Shaowen WangShaowen WangCyberInfrastructure and Geospatial Information CyberInfrastructure and Geospatial Information

Laboratory (CIGI)Laboratory (CIGI)Department of GeographyDepartment of Geography

andandNational Center for Supercomputing Applications (NCSA)National Center for Supercomputing Applications (NCSA)

University of Illinois at Urbana-ChampaignUniversity of Illinois at Urbana-Champaign

February 21 - March, 2011February 21 - March, 2011

Principles of GISPrinciples of GIS

Fundamental spatial conceptsFundamental spatial concepts

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Things We just Things We just LearnedLearned

DatabasesDatabases– DBMSDBMS

Data modelingData modeling– RelationalRelational– Object-orientedObject-oriented

Relational databasesRelational databases– SQLSQL– Extended RDBMSExtended RDBMS– Spatial data handlingSpatial data handling

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Spatial ConceptsSpatial Concepts

Euclidean geometry Euclidean geometry Sets of geometric elementsSets of geometric elements TopologyTopology

– NeighborhoodNeighborhood GraphGraph

– NodesNodes– EdgesEdges

Metric spaceMetric space

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Point ObjectPoint Object

Cartesian planeCartesian plane VectorVector

– NormNorm DistanceDistance AngleAngle

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Line ObjectLine Object

Parameterized representationParameterized representation LineLine Line segmentLine segment Half lineHalf line

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Polygonal ObjectsPolygonal Objects

PolylinePolyline– Simple closed polylineSimple closed polyline

PolygonPolygon– Convex polygonConvex polygon– Star-shaped polygonStar-shaped polygon

MonotoneMonotone– ChainChain– PolygonPolygon

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Triangulation Triangulation

DiagonalDiagonal Non-diagonal Non-diagonal TIN (Triangulated Irregular TIN (Triangulated Irregular

Network)Network)

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SetsSets

Organization of geometric objectsOrganization of geometric objects Creation of new geometric objectsCreation of new geometric objects

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ConceptsConcepts

ElementElement MembershipMembership

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Set CharacteristicsSet Characteristics

EqualityEquality SubsetSubset Power setPower set Empty setEmpty set CardinalityCardinality

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Set OperationsSet Operations

IntersectionIntersection UnionUnion DifferenceDifference ComplementComplement

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Types of SetsTypes of Sets

Specific useful setsSpecific useful sets– BooleansBooleans– IntegersIntegers– RealsReals– Real planeReal plane– Closed intervalClosed interval– Open intervalOpen interval– Semi-open intervalSemi-open interval

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Relations of SetsRelations of Sets

ProductProduct Binary relationBinary relation

– ReflexiveReflexive– SymmetricSymmetric– TransitiveTransitive

Equivalence relationEquivalence relation

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FunctionsFunctions

DomainDomain CodomainCodomain

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Function PropertiesFunction Properties

InjectionInjection– Inverse functionInverse function

SurjectionSurjection Bijection Bijection

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ConvexityConvexity

VisibilityVisibility Observation pointObservation point Convex hullConvex hull

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Topological SpacesTopological Spaces

Topological propertiesTopological properties TopologyTopology Point-set topologyPoint-set topology

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NeighborhoodNeighborhood

NeighborhoodsNeighborhoods– A collection of subsets of a given set A collection of subsets of a given set

of points of points SS TT1: Every point in 1: Every point in SS is in some neighbor is in some neighbor TT2: The intersection of any two 2: The intersection of any two

neighborhoods of any point neighborhoods of any point xx in S in S contains a neighborhood of contains a neighborhood of xx

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Usual TopologyUsual Topology

Euclidean planeEuclidean plane Open diskOpen disk Validate Validate T T 1 and 1 and T T 22

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Travel Time TopologyTravel Time Topology

Travel time relation Travel time relation – SymmetricSymmetric

Neighborhoods Neighborhoods – All time zonesAll time zones

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Near PointNear Point

XX – Subset of points in a topological spaceSubset of points in a topological space

xx– An individual point in the topological An individual point in the topological

spacespace Every neighborhood of Every neighborhood of xx contains contains

some point of some point of XX

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Properties of A Properties of A Topological SpaceTopological Space Open setOpen set Closed setClosed set ClosureClosure

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Properties of A Properties of A Topological SpaceTopological Space

Open setOpen set– Every point of a set can be surrounded by a Every point of a set can be surrounded by a

neighborhood that is entirely within the setneighborhood that is entirely within the set Closed setClosed set

– A set contains all its near pointsA set contains all its near points Closure (Closure (X X --))

– The union of a point set with the set of all its The union of a point set with the set of all its near pointsnear points

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Properties of A Properties of A Topological Space – Topological Space – continuedcontinued

Interior (Interior (X X oo) of a point set) of a point set– Consists of all points that belong to the set and Consists of all points that belong to the set and

are not near points of the complement of the are not near points of the complement of the setset

Boundary of a point set (Boundary of a point set (∂∂X)X)– Consists of all points that are near to both the Consists of all points that are near to both the

set and its complementset and its complement ConnectednessConnectedness

– Partition into two non-empty disjoint subsets: A Partition into two non-empty disjoint subsets: A and Band B

– Either A contains a point near BEither A contains a point near B– Or B contains a point near AOr B contains a point near A

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Future TopicsFuture Topics

Combinatorial topologyCombinatorial topology Network spacesNetwork spaces

– GraphGraph Metric spacesMetric spaces Fractal geometry Fractal geometry

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End of This Topic