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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