数字地图制图原理-Spatial-concepts

WorboysandDuckham(2004)GIS:AComputingPerspective,SecondEdition,CRCPress,Chapter3,Fundamentalspatialconcepts,WorboysandDuckham(2004)GIS:AComputingPerspective,SecondEdition,CRCPress,Geometry:providesaformalrepresentationoftheabstractpropertiesandstructureswithinaspaceInvariance:agroupoftransformationsofspaceunderwhichpropositionsremaintrueDistance-translationsandrotationsAngleandparallelism-translationsrotations,andscalings,Geometryandinvariance,WorboysandDuckham(2004)GIS:AComputingPerspective,SecondEdition,CRCPress,3.1,Euclideanspace,WorboysandDuckham(2004)GIS:AComputingPerspective,SecondEdition,CRCPress,EuclideanSpace,EuclideanSpace:coordinatizedmodelofspaceTransformsspatialpropertiesintopropertiesoftuplesofrealnumbersCoordinateframeconsistsofafixed,distinguishedpoint(origin)andapairoforthogonallines(axes),intersectingintheoriginPointobjectsLineobjectsPolygonalobjects,WorboysandDuckham(2004)GIS:AComputingPerspective,SecondEdition,CRCPress,Scalar:Addition,subtraction,andmultiplication,e.g.,(x1,y1)(x2,y2)=(x1x2,y1y2)Norm:Distance:|ab|=|a-b|Anglebetweenvectors:,ApointintheCartesianplaneR2isassociatedwithauniquepairofrealnumbera=(x,y)measuringdistancefromtheorigininthexandydirections.Itissometimesconvenienttothinkofthepointaasavector.,Points,既有大小，又有方向，例如力、力矩位移、速度、加速度等；用一条有方向的线段，即有向线段来表示矢量；有向线段的长度表示矢量的大小，有向线段的方向表示矢量的方向；如果一个矢量无限长，则它将平面划分成两个“半平面”，位于矢量右侧的半平面称为顺时针半平面，位于矢量左侧的半平面称为逆时针半平面；矢量的值为始末点的坐标值之差。,vectors,(x1,y1)+(x2,y2)=(x1+x2,y1+y2)(x1,y1)(x2,y2)=(x1x2,y1y2)s(x1,y1)=(s*x1,s*y1),Innerproduct(內积),axb=|a|b|sinnWherenisaunitvectorperpendiculartobothaandb|axb|=|a|b|sinInR2,a=(x1,y1),b=(x2,y2)Signed|axb|=x1y2-x2y1在求矢量的叉积时，遵循右手法则，即从a到b呈逆时针方向时，值为正，否则为负。,Crossproduct(叉积),矩阵是一个按行列组织的规则数组矩阵用矢量来表示行列式，一个22的矩阵的行列式等于其两对角线上元素交叉乘积的差。假设矢量a=(x1,y1),b=(x2,y2)，矩阵C的行列式记为|C|22的矩阵的行列式的绝对值等于由其两列矢量形成的平行四边形的面积,矩阵,EquationsPoint-vector,WorboysandDuckham(2004)GIS:AComputingPerspective,SecondEdition,CRCPress,Severalanalyticalformsoflines,直线方程的一般式,直线可以定义为R2空间上满足关系式AxByC0的所有坐标点的集合，记为：,连接点p(x1,y2)到点q(x2,y2)的直线参数用矢量法表示为：,连接点q(x2,y2)到p(x1,y1)点的直线参数用矢量法表示为：,参数的确定：？,过两点公式推导,参数的意义：,参数A，B，C都是标量，它们确定了直线的位置，直线的斜率m定义为（A/B），它表示x值的变化所引起的y值的变化。如果两条直线的斜率相同，则这两条直线相互平行。Y轴上的截距h定义为（C/B），它表示x0时，y的值。X轴上的截距定义为（C/A），它表示y0时，x的值（直线与X轴交点的x值）；（A2B2）1/2是连接p和q的矢量的模。参数C表示的是由p和q形成的平行四边形的面积，如果p与q方向相同或相反，则C为0。,直线L的标准化，是参量除以矢量的模：,直线方程标准化,WorboysandDuckham(2004)GIS:AComputingPerspective,SecondEdition,CRCPress,Thelineincidentwithaandbisdefinedasthepointseta+(1)b|RThelinesegmentbetweenaandbisdefinedasthepointseta+(1)b|0,1Thehalflineradiatingfrombandpassingthroughaisdefinedasthepointseta+(1)b|0,直线方程的点-矢表示法,计算机中的线大多是以离散的矢量形式表示，方程形式往往用于一些几何计算中。直线方程表示法和点-矢表示法各有优势对于某些计算，直线的方程表示比较有优势对于另一些计算，点-矢表示比较有优势思考：直线的不同表示方法如何用来表达空间数据？试讨论不同表示方法在表达空间数据时的优劣,WorboysandDuckham(2004)GIS:AComputingPerspective,SecondEdition,CRCPress,WorboysandDuckham(2004)GIS:AComputingPerspective,SecondEdition,CRCPress,Polygonalobjects,ApolylineinR2isafinitesetoflinesegments(callededges)suchthateachedgesend-pointissharedbyexactlytwoedges,exceptpossiblyfortwopoints,calledtheextremesofthepolyline.Ifnotwoedgesintersectatanyplaceotherthanpossiblyattheirend-points,thepolylineissimple.Apolylineisclosedifithasnoextremepoints.A(simple)polygoninR2istheareaenclosedbyasimpleclosedpolyline.Thispolylineformstheboundaryofthepolygon.Eachend-pointofanedgeofthepolylineiscalledavertexofthepolygon.AconvexpolygonhaseverypointintervisibleAstar-shapedorsemi-convexpolygonhasatleastonepointthatisintervisible,WorboysandDuckham(2004)GIS:AComputingPerspective,SecondEdition,CRCPress,Polygonalobjects,WorboysandDuckham(2004)GIS:AComputingPerspective,SecondEdition,CRCPress,PolygonalObjects,Monotonechain:thereissomelineintheEuclideanplanesuchthattheprojectionoftheverticesontothelinepreservestheorderingofthelistofpointsinthechainMonotonepolygon:iftheboundarycanbesplitintotwopolylines,suchthatthechainofverticesofeachpolylineisamonotonechainTriangulation:partitioningofthepolygonintotrianglesthatintersectonlyattheirmutualboundaries,WorboysandDuckham(2004)GIS:AComputingPerspective,SecondEdition,CRCPress,Polygonobjects,monotonepolygon,WorboysandDuckham(2004)GIS:AComputingPerspective,SecondEdition,CRCPress,Transformations,TransformationspreserveparticularpropertiesofembeddedobjectsEuclideanTransformationSimilaritytransformationsAffinetransformationsProjectivetransformationsTopologicaltransformation,AnEuclideantransformationiseitheratranslation,arotation,orareflectionTranslation:throughrealconstantsaandb(x,y)-(x+a,y+b)Rotation:throughangleaboutorigin(x,y)-(xcos-ysin,xsin+ycos)Reflection:inlinethroughoriginatangletox-axis(x,y)-(xcos2+ysin2,xsin2-ycos2)角度、距离和连通性均不变,WorboysandDuckham(2004)GIS:AComputingPerspective,SecondEdition,CRCPress,Similaritytransformation.包括了等比缩放(Scaling)，因此，距离不再固定不变。但角度保持不变，所以方向不变。Affinetransformation.包括缩放和错切(shearing)变换，距离与角度都不再保持不变，但是平行关系与线性状态保持不变,WorboysandDuckham(2004)GIS:AComputingPerspective,SecondEdition,CRCPress,Projectivetransformation.距离、角度甚至平行关系都不再保持不变，因此，目标的面积与形状都可能改变。,WorboysandDuckham(2004)GIS:AComputingPerspective,SecondEdition,CRCPress,Topologicaltransformation.橡皮变换，距离、角度、平行关系均不保持，保持几何目标间的拓扑性质如连通性，邻接关系等。,WorboysandDuckham(2004)GIS:AComputingPerspective,SecondEdition,CRCPress,思考,空间关系绝对关系，相对关系的差别相对关系在不同变换下的性质,WorboysandDuckham(2004)GIS:AComputingPerspective,SecondEdition,CRCPress,二维基本几何变换平移、旋转、缩放、错切,(x,y)为变换前点坐标(x,y)为变换后坐标T为变换矩阵，且变换矩阵中a,b,c,d可取不同的值，从而实现不同的变换，以达到对图形进行变换的目的。,平移变换用于移动坐标系的原点变换前后的坐标必须满足因子Tx为正，原点往左移，Tx为负，则往右移。当因子Ty为正，原点往下移，Ty为负，则往上移。,平移变换与齐次坐标,把2x2矩阵扩充为3x2矩阵？把点矢量也作扩充，将x,y扩充为x,y,1，即把点集矩阵扩充为nx3阶矩阵。令变换矩阵中的b,c=0，a,d=1，就得到平移变换,齐次坐标,齐次坐标是笛卡尔坐标的扩展，它将n维坐标变为n+1维坐标：二维点（x,y）的齐次坐标是（hx,hy,h），h是任意非零标量,（x,y）的齐次转换是一个经过原点和点（x,y,1）的射线轨迹用三维矢量表示二维矢量，将（x,y,1）看作Z=1平面上的点。经此扩充后，图形落在Z=1的平面上。它对几何图形的性质没有影响,旋转变换,指图形的放置围绕原点旋转角，且逆时针为正，顺时针为负。旋转变换矩阵为：对点进行旋转变换：,缩放变换,平移和旋转变换都保持二维空间上目标变换时的距离及大小不变，但缩放变换会改变坐标系的单位长度（距离）当Sx和Sy小于1时，几何目标即被缩小；当它们大于1时，几何目标被放大；当SxSy时，缩放后将保持原有的方向和大小（等比缩放）。如果SxSy，几何目标将在某一方向上变形或拉伸,错切变换,错切变换矩阵，shx，shy之一为0：沿X向错切：令shy0沿Y向错切：令shx0此处的错切公式是对第I象限点而言，其余象限的点的错切方向应作相应改变。,x=x+shx*yy=shy*x+y,二维组合变换,由多种基本变换组合而成的变换称之为组合变换，相应的变换矩阵称为组合变换矩阵点p先平移后缩放点p先缩放后平移点的转换可用于操作地图目标。每个转换从原始点(x,y)产生新的点(x,y),Applicationofthetransformations,PlacementandmanipulationofsymbolsondigitalmapsMappinggeographicfeaturesonscreen(windows)Transformationbetweenlogicalcoordinatesystems(e.g.geographiccoordinatesystem)andthecoordinatesystemofadisplaydevice,WorboysandDuckham(2004)GIS:AComputingPerspective,SecondEdition,CRCPress,-90,90,-180,180,X,Y,WorboysandDuckham(2004)GIS:AComputingPerspective,SecondEdition,CRCPress,3.2,Set-basedgeometryofspace,WorboysandDuckham(2004)GIS:AComputingPerspective,SecondEdition,CRCPress,Sets,Thesetbasedmodelinvolves:Theconstituentobjectstobemodeled,calledelementsormembersCollectionofelements,calledsetsTherelationshipbetweentheelementsandthesetstowhichtheybelong,termedmembershipWewritesStoindicatethatanelementsisamemberofthesetS,WorboysandDuckham(2004)GIS:AComputingPerspective,SecondEdition,CRCPress,Sets,Alargenumberofmodelingtoolsareconstructed:EqualitySubset:STPowerset:thesetofallsubsetsofaset,P(S)Emptyset;Cardinality:thenumberofmembersinaset#SIntersection:STUnion:STDifference:STComplement:elementsthatarenotintheset,S,WorboysandDuckham(2004)GIS:AComputingPerspective,SecondEdition,CRCPress,Distinguishedsets,WorboysandDuckham(2004)GIS:AComputingPerspective,SecondEdition,CRCPress,Relations,Product:returnsthesetoforderedpairs,whosefirstelementisamemberofthefirstsetandsecondelementisamemberofthesecondsetBinaryrelation:asubsetoftheproductoftwosets,whoseorderedpairsshowtherelationshipsbetweenmembersofthefirstsetandmembersofthesecondsetReflexiverelations:whereeveryelementofthesetisrelatedtoitselfSymmetricrelations:whereifxisrelatedtoythenyisrelatedtoxTransitiverelations:whereifxisrelatedtoyandyisrelatedtozthenxisrelatedtozEquivalencerelation:abinaryrelationthatisreflexive,symmetricandtransitive,WorboysandDuckham(2004)GIS:AComputingPerspective,SecondEdition,CRCPress,Functions,Function:atypeofrelationwhichhasthepropertythateachmemberofthefirstsetrelatestoexactlyonememberofthesecondsetf:S-T,WorboysandDuckham(2004)GIS:AComputingPerspective,SecondEdition,CRCPress,Functions,Injection:anytwodifferentpointsinthedomainaretransformedtotwodistinctpointsinthecodomainImage:thesetofallpossibleoutputsSurjection:whentheimageequalsthecodomainBijection:afunctionthatisbothasurjectionandaninjection,Domain,codomainandimage,WhatcangointoafunctioniscalledtheDomain(人为指定)WhatmaypossiblycomeoutofafunctioniscalledtheCodomainWhatactuallycomesoutofafunctioniscalledtheImage(Range)Q:Isthefunctiony=x2injective?,WorboysandDuckham(2004)GIS:AComputingPerspective,SecondEdition,CRCPress,TheDomainisanessentialpartofthefunction.ChangetheDomainandwehaveadifferentfunction.,Inversefunctions,InjectivefunctionshaveinversefunctionsProjectionGivenapointintheplanethatispartoftheimageofthetransformation,itispossibletoreconstructthepointonthespheroidfromwhichitcameExample:AnewfunctionwhosedomainistheimageoftheUTMmapstheimagebacktothespheroid,Whatisthedomain,codomainandimageinthisexample?,WorboysandDuckham(2004)GIS:AComputingPerspective,SecondEdition,CRCPress,f:(lon,lat)-(x,y),f-1:(x,y)-(lon,lat),WorboysandDuckham(2004)GIS:AComputingPerspective,SecondEdition,CRCPress,Convexityapropertyofset-basedgeometry,AsetisconvexifeverypointisvisiblefromeveryotherpointwithinthesetLetSbeasetofpointsintheEuclideanplaneVisible:PointxinSisvisiblefrompointyinSifeitherx=yor;itispossibletodrawastraight-linesegmentbetweenxandythatconsistsentirelyofpointsofS直线的广义理解：两点之间距离最短的线，取决于空间特性GreatcirclearcsinasphereGeodesics测地线,WorboysandDuckham(2004)GIS:AComputingPerspective,SecondEdition,CRCPress,Convexity,Observationpoint:ThepointxinSisanobservationpointforSifeverypointofSisvisiblefromxSemi-convex:ThesetSissemi-convex(star-shapedifSisapolygonalregion)ifthereissomeobservationpointforSConvex:ThesetSisconvexifeverypointofSisanobservationpointforS,WorboysandDuckham(2004)GIS:AComputingPerspective,SecondEdition,CRCPress,Convexity,Visibilitybetweenpointsx,y,andz,WorboysandDuckham(2004)GIS:AComputingPerspective,SecondEdition,CRCPress,3.3,TopologyofSpace,拓扑空间,拓扑空间（topologicalspace）是欧几里得空间的一种推广，具有拓扑结构的集合。如果对一个非空集合X给予适当的结构，使之能引入微积分中的极限和连续的概念，这样的结构就称为拓扑结构，具有拓扑结构的空间称为拓扑空间。给定任意一个集合X，对它的每一点赋予一种确定的邻近结构便成为一个拓扑空间构造邻近结构有多种方法，如邻域系、开集系、闭集系、闭包系、内部系等不同方法。,拓扑学要研究的是世界中存在多少种不同的曲面。一个基本的假设是所有的曲面都是由理想的弹性膜做成，可以随意延伸和收缩，但不允许折叠和撕裂。这种延伸和收缩就是拓扑变换拓扑学研究拓扑变换下不变的性质,拓扑学基本观念,WorboysandDuckham(2004)GIS:AComputingPerspective,SecondEdition,CRCPress,Topology,Topology:“studyofform”;concernspropertiesthatareinvariantundertopologicaltransformationsIntuitively,topologicaltransformationsarerubbersheettransformations,拓扑学主要分为点集拓扑和代数拓扑两大分支。点集拓扑集中讨论由点组成的集合的拓扑性质，如邻域、邻接、开集和闭集等。一些重要的拓扑关系，如内部、边界、外部等均可在点集拓扑中定义。代数拓扑的一些理论，如单纯复形(SimplicialComplex)可以应用到空间数据模型中。虽然在今天的空间数据库中还难以见到，但在一些研究工作中已经把代数拓扑运用到空间系统的概念模型中。,拓扑学的分类,点集拓扑基本概念,邻域(Neighborhood)开集(Openset)闭包(Closure)闭集(Closedset)内部(Interior)、边界(Boundary)、外部(Exterior)关联(Incidence)与邻接(Adjacency),邻域（Neighborhood）,定义：设(X,d)为度量空间，xX,0且R，则称X的子集B(x,)=yX|d(y,x)0ifsandtaredistinctpointsandd(s,t)=0ifsandtareidenticalForeachpairs,tinS,thedistancefromstotisequaltothedistancefromttos,d(s,t)=d(t,s)Foreachtripes,t,uinS,thesumofthedistancesfromstotandfromttouisalwaysatleastaslargeasthedistancefromstou,度量空间(MetricSpace)，在数学中是指一个集合X，并且该集合中的任意元素之间的距离是可定义的。设X为一个集合，一个映射d：XXR。若对于任何x,y,z属于X，有（I）（正定性）d(x,y)0,且d(x,y)=0当且仅当x=y；（II）（对称性）d(x,y)=d(y,x)；（III）（三角不等式）d(x,z)d(x,y)+d(y,z）则称d为集合X的一个度量（或距离）。称偶对（X,d）为一个度量空间，或者称X为一个对于度量d而言的度量空间。,度量空间,WorboysandDuckham(2004)GIS:AComputingPerspective,SecondEdition,CRCPress,Distancesdefinedontheglobe,Metricspace,Metricspace,Metricspace,Quasimetric,欧氏空间(EuclideanSpace),设d为定义在集合Rn上的距离函数，