-simplex

1-simplex

2-simplex

3-simplex

Simplicial complex

Simplexes and Simplicial Complex


2 2

Simplicial Complex Simplicial Complex


Cell

A p-dimensional cell (or p-cell ) is a set which is homeomorphic to thep -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.


0-dimensional unit cell




0-cell

1-cell

2-cell

3-cell




0-cell

1-cell

2-cell

3-cellCell complex

Cells and Cell Complex


Cell decompositionof a 2-dimensional space

1-dimensional skeleton 0-dimensional skeleton

Cell Decomposition


Start with 0-cells

Gluing of 1-cells Gluing of 2-cells

0X

1X 2X

Generation of a cell complex


h1M

2M

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Topological Mapping


Topological Consistency Constraints• 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.


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Consistency Constraints: Example




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Closed Boundary Criterion

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1


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Node Criterion (“Umbrella”)

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Properties of Space for Spatial Data

Three-dimensional Euclidean spaceVector spaceMetric space (Euclidean metric)Topological space (topology induced by Euclidean metric)The topological space is structured by simple sub-spaces (simplexes, cells, and their complexes)


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 A).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.

A


A

interior

A

boundary

A

closure

A

exterior

Topolical Invariants


Topological RelationshipsRelationships between two regions can be determined based on the intersection of their boundaries and interiors (4-intersection).

A B

BABA

BABABAI

),(4




disjoint

meet

equal

inside

covered by

contains

covers

overlap

Spatial Relationships Between Simple Regions


9-Intersection

––––

–

–

9 ),(

BABABA

BABABA

BABABA

BAI


Order Structures


Order Structures

Partially ordered sets and latticesRelationships of containment and inclusion

“… is contained in …”“ … contains …”“ … subset of …”“ … less than or equal …”

Monotone functions


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(2) x y and y x implies x = y(3) x y and y z imply x z

Example: numbers with ( ), sets with ()© Wolfgang Kainz 156

Order Relationships

A

B

CD

E




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?


Order Relationships:Order Diagram

BCD

E

A A

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.


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 all s 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 ).


Example: lower bounds

A

B C

D E

Lower bounds of B are B, D and E.

BB



A

B C

D E

Lower bounds of C are C, D and E.

CC





A

B C

D E Lower bounds of {B,C} are D and E.

CBB C


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 sup{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 ywe write inf{x, y} or x y (“x join y ”).


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.


Order Relationships:Normal Completion

poset

A

B C

D E

lattice

A

B C

D E

X

{ }


Order Relationships:Geometric Interpretation

A

B C

D E

X

{ }

BCD

E

A

X


Monotone functionsM and N are two posets. A functionf : MN is called a monotone function(or order preserving), if x y in Mimplies that f (x ) f (y ) in N.The function is an order-embeddingwhen f is injective.If f is bijective and monotone with a monotone inverse, it is called an order isomorphism.




c

A B

da b

Example: subdivision of land


Field b

Land

Zone A Zone B

Field dField a

Land

Zone A Zone B

Field a Field bField c Field d Field c

Example: hierarchy

c

A B

da b


c

A B

da b

Example: poset representation

c

A B

da b


A, B, a, b, c, d

c d

c, d

{ }

a b

B, b, c, dA, a, c, d

Example: normal completion


A B

c d

X

L

ø

a b

Example: resulting lattice

A B

a bc

dX


X

Y

Example: two overlapping areas




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Simplicial complex and poset

B IJHGA C FED

dc ba n ml k r qpohg fe sji

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X Y


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Simplicial complex and normal completion

B IJHGA C FED


1 09876 5432

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Ø


Simplicial complex and normal completion

B IJHGA C FED


1 09876 5432

X Y

U

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Ø

Z

X

Y


B IJHGA C FED

dc ba n m r qpog fe sji

1 09876 5432

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l kh

i

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k

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E

FZ

Boundary


B IJHGA C FED


1 09876 5432

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A

C

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a

c d

Neighborhood


B IJHGA C FED


1 09876 5432

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CB

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Neighborhood




B IJHGA C FED


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E

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Touch


B IJHGA C FED


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Containment


Lattices and Spatial Objects

B IJHGA C FED


1 09876 5432

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Geometry and Topology

Object Level


Comparison of topological and order relationships

Order relation Topological relationA B inside(A,B)

contains(B,A)equal(A,B)coveredBy(A,B)covers(B,A)

A and B not comparable disjoint(A,B)meet(A,B)overlap(A,B)

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