spatial relations.docx
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spatial relations.docx
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spatialrelations
Gettingstarted
Understandingspatialrelations
Aprimaryfunctionofageographicinformationsystemisdeterminingthespatialrelationshipsbetweenfeatures.Thedistanceseparatingahazardouswastedisposalsiteandhospital,school,orhousingdevelopmentisanexampleofaspatialrelationship.
PredicatesareBooleanfunctionsthatreturnTRUEifatestpassesandFALSE,otherwise,todetermineifaspecificrelationshipexistsbetweenapairofgeometries.Otherfunctionsreturnavalueasaresultofaspatialrelationship.Theresultreturnedbythedistance function,thespaceseparatingtwogeometries,isadoubleprecisionnumber.Alternatively,functionslikeintersection returnageometryastheresultofcombiningtwogeometries.
Predicates
Predicatesreturnt(TRUE)ifacomparisonmeetsthefunctionscriteria;otherwise,theyreturnf(FALSE).Predicatesthattestforaspatialrelationshipcomparepairsofgeometrythatcanbeadifferenttypeordimension.
PredicatescomparetheXandYcoordinatesofthesubmittedgeometries.TheZcoordinates andmeasure values,iftheyexist,areignored.GeometriesthathaveZcoordinatesormeasurescanbecomparedwiththosethatdon't.
TheDimensionallyExtended9IntersectionModel (DE-9IM)developedbyClementini,etal.,dimensionallyextendsthe9IntersectionModelofEgenhoferandHerring.DE-9IMisamathematicalapproachthatdefinesthepair-wisespatialrelationshipbetweengeometriesofdifferenttypesanddimensions.Thismodelexpressesspatialrelationshipsamongalltypesofgeometryaspair-wiseintersectionsoftheirinterior,boundary,andexteriorwithconsiderationforthedimensionoftheresultingintersections.
Predicates:
Equal
Disjoint
Intersects
Touch
Overlap
Cross
Within
Contains
RelatedConcepts:
Minimumdistance
IntersectionofGeometries
DifferenceofGeometries
UnionofGeometries
Symmetricdifferenceofgeometries
Example:
Givengeometriesaandb,I(a),B(a),andE(a)representtheinterior,boundary,andexteriorofa,andI(b),B(b),andE(b)representtheinterior,boundary,andexteriorofb.TheintersectionsofI(a),B(a),andE(a)withI(b),B(b),andE(b)producesa3-by-3matrix.Eachintersectioncanresultingeometriesofdifferentdimensions.Forexample,theintersectionoftheboundariesoftwopolygonscouldconsistofapointandalinestring,inwhichcasethedimfunctionwouldreturnthemaximumdimensionof1.
Thedimfunctionreturnsavalueof-1,0,1,or2.The-1correspondstothenullsetthatisreturnedwhennointersectionwasfoundordim(Æ).
Interior Boundary Exterior
Interior dim(I(a)ÇI(b)) dim(I(a)ÇB(b)) dim(I(a)ÇE(b))
Boundary dim(B(a)ÇI(b)) dim(B(a)ÇB(b)) dim(B(a)ÇE(b))
Exterior dim(E(a)ÇI(b)) dim(E(a)ÇB(b)) dim(E(a)ÇE(b))
TheresultsofthespatialrelationshippredicatescanbeunderstoodorverifiedbycomparingtheresultsofthepredicatewithapatternmatrixthatrepresentstheacceptablevaluesfortheDE-9IM.
Thepatternmatrixcontainstheacceptablevaluesforeachoftheintersectionmatrixcells.Thepossiblepatternvaluesare:
TAnintersectionmustexist;dim=0,1,or2.
FAnintersectionmustnotexist;dim=-1.
*Itdoesnotmatterifanintersectionexistsornot;dim=-1,0,1,or2.
0Anintersectionmustexistanditsmaximumdimensionmustbe0;dim=0.
1Anintersectionmustexistanditsmaximumdimensionmustbe1;dim=1.
2Anintersectionmustexistanditsmaximumdimensionmustbe2;dim=2.
Eachpredicatehasatleastonepatternmatrix,butsomerequiremorethanonetodescribetherelationshipsofvariousgeometrytypecombinations.
ThepatternmatrixoftheWithinpredicateforgeometrycombinationshasthefollowingform:
b
Interior
Boundary
Exterior
Interior
T
*
F
a
Boundary
*
*
F
Exterior
*
*
*
Simplyput,theWithinpredicatereturnstruewhentheinteriorsofbothgeometriesintersect,andtheinteriorandboundaryofadoesnotintersecttheexteriorofb. Allotherconditionsdonotmatter.
Equal
Equal returnst(TRUE)iftwogeometriesofthesametypehaveidenticalX,Ycoordinatevalues.
GeometriesareequaliftheyhavematchingX,Ycoordinates.
TheDE-9IMpatternmatrixforequalityensuresthattheinteriorsintersectandthatnopart
interiororboundaryofeithergeometryintersectstheexterioroftheother.
b
Interior
Boundary
Exterior
Interior
T
*
F
a
Boundary
*
*
F
Exterior
F
F
*
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Disjoint
Disjoint returnst(TRUE)iftheintersectionofthetwogeometriesisanemptyset.
Geometriesaredisjointiftheydonotintersectoneanotherinanyway.
Thedisjointpredicatespatternmatrixsimplystatesthatneithertheinteriorsnortheboundariesofeithergeometryintersect.
b
Interior
Boundary
Exterior
Interior
F
F
*
a
Boundary
F
F
*
Exterior
*
*
*
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Intersects
Intersects returnst(TRUE)iftheintersectiondoesnotresultinanemptyset.Intersectsreturnstheexactoppositeresultofdisjoint.
TheintersectspredicatewillreturnTRUEiftheconditionsofanyofthefollowingpatternmatricesreturnsTRUE.
TheintersectspredicatereturnsTRUEiftheinteriorsofbothgeometriesintersect.
b
Interior
Boundary
Exterior
Interior
T
*
*
a
Boundary
*
*
*
Exterior
*
*
*
TheintersectspredicatereturnsTRUEiftheboundaryofthefirstgeometryintersectstheboundaryofthesecondgeometry.
b
Interior
Boundary
Exterior
Interior
*
T
*
a
Boundary
*
*
*
Exterior
*
*
*
TheintersectspredicatereturnsTRUEiftheboundaryofthefirstgeometryintersectstheinteriorofthesecond.
b
Interior
Boundary
Exterior
Interior
*
*
*
a
Boundary
T
*
*
Exterior
*
*
*
TheintersectspredicatereturnsTRUEiftheboundariesofeithergeometryintersect.
b
Interior
Boundary
Exterior
Interior
*
*
*
a
Boundary
*
T
*
Exterior
*
*
*
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Touch
Touch returnst(TRUE)ifnoneofthepointscommontobothgeometriesintersecttheinteriorsofbothgeometries.Atleastonegeometrymustbealinestring,polygon,multilinestring,ormultipolygon.
TouchreturnsTRUEifeitherofthegeometries'boundariesintersectorifonlyoneofthegeometry'sinteriorsintersectstheother'sboundary.
ThepatternmatricesshowusthatthetouchpredicatereturnsTRUEwhentheinteriorsofthegeometrydon'tintersectandtheboundaryofeithergeometryintersectstheothersinteriororboundary.
ThetouchpredicatereturnsTRUEiftheboundaryofonegeometryintersectstheinterioroftheotherbuttheinteriorsdonotintersect.
b
Interior
Boundary
Exterior
Interior
F
T
*
a
Boundary
*
*
*
Exterior
*
*
*
ThetouchpredicatereturnsTRUEiftheboundaryofonegeometryintersectstheinterioroftheotherbuttheinteriorsdonotintersect.
b
Interior
Boundary
Exterior
Interior
F
*
*
a
Boundary
T
*
*
Exterior
*
*
*
ThetouchpredicatereturnsTRUEiftheboundariesofbothgeometriesintersectbuttheinteriorsdonot.
b
Interior
Boundary
Exterior
Interior
F
*
*
a
Boundary
*
T
*
Exterior
*
*
*
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Overlap
Overlap comparestwogeometriesofthesamedimensionandreturnst(TRUE)iftheirintersectionsetresultsinageometrydifferentfrombothbutofthesamedimension.
Overlapreturnst(TRUE)onlyforgeometriesofthesamedimensionandonlywhentheirintersectionsetresultsinageometryofthesamedimension.Inotherwords,iftheintersectionoftwopolygonsresultsinpolygon,thenoverlapreturnst(TRUE).
Thispatternmatrixappliestopolygon/polygon,multipoint/multipointandmultipolygon/multipolygonoverlays.ForthesecombinationstheoverlappredicatereturnsTRUEiftheinteriorofbothgeometriesintersectstheothersinteriorandexterior.
b
Interior
Boundary
Exterior
Interior
T
*
T
a
Boundary
*
*
*
Exterior
T
*
*
Thispatternmatrixappliestolinestring/linestringandmultilinestring/multilinestringoverlaps.Inthiscasetheintersectionofthegeometriesmustresultinageometrythathasadimensionof1(anotherlinestring).Ifthedimensionoftheintersectionoftheinteriorshadresultedin0(apoint)theoverlappredicatewouldreturnFALSE;however,thecrosspredicatewouldhavereturnedTRUE.
b
Interior
Boundary
Exterior
Interior
1
*
T
a
Boundary
*
*
*
Exterior
T
*
*
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Cross
Cross returnst(TRUE)iftheintersectionresultsinageometrywhosedimensionisonelessthanthemaximumdimensionofthetwosourcegeometriesandtheintersectionsetisinteriortobothsourcegeometries.Crossreturnst(TRUE)foronlymultipoint/polygon,multipoint/linestring,linestring/linestring,linestring/polygon,andlinestring/multipolygoncomparisons.
Crossreturnst(TRUE)ifthedimensionoftheintersectionisonelessthanthemaximumdimensionofthesourcegeometriesandtheinteriorsofbothgeometriesareintersected.
Thiscrosspredicatepatternmatrixappliest
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