International Journal of Geo-Information
Article 3D Cadastral Data Model Based on Conformal Geometry Algebra
Ji-yi Zhang 1, Peng-cheng Yin 2, Gang Li 1,*, He-he Gu 1, Hua Zhao 1 and Jian-chun Fu 1
1 School of Environmental Science and Spatial Informatics, China University of Mining and Technology, Xuzhou 221008, China; [email protected] (J.-Y.Z.); [email protected] (H.-H.G.); [email protected] (H.Z.); [email protected] (J.-C.F.) 2 Bureau of Land and Resources of Xuzhou, Xuzhou 221006, China; [email protected] * Correspondence: [email protected]; Tel.: +86-136-0520-5938; Fax: +86-516-8359-1333
Academic Editors: François Anton and Wolfgang Kainz Received: 3 October 2015; Accepted: 5 February 2016; Published: 19 February 2016
Abstract: Three-dimensional (3D) cadastral data models that are based on Euclidean geometry (EG) are incapable of providing a unified representation of geometry and topological relations for 3D spatial units in a cadastral database. This lack of unification causes problems such as complex expression structure and inefficiency in the updating of 3D cadastral objects. The inability of current cadastral data models to express cadastral objects in a unified manner can be attributed to the different expressions of dimensional objects. Because the hierarchical Grassmann structure corresponds to the hierarchical structure of dimensions in conformal geometric algebra (CGA), geometric objects in different dimensions can be constructed by outer products in a unified expression form, which enables the direct extension of two-dimensional (2D) spatial representations to 3D spatial representations. The multivector structure in CGA can be employed to organize and store different dimensional objects in a multidimensional and unified manner. With the advantages of CGA in multidimensional expressions, a new 3D cadastral data model that is based on CGA is proposed in this paper. The geometries and topological relations of 3D spatial units can be represented in a unified form within the multivector structure. Detailed methods for 3D cadastral data model design based on CGA and data organization in CGA are introduced. The new cadastral data model is tested and analyzed with experimental data. The results indicate that the geometry and topological relations of 3D cadastral objects can be represented in a multidimensional manner with an intuitive topological structure and a unified dimensional expression.
Keywords: data model; CGA; 3D cadastre; outer production; topological relation
1. Introduction Due to an increasing demand for land use in urban areas, the future direction of urban development involves extending the current land use to higher spaces and the underground. With increasingly complex three-dimensional (3D) structures, the traditional two-dimensional (2D) cadastre is becoming more limited for handling complex 3D cadastral objects in urban areas. A 3D cadastre is needed to address these limitations. Several countries and regions have begun to register and manage complex 3D cadastral objects via urban cadastral management [1–6]. Relative theories and technologies, including organization, storage and query of 3D spatial units, have recently gained attention. A 3D cadastral data model focuses on methods to organize, store and represent spatial representations for 3D cadastral objects in a cadastral database. Because land ownership boundaries and rights should be accurately defined in a cadastral registration, a large percentage of cadastral
ISPRS Int. J. Geo-Inf. 2016, 5, 20; doi:10.3390/ijgi5020020 www.mdpi.com/journal/ijgi ISPRS Int. J. Geo-Inf. 2016, 5, 20 2 of 16 data models represent the structures of 3D spatial units by describing topological relations among composed elements [7–9]. The majority of existing data representation models of traditional 3D cadastres are topological-based [7]. Several topological-based 3D cadastral data models, which are based on Euclidean Geometry (EG), have been developed for 3D cadastral management [7–13]. Because EG space lacks a unified form in dimensional expansion, the extension from a classical 2D cadastre to a 3D cadastre that is based on EG is not direct or simple. Representations of 3D spatial units are more complicated than representations of 2D spatial units in EG space [14], which is one of the main obstacles in 3D cadastre implementation. EG-based cadastral data models organize, store and represent geometry according to their topological relations, which are contained in 3D cadastral objects in a cadastral database. The topological structures and relations are logged and maintained during the entire procedure of data storage, relationship representation, operation, division of rights, and validation [13]. Universally accepted 3D cadastral data models have not been constructed for efficient complex spatial representations. How to store and represent spatial representations efficiently for 3D cadastral objects in a database remains a challenge in 3D cadastre development [15]. One of the main issues caused by the division of multidimensionality in EG space is the separation between the representation of geometry and the representation of algebra. Geometric elements with higher dimensions in EG space cannot be expressed directly and algebraically by their geometric components, which have lower dimensions. A separation exists between the expression of geometry and the expression of algebra [16], which produces the diversity of distinct expressions and computational structures in objects of different dimensions. EG-based 3D data models focus on the description of topological relations among different dimensional construction elements of 3D objects [7]. These methods for spatial representations are not intuitive. They cannot handle a complex hierarchical structure, especially the dimensional hierarchical structure in a 3D cadastre as an entity. Topological relations among different dimensions, such as points, boundary lines, boundary faces and spatial units, are commonly employed to indirectly realize the spatial representations of 3D objects [17–20]. For instance, lines and surfaces are expressed by the descriptions of construction points and lines, respectively, whereas a 3D spatial unit is represented by recording of relative surfaces. The principle of EG-based data models is a 3D extension of the topological relations among points, lines and planes in 2D space. In these data models, the geometries only comprise symbols that are logged or described in the database instead of the real computational primitives that are represented algebraically. The topological relations in EG-based data models resemble the connection structures between these symbolized geometries [21–23]. Due to the complexity of 3D objects and the limitations of EG in complex 3D object representation, EG-based data models cannot realize the integration of geometry and topology as an entity. Although several studies have attempted to integrate geometry and topology [24–26], none have achieved this goal due to the limitation of EG. New mathematical theories are needed to integrate geometric and algebraic representations as an entity. In this paper, conformal geometric algebra (CGA) is introduced as the theoretical foundation of 3D cadastral spatial representations to solve problems in cadastral development from a 2D structure to a 3D structure. As a branch of Clifford algebra [27,28], CGA can integrate the geometric and algebraic representations of spatial objects. CGA can also solve geometric problems directly in an algebraic way [29]. Compared with EG, different dimensional elements in CGA can be expressed in a unified form by outer products consistent with the Grassmann structure, as shown in Figure1, which indicates that the geometric structure of spatial units is consistent with the Grassmann dimensional structure. Several achievements that introduce CGA into geographic information systems (GISs) research are as follows: a multi-dimensional integrated GIS model [23,30], a spatial analysis computation [31–33], and an expression of topological relations and operations integration [34,35]. Yuan et al (2014) suggest that CGA has the potential to become the fundamental theory and technology for constructing a multidimensional unified GIS [35]. Therefore, it has high potential to serve as a solid foundation for 3D cadastral data modeling. ISPRS Int. J. Geo-Inf. 2016, 5, 20 3 of 16 ISPRS Int. J. Geo-Inf. 2016, 5, 20 3 of 16
Grade 2
Grade 3 e Grade 3
Grade 4 e Grade 4
HyperSphere Grade 5 e
e Infinite Point Point Line Cirle Plane Sphere Point Pair FigureFigure 1. Grassmann 1. Grassmann structure structure that that contains contains dimensional dimensional elements elements,, where where ^ denotes ˆ denotes an anouter outer product. product.
ThisThis paper paper is isorganized organized as as follows: follows: In In Section Section 2,2 ,we we review review the the fundamental fundamental idea of constructing a a 3D cadastral datadata modelmodel with with CGA. CGA. The The construction construction and and development development of theof 3Dthe cadastral3D cadastral data data model modelare presented are presented in Section in Section3. Both 3. Both the theoretical the theoretical and theand technologicalthe technological implementations implementations of the of datathe datamodel model are discussed.are discussed. In Section In Section4, we test4, we the test CGA-based the CGA 3D-based cadastral 3D cadastral data model data using model real-world using realcase-world studies. case The studies. characteristics, The characteristics, improvements improvements and future and directions future directions of the 3D cadastralof the 3D datacadastral model dataare model discussed are discussed in Section 5in. WeSection state 5. our We conclusions state our conclusions in Section6 in. Section 6.
2. 2.Basic Basic Idea Idea TheThe unification unification of ofgeometric geometric and and topological topological representations representations is isone one of ofthe the objectives objectives in incadastral cadastral datadata modeling. modeling. The The key key to integrating to integrating geometry geometry and andtopology topology is the is integrated the integrated representatio representationn of the of geometricthe geometric structure structure and the and dimensional the dimensional structure structure of geometri of geometricc elements elements of different of different dimensions dimensions.. ThisThis representation representation is not is not possible possible in EG in EGbecause because traditional traditional Euclidean Euclidean space space lacks lacksconsistency consistency in the in representationthe representation of geometries of geometries of different of different dimen dimensionssions [14] [.14 To]. To achieve achieve synchronization synchronization between between geometrygeometry and and topology topology at atdifferent different dimensions, dimensions, a distinct a distinct and and unified unified mathematical mathematical representation representation andand data data organization organization structure structure are are necessary. necessary. CGACGA is is one one of the the most most powerful powerful geometric geometric algebra algebra (GA) (GA) systems. systems. Compared Compared with other with geometric other geometricsystems, systems, such as Euclidean such as Euclidean geometry andgeometry projective and geometry, projective CGA geometry, has the CGA unique has advantages the unique of advantagessimple and of intuitive simple and structure, intuitive distinct structure geometric, distinct meaning geometric and meaning unified characteristicsand unified characteristics with regard to withthe expressionregard to theof the expression geometry of [35 – the38 ]. geometry Because dimensional[35–38]. Because and topological dimensional structures and topological in CGA are structuresconsistent in with CGA the are Grassmann consistent dimensional with the Grassmann structure, dimensional different dimensional structure elements, different can dimensional be expressed elementsdirectly can via outerbe expressed products directly with conformal via outer points. product Thes with structure conformal of the Grassmannpoints. The hierarchy structure is of similar the Grassmannto the topological hierarchy structures is similar of differentto the topological dimensional structures objects [of39 ],different which enables dimensional the construction objects [39 of], a whichunified enable representations the construction for different of a dimensionalunified representation geometries for in CGAdifferent [40]. dimensional The unified representationgeometries in of CGAthe geometric [40]. The constructionunified representation is purely algebraic, of the whichgeometric indicates construction that geometric is purely objects algebraic, can be expressed which indicatesas an algebraic that geometric entity [ 41objects]. With can the be expression expressed ofas thean structurealgebraic ofentity the geometric[41]. With objectsthe expression in CGA, of the thetopological structure of relations the geometri are implicitlyc objects contained in CGA, the intheir topological algebraic relations expression are implicitly [42]. In this contained manner, in the theirunified algebraic representation expression of geometries[42]. In this and manner, topological the relations unified can representation be achieved. of geometries and topologicalFundamental relations can primitives be achieved. (e.g., points, point pairs, lines, circles, planes and spheres) in CGA canFundamental be represented primitives consistently (e.g., point usings, thepoint outer pairs, product lines, circles, with the planes integration and spheres of geometries) in CGA can and betopological represented relations. consistently However, using the the representation outer product of arbitrary with the geometries integration may of not geometries fit a hierarchal and topologicalGrassmann relations structure.. However, To ensure the representation compatibility withof arbitrary an existing geometries GIS data may model, not fit a a 3Dhierarchal cadastral Grassmann structure. To ensure compatibility with an existing GIS data model, a 3D cadastral data model with a complex-simplex structure is formed. For example, complex geometries should be
ISPRS Int. J. Geo-Inf. 2016, 5, 20 4 of 16 ISPRS Int. J. Geo-Inf. 2016, 5, 20 4 of 16 composeddata model of with elementary a complex-simplex geometries. structure In this is combination, formed. For example, topological complex structures geometries should should also be encodedcomposed and of elementary represented geometries. in the expression In this combination, of geometric topological structures. structures Although should CGA also includes be encoded six primitives,and represented only in points, the expression lines and of planes geometric are employed structures. in Although the spatial CGA representations includes six primitives, to fit the characteristicsonly points, lines of and accurate planes cadastral are employed boundary in the representation. spatial representations Because to the fit elementalthe characteristics geometri ofc objectsaccurate in cadastralCGA have boundary different representation.dimensions, a un Becauseified representation the elemental structure geometric that objects represents in CGA different have dimensionaldifferent dimensions, objects is apreferable. unified representation structure that represents different dimensional objects is preferable.The multivector structure, which is the fundamental algebraic structure in CGA, combines differentThe dimensional multivector structure,objects. In whicha multivector is the fundamentalstructure, different algebraic dimen structuresional geometric in CGA, combineselements thatdifferent comprise dimensional the complete objects. object In a are multivector represented structure, as blade different subspaces dimensional of the CGA geometric according elements to the gradesthat comprise (i.e., dimensions) the complete of object the objects. are represented The structures as blade of subspaces the objects of are the CGAcoded according in the weight, to the amplitudegrades (i.e. , dimensions)and orientation of the of objects. the com Theponents structures of of the the objectsmultivectors are coded [33]. in the In weight,this manner, amplitude the representationand orientation of of geometries the components in a multivector of the multivectors structure [can33]. be In expressed this manner, algebraic the representationally. With well of- definedgeometries data in structures a multivector (e.g., structure the hierarchal can be expressed structure algebraically.according to Withthe dimensions well-defined [3 data5]), consistent structures (e.g.,topological the hierarchal relations structure can be maintained according even to the if dimensionscomplex geometries [35]), consistent are represented. topological For relationsexample, canwe canbe maintained represent complex even if complex 3D cadastr geometriesal spatial are units represented. according Forto the example, combination we can of represent points, lines complex and planes.3D cadastral In a spatial multivector units according structure, to the the top combinationological relations of points, among lines these and planes. different In adimensional multivector elementsstructure, can the be topological formulated relations using the among coefficients these differentof the blades dimensional in the multivectors, elements canand be the formulated geometric constructionusing the coefficients relations can of the be bladescompletely in the investigated multivectors, using and the the structures geometric of constructionthe multidimensional relations ocanrganization be completely structures. investigated The multivector using the structure structures provide of the multidimensionals a unified and ideal organization structure for structures. storing Thedifferent multivector dimensional structure objects provides by integrating a unified geometric and ideal structureand topological for storing structures different in dimensionalan algebraic mannerobjects by. integrating geometric and topological structures in an algebraic manner. We developed a multidimensional-unifiedmultidimensional-unified 3D cadastralcadastral data model using CGA. The premise of the construction of the data model is illustrated in Figure 2 2.. InIn thethe firstfirst phase,phase, 3D3D cadastralcadastral objectsobjects areare decomposed into geometric and topological components from which we can obtain simplex elements that are composedcomposed of complex objects. Conformal points can be obtained from original Euclidean point sets viavia expressionexpression inin CGACGA in in the the second second phase. phase. These These simplex simplex elements, elements, which which are are employed employed to toconstruct construct 3D 3D cadastral cadastral spatial spatial units, units, can can be consistentlybe consistently represented represented by conformalby conformal point point sets sets using using the theouter outer product product operation operation in CGA in inCGA the thirdin the phase. third Complex phase. Complex 3D cadastral 3D cadastral spatial units spatial are expressed units are viaexpressed the multivector via the m structure,ultivector in structure, which different in which dimensional different dimension constructional construction simplexes are simplexes integrated are to integratedrealize a unified to realize expression a unified for exp geometricression for information geometric and information topological and relations. topological relations.
3D Cadastral Objects Geometry and Topology Unified Representation in 3D Cadastral Object CGA (MultiVector)
CGA Geometric Primitives Geometry Topology Representation Division Relations (GeoPrim)
Geometric embedding Geometric Primitives CGA Point Sets and Outer Product
Figure 2.2. Processing flow flow of the CGA (conformal(conformal geometricgeometric algebra)algebra) - -basedbased 3D (three-dimensional)(three-dimensional) cadastral data model model..
ISPRS Int. J. Geo-Inf. 2016, 5, 20 5 of 16
3. Methods
3.1. Geometrically and Topologically Unified Representation in CGA Using nonlinear spherical projection, the Euclidean space is embedded in a high-dimensional space, which is subject to a homogeneous projection transformation to obtain the conformal model of 4,1 Euclidean geometry [33,43], where e1, e2, e3, e0, e8 are the five vectors that support the space R and the x, y, z coordinates of the Euclidean space are the coefficients of the e1, e2, e3 vectors, respectively. In this paper, e0 denotes the basic coordinate origin, whose coefficient is normally kept constant in CGA, and e8 is the infinity point, whose coefficients can be calculated by Euclidean coordinates. The outer product of two arbitrary vectors in CGA is defined as follows: Definition 1: If a and b are two vectors in the conformal space R4,1, the outer product is defined as 1 a ^ b “ pab ´ baq 2 where both ab and ba are the geometric products of vectors a and b [41], which produce scalars and bivectors. This definition indicates that the dimensions of geometric objects can be encoded based on the outer product operation. The geometric objects of different dimensions in the conformal space are constructed by the outer product, which can realize the uniform expression form for the different dimensional geometric objects. By introducing the two additional dimensions of the origin and the infinity point, the geometric object dimensional structure completely corresponds to the Grassmann structure in the conformal space [30]. The expression ability of geometric entities in CGA has been significantly expanded. These features enable the geometric objects of different dimensions in CGA to be directly constructed via the outer product of conformal point sets, which is shown in Table1. For example, the outer product of two conformal points, which comprises a one-dimensional (1D) subspace in CGA, can directly express 2D subspace point pairs. Conformal lines, which comprise a 3D subspace, can be represented by a 2D subspace outer product with the infinity point e8, whereas a plane is expressed by the outer product of three conformal points and the infinity point e8. Table1 lists the basic geometric objects, which are expressed by the outer product of the conformal points in CGA. As shown in the table, different dimensional geometric objects follow a uniform manner of expression in CGA. The line and circle and the plane and sphere in CGA achieve a unified expression form and geometric meaning, respectively [34,44]. A multivector is a basic mathematical element that can simultaneously integrate multiple dimensional objects in CGA. The multidimensional unified expression of a common geometric object by a multivector in conformal space is defined as follows: 4,1 Definition 2: If Pi are points in conformal space R , then the multivector A, which is composed of points, lines and planes in Cl4,1, is expressed as n n n A “ tPiu ` Pi ^ Pj ^ e8 ` Pi ^ Pj ^ Pk ^ e8 i“1 ÿ i“ÿ1,j“1 ( i“1,jÿ“1,k“1 ( i‰j i‰j‰k where n is the number of points and {} represents the sets of geometric elements with the same dimensions. The multivector definition indicates that the geometric elements in different dimensions are connected by addition. According to Definition 2, the integration and unification of different geometric dimensions can be realized via the multivector structure in CGA. Simplex objects, such as points, lines and surfaces, can be integrated within a multivector to express complex 3D geometric objects. Elements of different dimensions are independent in the multivector structure. With the multivector structure, multidimensional fusion expression for 3D complex geometric objects can be realized and topological ISPRSISPRSISPRS Int. Int.Int. J. Geo-Inf.J.J. GeoGeo--IInf.2016 201, 65,, 5 20,, 20 6 of6 16of 16 ISPRSISPRSISPRS Int. Int.Int. J. Geo J.J. GeoGeo-Inf.--IInf. nf.201 2012016, 656,,, 525,0, 2 200 6 of66 of16of 16 16 ISPRS Int. J. Geo-Inf. 2016, 5, 20 6 of 16 relations among different dimensional construction elements are distinctly expressed by the relationsrelationsrelations among among among different differentdifferent dimension dimension dimensionalal al construction construction construction elements elements elements are are are distinctly distinctly distinctly express expressexpressededed by byby the the the relationstopologicarelationstopologica among amongll hierarchy different different that dimensional is constructed dimension constructional from construction multidimensional elements elements are distinctly components. are expressed distinctly byexpress the topologicaled by the topologicatopologicatopological hierarchyll hierarchyhierarchy that thatthat is isisconstructed constructedconstructed from fromfrom multidimensional multidimensionalmultidimensional components. components.components. hierarchytopologica thatl hierarchy is constructed that is from constructed multidimensional from multidimensional components. components. Table 1. Outer product expression of geometricic objects inin CGA TableTableTable 1. 1.Outer1. OuterOuter product productproduct expression expressionexpression of of ofgeometr geometrgeometric icobjectsic objectsobjects in in inCGA CGACGA TableTable 1. Outer1. Outer product product expression expression of geometricof geometr objectsic objects in CGA.in CGA Object Drawing Outer production Grade ObjectObjectObject DrawingDrawingDrawing OuterOuterOuter production productionproduction GradeGradeGrade Object Drawing Outer production Grade PointObject Drawing Outer ProductionNull Grade 1 Point Null 1 PointPointPoint NullNullNull 1 11 PointPoint NullNull 1 1 Point pair 푃푃 = 푃1^푃2 2 Point pair 푃푃 = ^푃1^푃2 2 PointPointPoint pair pair pair PP푃푃푃푃“=P=푃11^푃11푃P^2 푃22 2 2 2 Point pair 푃푃 = 푃1^푃2 2 Line “퐿 =^푃1^푃^2^푒∞ 3 LineLine L 퐿P=1 푃P1^2 푃2e^8푒∞ 3 3 LineLineLine 퐿 =퐿퐿 ==푃1푃^푃11푃^^2푃^푃22푒^^∞푒푒 ∞∞ 3 33 Line 퐿 = 푃1^푃2^푒∞ 3
Circle 퐶 =^푃 ^푃^ ^푃 3 CircleCircle C “퐶P=1 푃P1^2푃2P^3푃3 3 3 CircleCircleCircle 퐶 퐶=퐶 ==푃 푃^푃1푃^^푃^푃2푃^^푃 푃3 3 33 Circle 퐶 =1푃1^2푃2^3푃3 3 1 2 3
Plane 푃 = 푃 ^푃 ^푃 ^푒 4 Plane 푃 =^푃1^^푃2^푃^3^푒∞ 4 PlanePlanePlanePlane P 푃“푃=푃P1==푃 푃^푃P1푃2^^푃^푃2P푃^^3푃^푃3푒^e^8푒푒 ∞∞ 4 4 44 Plane 푃 =1푃1^2푃2^3푃3^∞푒 4 1 2 3 ∞
Sphere 푆 = 푃1^푃2^푃3^푃4 4 Sphere 푆 =^푃1^^푃2^푃^3^푃4 4 SphereSphereSphereSphere S “푆 =푆푆P1==푃1푃^푃P11푃2^^2푃^푃2P2푃^^3푃^푃33푃^P^4푃4 푃44 4 4 44 Sphere 푆 = 푃1^푃2^푃3^푃4 4
3.2.. 3D Cadastral Object Data Model Based on CGA 3.2.3.23.23.2. 3D .. 3D3D Cadastral CadastralCadastral Object ObjectObject Data DataData Model ModelModel Based BasedBased on onon CGA CGACGA 3.2. 3D Cadastral Object Data Model Based on CGA This section introducesintroduces aa detailed description of the expression method of 3D cadastral objects ThisThisThis section sectionsection introducesintroduces introducesintroduces aa detailed adetaileda detaileddetailed description description descriptiondescription of of theof ofthe thethe expression expression expressionexpression method method methodmethod of of 3D of of3D cadastral3D3D cadastral cadastralcadastral objects objects objectsobjects in inin CGA. CGA.This First, First,section the the introduces relationship relationship a detailed between between description the the main main of research research the expression contents contents method and and thethe of landland3D cadastral administration administration objects CGA.inin in CGA. CGA. CGA. First, First, First, First,the relationshipthe the the relationship relationship relationship between between between between the main the the the mainresearch main main research research research contents contents contents contents and the and land and and the administrationthethe land landland administration administration administration domain domainin CGA. model First, (LADM) the relationship [45] is discussed. between Second, the main the research concept and contents scope and of a the3D cadastralland administration spatial unit modeldomaindomaindomain (LADM) model modelmodel (LADM) [(LADM)(LADM)45] is discussed. [4 [5[44]5 5is]] isisdiscussed. discussed.discussed. Second, Second, Second,Second, the concept the thethe concept conceptconcept and scope and andand scope of scopescope a3D of of ofa cadastral 3Daa 3D3D cadastral cadastralcadastral spatial spatial spatialspatial unit unit are unitunit aredomain defined model as (LADM) semantics. [45 ] Third, is discussed. a method Second, to realizethe concept a unified and scope multidimensional of a 3D cadastral expression spatial unit of definedareareare defined defineddefined as semantics. as asas semantics. semantics. semantics. Third, Third,a methodThird, Third, a a amethod to method method realize to a to unifiedto realize realize realize multidimensional a a aunified unified unified m multidimensional multidimensionalultidimensional expression of expression geometry expressionexpression and of of of geometryare defined and as topology semantics. for 3D Third, cadastral a method objects, to which realize is abased unified on CGA, multidimensional isis introduceintroduced. expression We obtained of topologygeometrygeometrygeometry forand andand 3D topology topology cadastraltopology for for objects,for 3D 3D3D cadastral cadastralcadastral which objects, is objects,objects, based which on whichwhich CGA, is isisbased is basedbased introduced. on onon CGA, CGA,CGA, Weis isisintroduce obtainedintroduceintroduced. ad. d.We 3D WeWe cadastralobtained obtainedobtained ageometry 3D cadastral and spatialtopology unit for from 3D cadastralthe abstraction objects, of whichboth the is legallegalbased andand on physphysCGA,icalical is introducelevelslevels of thethed. We3D3D cadastralcadastralobtained spatiala 3Daa 3D3D cadastral cadastralcadastral unit from spatial spatialspatial the unit abstraction unitunit from fromfrom the thethe ofabstraction abstractionabstraction both the legalof of ofboth bothboth and the thethe physicallegal legallegal and andand levels phys physphysical oficalical thelevels levelslevels 3D of cadastral of ofthe thethe 3D 3D3D cadastral cadastral objects.cadastral objectsa 3D cadastral.. The 3D spatial cadastral unit spatial from the unit abstraction is decomposed of both toto the obtainobtain legal aa and seriesseries phys ofof ical geometricgeometric levels of simplexsimplex the 3Des cadastral,, which Theobjectsobjectsobjects 3D. cadastralThe.. TheThe 3D 3D3D cadastral cadastral spatialcadastral unitspatial spatialspatial is decomposed unit unitunit is isisdecomposed decomposeddecomposed to obtain to to atoobtain series obtainobtain a of aseriesa geometricseriesseries of of ofgeometric geometricgeometric simplexes, simplex simplexsimplex whicheses,es canwhich,, whichwhich be canobjects be . representedThe 3D cadastral by different spatial unit dimension is decomposedal blades to obtaininin CGA CGA a series.. To achieveof geometric the multidimensional simplexes, which representedcancancan be be be represented represented represented by different by by by dimensionaldifferent differentdifferent dimension dimension dimension blades inalal CGA.alblades bladesblades To inachieve in in CGA CGA CGA. the .T. oTT multidimensional oo achieve achieve achieve the the the multidimensional multidimensionalmultidimensional expression for expressioncan be represented for the complex by different 3D cadastral dimension spatialal unit,blades the inm ultivector CGA. To,, whichwhich achieve isis aa the spatialspatial multidimensional mathematical theexpressionexpressionexpression complex for for 3Dfor the the cadastralthe complex complexcomplex spatial 3D 3D3D cadastral ccadastral unit,adastral the spatial spatialspatial multivector, unit, unit,unit, the thethe whichm multivectormultivectorultivector is a spatial, which,, whichwhich mathematical is isisa spatialaa spatialspatial mathematical mathematical structuremathematical in structureexpression inin for CGA,CGA, the complexisis employedemployed 3D c.. adastral TheThe 3D3D spatialcadastralcadastral unit, datadata the model,model, multivector whichwhich, which isis basedbased is a onon spatial CGA,CGA, mathematical isis shownshown inin CGA,structurestructurestructure is employed. in in inCGA, CGA,CGA, is The isisemployed employedemployed 3D cadastral. The.. TheThe 3D data 3D3D cadastral model,cadastralcadastral whichdata datadata model, is model,model, based which which onwhich CGA, is isisbased based isbased shown on onon CGA, inCGA,CGA, Figure is isisshown shown3shown. Last, in in in Figurestructure 3. Last, in CGA, the structure is employed of 3D. The cadastral 3D cadastral spatial unitdata data model, storage which organization is based on inin CGA, CCGA isis designedshown in.. theFigureFigureFigure structure 3. 3.3.Last, Last,Last, of the 3Dthethe structure cadastral structurestructure of spatial of of3D 3D3D cadastral cadastralcadastral unit data spatial spatialspatial storage unit unitunit organization data datadata storage storagestorage inorganization organization CGAorganization is designed. in in inC GACCGAGA is isisdesigned designeddesigned. .. Figure 3. Last, the structure of 3D cadastral spatial unit data storage organization in CGA is designed. LeegallSpattiiallUniitt PhyssiiccallSpattiiallUniitt LeLgLaeelggSaapllSaSptpiaaatltiUiaanllUUitnniitt PhPyPhshiycysasilicScaapllSaSptpiaaatltiUiaanllUUitnniitt LegalSpatialUnit PhysicalSpatialUnit
SpattiiallUniitt SpSaSptpiaaatltiUiaanllUUitnniitt deSeccpoamtipaolUssiinttiiiotn decddoeemccoopmmosppiotoisosiitntiioonn decomposition
BoundarryFaccee BoundarryLiineess Poiinttss EucclliideeanPoiinttSeettss BoBuBonoudunandrdyaaFrryayFcFeaaccee BoBuBonoudunandrdyaaLrryiynLLeiisnneess PoPiPnootisinnttss EuEcEluuidccleliiaddneeaPanonPiPnootiSinnetttSSseettss BoundaryFace BoundaryLines Points EuclideanPointSets CGA Exprreessssiion CGA Exprreessssiion CGA Exprreessssiion CGA Exprreessssiion CGCCAGG EAAx EpErxxeppsrsreiesossnsiioonn CGCCAGG EAAx EpErxxeppsrsreiesossnsiioonn CGCCAGG EAAx EpErxxeppsrsreiesossnsiioonn CGCCAGG EAAx EpErxxeppsrsreiesossnsiioonn CGA Expression outer CGA Expression outer CGA Expression CGA Expression Quadvector outterr Bivector outterr Vector CGAPointSets Quadvecttor ouptoroeourudt teuerrc t Biivecttor ouptoroeourudt teuerrc t Vecttor CGAPoiinttSetts QuQQauduavadedcvvteoeccrttoorr prrooudtuecr tt BivBBeicivvteoeccrttoorr prrooudtuecr tt VeVcVteoeccrttoorr CGCCAGGPAAoPiPnootiSinnetttSSseettss Quadvector proppdrruoocddtuucctt Bivector proppdrruoocddtuucctt Vector CGAPointSets product product
Mullttiiveeccttorr MuMMltuiuvllteticivvteoeccrttoorr rreeMccounltssittvrrueccttiioorn recroreenccsootnrnusstctrrtuiuocctntiioonn reconstruction CGASpattiiallUniitt CGCCAGGSAApSaSptpiaaatltiUiaanllUUitnniitt CGASpatialUnit
Figure 3.. 3D cadastral data model based on CGA.. FigureFigureFigure 3.3. 33D3.. 3D3D ca cadastral cadastralcadastraldastral data datadata model modelmodel based basedbased on onon CGA.CGA CGACGA. .. Figure 3. 3D cadastral data model based on CGA. 3.2.1.. Initial Conditions 3.2.13.2.13.2.1. Initial.. InitialInitial Conditions ConditionsConditions 3.2.1 . Initial Conditions
ISPRS Int. J. Geo-Inf. 2016, 5, 20 7 of 16
ISPRS Int. J. Geo-Inf. 2016, 5, 20 7 of 16 3.2.1. Initial Conditions The Land Administration Domain Model (LADM), which includes a 3D cadastre, is a conceptual model for cadastral management [46]. [46]. The LADM has obtained the certificationcertification of the International Organization for Standardization (ISO).(ISO). The LADM contains four parts,parts, as shownshown in Figure4 4:: aa partyparty (LA_Party);(LA_Party); rights,rights, restrictions,restrictions, r responsibilitiesesponsibilities ( (LA_RRR);LA_RRR); a b basicasic administrative u unitnit ( (LA_BAUnit);LA_BAUnit); and a spatialspatial unit (LA_SpatialUnit).(LA_SpatialUnit). The main research contents of this paper belongsbelongs to the spatialspatial unit in the LADM. It is primarily based on CGA theory to investigate data organization andand the relationship betweenbetween geometry geometry and and topology, topology, which which is integratedis integrated in the in expressionthe expression for 3D for spatial 3D spatial units. Theunits construction. The construction of a 3D of cadastral a 3D cadastral data model data that model is based that on is CGAbased is on an extensionCGA is an of extension the spatial of units the inspatial the LADM.units in Thethe relationshipLADM. The betweenrelationship the between CGA-based the dataCGA model-based anddata themodel LADM and isthe depicted LADM inis Figuredepicted4. in Figure 4.
LA_Party
LA_RRR
LA_BAUnit
LA_SpatialUnit
Spatial Unit:: Spatial Unit:: LA_LegalSpaceUtilityNetwork LA_LegalSpaceBuildingUnit
CGA-based spatial data representations
Figure 4 4.. RelationshipRelationship between between a CGA a CGA-based-based data model data model and LADM and LADM(Land Administration (Land Administration Domain DomainModel) Model).
Prior to building a 3D cadastral data model, a description of the co conceptncept of a basic unit in a 3D cadastre—thecadastre—the 3D cadastral spatial unit—isunit—is needed. A 3D cadastral spatial unit is a combinedcombined object that is abstracted from a legal object and a 3D physical object in the real world. It is the smallest unit in 3D 3D cadastral cadastral mana management.gement. From From a legal a legal perspective, perspective, a 3D a cadastral 3D cadastral spatial spatial unit is unit a closed is a closed 3D space 3D spacewith legally with legally binding binding and distinct and distinct 3D boundaries 3D boundaries.. The differen The differencece among among ownership ownership rightsrights is the ismain the maincriterion criterion for distinguishing for distinguishing the boundary the boundary of a of cadastral a cadastral spatial spatial unit unit.. The The internal internal consistency consistency of ofownership ownership rights rights is the is premise the premise of 3D of cadastral 3D cadastral management. management. From the From point the of point view of real view physical of real physicalobjects, 3D objects, cadastral 3D cadastral spatial spatial units unitscomprise comprise the abstract the abstract expression expression of of a a realistic realistic 3D 3D cadastral management object according t too the needs of cadastralcadastral management.management. 3D spatial units of physical objects can be an independent building that is located on, below below,, or above ground. I Itt can also be a semi-enclosedsemi-enclosed building (e.g., a sports stadium) that needs to be abstract abstracteded to obtain the 3D ownership ownership boundary according to relative cadastral laws.laws. 3D spatial units may simultaneously contain many buildings withwith identicalidentical ownership,ownership, such such as as a a factory factory spatial spatial unit unit that that contains contains multiple multiple workshops workshops or aor villa a villa that that contains contains houses, houses, courtyards courtyards and and an an underground underground garage. garage.
3.2.2.3.2.2. 3D Cadastral Object Reconstruction in CGA The premise ofof aa 3D3D cadastral cadastral data data model model is is the the abstract abstract expression expression of 3Dof 3D cadastral cadastral entities entities in the in realthe real world. world. The The geometric geometric decomposition decomposition of complex of complex 3D cadastral3D cadastral objects object is neededs is needed to organize to organize and storeand store these these objects objects in a cadastral in a cadastral database. database The representation. The representation of a current of a 3D current cadastral 3D data cadastr modelal data and cadastralmodel and requirements cadastral requirements for distinct boundary for distinct ownership boundary are ownership both considered are both in 3D considered cadastral inobject 3D cadastral object decomposition. In this paper, 3D cadastral objects can be decomposed into four geometric objects as follows: points, lines, surfaces, and spatial units. By further decomposition of
ISPRS Int. J. Geo-Inf. 2016, 5, 20 8 of 16 decomposition. In this paper, 3D cadastral objects can be decomposed into four geometric objects as ISPRS Int. J. Geo-Inf. 2016, 5, 20 8 of 16 follows: points, lines, surfaces, and spatial units. By further decomposition of these geometric objects, we obtainthese threegeometric types objects, of basic we geometricobtain three primitives: types of basic nodes, geometric boundary primitives: lines, nodes, and boundary boundary faces.lines, Theand complexboundary 3Dfaces cadastral. objects are represented by these different dimensional basic primitives, which shouldThe complex be expressed 3D cadastral in CGA objects to realize are represented a consistent by these representation different dimensional form. Different basic primitives, dimensional basicwhich geometric should simplexes be expressed in in CGA CGA can to realize be directly a consistent constructed representation by the form. outer Different product dimensional of conformal pointbasic sets, geometric which is simplexes illustrated in inCGA Table can1 be. However, directly constructed these simplexes by the outer in Table product1 do of not conformal have boundary point constraints,sets, which which is areillustrated very important in Table 1. in However, cadastral these registration simplexes and in management. Table 1 do not Boundary have boundary limitations for theseconstraints, basic simplexes, which are whichvery important are expressed in cadastral by conformal registration point and sets, management. are needed Boundary to accurately limitations for these basic simplexes, which are expressed by conformal point sets, are needed to describe the boundaries of 3D cadastral objects. The coordinates of these point sets can be employed to accurately describe the boundaries of 3D cadastral objects. The coordinates of these point sets can be limitemployed the boundaries to limit of the corresponding boundaries of corresponding geometric objects, geometric which objects, is defined which asis defined follows: as follows: Definition 3: If GeoPrimxky is a basic geometric primitive of the k-dimensional subspace in Cl4,1 Definition 3: If 퐺푒표푃푟푖푚<푘> is a basic geometric primitive of the k-dimensional subspace in and tPiu denotes the conformal point sets that compose GeoPrimxky, then the geometric representation 퐶푙4,1 and {푃푖} denotes the conformal point sets that compose 퐺푒표푃푟푖푚<푘> , then the geometric of GeoPrimxky by the outer product can be defined as follows: representation of 퐺푒표푃푟푖푚<푘> by the outer product can be defined as follows: P k “ 1 Pk 1 ¨P1 ^ P2 k “ 2 ˛ P12 P k 2 GeoPrimăką “ tP1, P2, ..., Pnu GeoPrimkn˚= { P‹12 , P ,..., P } ˚P1 ^ PP2 ^ ee8 k k “3 3 ‹ ˚ 12 ‹ ˚‹ ˚PP1^ PP 2 ^ PP 3 ^ ee8 k k “4 4 ‹ ˚ 1 2 3 ‹ ˝ ‚ where n is the number of points that compose GeoPrim . where n is the number of points that compose 퐺푒표푃푟푖푚x
P1 Plane: P1 P 2 P 3 e
P2 P5 Facet
P4 P3
GeoPrim4 ( P 1 P 2 P 3 e ){ P 1 , P 2 , P 3 , P 4 , P 5 }
Figure 5. Primitive facet represented in 퐶푙 . Figure 5. Primitive facet represented in Cl4,14,1.
ConsiderConsider a primitive a primitive facet facet that that is is composed composed of five five points, points, as as shown shown in inFigure Figure 5. The5. The plane plane that that contains the facet in 퐶푙4,1 can be represented by 푃1^^푃2^^푃3^푒^∞, whereas the boundary of the facet is contains the facet in Cl4,1 can be represented by P1 P2 P3 e8, whereas the boundary of the facet restricted by the five composing points. Figure 5 and the definition of 퐺푒표푃푟푖푚 indicate that the is restricted by the five composing points. Figure5 and the definition of GeoPrim<푘> indicate that basic geometric simplexes are directly represented by the outer product of conformal pointxky sets. The the basicgeometric geometric ranges simplexes of these simplexes are directly are restricted represented by the by sequence the outer of product point sets of conformalthat compose point the sets. The geometriccorresponding ranges boundary of these simplexes. simplexes In this are manner, restricted we byrealize the sequencethe accurate of and point united sets representation that compose the correspondingfor different boundary dimensional simplexes. simplexes In in this CGA. manner, Complex we geometric realize the objects accurate in CGA and can united be expressed representation by for differentintegrating dimensional different dimensional simplexes simplexes. in CGA. ComplexTo realize the geometric expression objects of complex in CGA 3D cancadastral be expressed objects by integratingas an entity, different the mathematical dimensional structure simplexes. of a To multivector realize the in expression CGA is employed of complex to realize 3D cadastral the unified objects as anexpression entity, the of mathematical multidimensional structure simplexes, of a which multivector is defined in as CGA follows: is employed to realize the unified expression of multidimensional simplexes, which is defined as follows:
ISPRS Int. J. Geo-Inf. 2016, 5, 20 9 of 16 ISPRS Int. J. Geo-Inf. 2016, 5, 20 9 of 16
퐺푒표푃푎푟푐푒푙 퐶푙 DefinitionDefinition 4: If4: GeoParcelIf is an is arbitraryan arbitrary 3D 3D cadastral cadastral spatial spatial unit unit in CGAin CGACl 4,1,4 the,1, the geometry geometry and topologyand topology of this 3Dof this cadastral 3D cadastral object object can be can expressed be expressed by the by following the following algebraic algebraic formula formula in CGAin CGA
MultiVector “ tGeoPrim u ` tGeoPrimă ąu ` tGeoPrim u MultiVectorăGeoParcelGeoParcelą {}{}{} GeoPrimă 11ą GeoPrim 33 GeoPrim ă 4 4ą where tGeoPrimx1yu, tGeoPrimx3yu, tGeoPrimx4yu are simplex sets that compose GeoParcel. where {퐺푒표푃푟푖푚 }, {퐺푒표푃푟푖푚 }, {퐺푒표푃푟푖푚 } are simplex sets that compose 퐺푒표푃푎푟푐푒푙. It should be noted<1> that GeoPrim<3>x0y, GeoPrim
A B C D E F G H
GeoPrim1 E G C G … A C A A
MultiVectorLine {} GeoPrim 13 GeoPrim
C B G F …
A D C B
MultiVectorFacet {}{} GeoPrim 1 GeoPrim 3 GeoPrim 4 G F
H E C B
D A MultiVector{}{}{} GeoPrim GeoPrim GeoPrim GeoParcel 1 3 4
FigureFigure 6. Process6. Process of of integrating integrating the the representation representation ofof thethe geometrygeometry and and topology topology of of a a3D 3D cadastral cadastral spatialspatial unit unit in CGA.in CGA.
TheThe construction construction process process of of a a 3D 3D cadastral cadastral datadata modelmodel thatthat is based based on on CGA CGA reflects reflects a adistinct distinct topologicaltopological hierarchy hierarchy relationship relationship among among different different dimensional dimensional geometry geometry objects. objects. A cuboid A cuboid is selected is to representselected to a represent 3D cadastral a 3D spatial cadastral unit, spatial which unit, is illustrated which is illustrated in Figure6 in. The Figure construction 6. The construction process of theprocess multidimensional of the multidimensional unified expression unified forexpression the 3D for cadastral the 3D objectcadastral is described. object is described. The 3D cadastralThe 3D object,cadastral which object, is represented which is byrepresented the multivector by the structure,multivector is structure, a complex isstructure a complex that structure is composed that is of points, boundary lines and boundary facets. All components that are contained in the multivector ISPRS Int. J. Geo-Inf. 2016, 5, 20 10 of 16
ISPRS Int. J. Geo-Inf. 2016, 5, 20 10 of 16 are independent. Topological relations among different dimensional components are obtained in the processcomposed of different of points, dimensional, boundary geometriclines and boundary and simplex facets. constructions All components by that the are outer contained product. in the The 3D cadastralmultivector spatial are unit independent. is at the top Topological of the hierarchical relations structure. among different dimensional components are obtained in the process of different dimensional, geometric and simplex constructions by the outer 3.3. 3Dprod Cadastraluct. The Data3D cadastral Organization spatial Basedunit is on at CGAthe top in of A the Database hierarchical structure. The 3D cadastral data structure based on CGA is organized in a cadastral database, as shown in 3.3. 3D Cadastral Data Organization Based on CGA in A Database Figure7. The data storage contains four types of data, including points, boundary lines, boundary faces andThe spatial 3D cadastral units. Thedata conformalstructure based point on setsCGA that is organized compose in thea cadastral left 3D database, cadastral as spatial shown in unit in Figure 7. The data storage contains four types of data, including points, boundary lines, boundary Figure7 can be obtained from Euclidean point sets by embedding in CGA. Because the conformal faces and spatial units. The conformal point sets that compose the left 3D cadastral spatial unit in space contains five basis vectors, the coordinates x, y and z of the original point become coefficients of Figure 7 can be obtained from Euclidean point sets by embedding in CGA. Because the conformal the basisspace vector containse1, fivee2, e basis3, respectively. vectors, the Becausecoordinates normally x, y and the z of coefficient the original of pointe0 is become the constant coefficients 1, it is not storedof inthe the basis point vector table 푒 and, 푒 , 푒 the, respectively. coefficient ofBecausee8 can normally be computed the coefficient with the of Euclidean푒 is the constant coordinates 1, it x, y 1 2 3 0 and zis [ 41 not]. Because stored in different the point dimensional table and the geometric coefficient objects of 푒 cancan be be constructed computed with from the conformal Euclidean point ∞ sets bycoordinates the outer x,product, y and z [41 only]. Because the original different coordinates dimensional are geometric stored objects in the can point be table.constructed The boundaryfrom line andconformal the boundary point sets face by the tables outer store product, point only IDs, the which original are coordinates employed are in stored connection in the point with table. the point table.The Relative boundary point line IDs and are the stored boundary in both face thetables boundary store point line IDs, and which boundary are employed face tables in connection in a specified orderwith to define the point the table. range Relative of corresponding point IDs are objects.stored in Toboth ensure the boundary compatibility line and with boundary the existing face tables method in a specified order to define the range of corresponding objects. To ensure compatibility with the in which lines form facets, the boundary line IDs that compose boundary faces are also stored in the existing method in which lines form facets, the boundary line IDs that compose boundary faces are corresponding boundary face table. Direction information about boundary lines and boundary faces also stored in the corresponding boundary face table. Direction information about boundary lines are storedand boundary in each table faces inare the stored form in of each a vector. table in The the 3D form cadastral of a vector. spatial The unit 3D cadastral is organized spatial by unit storing is the constituentorganized boundary by storing face the IDs. constituent boundary face IDs.
CGA::Point PID E1 E2 E3 E_inf 1 0.00 0.00 2.00 2.00E+00 2 0.00 2.00 2.00 4.00E+00 … … … … …
CGA::Boundary line AID PID_S PID_E DIRECTION … 1 1 10 v1 … 2 2 9 v2 … … … … … …
CGA::Boundary face FID AID_QUE PID_BOUND DIRECTION … 1 1,2,7,… 1,2,4,… v1 … 2 1,3,6,… 1,10,11,… v2 … … … … … …
CGA::SpatialUnit VID FID_QUE SpatialUnit … 1 1,2,3,… 18.13 … … … … …
FigureFigure 7. 3D 7. 3D cadastral cadastral data data organization organization in a a database database that that is isbased based on onCGA CGA.
The characteristics of the 3D cadastral data organization and correlation method are similar to Thethe characteristics characteristics of the of themultivector 3D cadastral structure, data which organization is not only and a mathematic correlational structure method but are also similar a to the characteristicsdata storage structure. of the multivector Based on the structure, design of whichpoint, boundary is not only line, a mathematicalboundary face and structure spatial but unit also a datadata storage storage structure. structures, Based conformal on the points design and of lines point, are boundary stored in the line, boundary boundary face face table. and Boundary spatial unit datafaces, storage which structures, are employed conformal to construct points the and 3D lines cad areastral stored spatial in unit, the boundarycan be represented face table. by Boundaryouter faces, which are employed to construct the 3D cadastral spatial unit, can be represented by outer products with conformal point sets, as shown in definition 3, or represented by the boundary line ISPRS Int. J. Geo-Inf. 2016, 5, 20 11 of 16
ID that belongs to the boundary face. Similar to the multivector structure that integrates different dimensional objects, the boundary face table also contains all primitives that are employed to compose the table. These characteristics are also listed in the spatial unit table, which stores boundary face IDs to realize multidimensional object storage. In this manner of data organization, the complex 3D cadastral spatial unit can be easily expressed and stored in the multivector structure. The topological relations among different dimensional constructed objects can be clearly described in the hierarchical data storage and the multivector structure, whereas the geometric information can be represented within the multivector by the outer product. The geometry and topology of 3D cadastral objects can be expressed in a multidimensional unified form that is integrated in the multivector structure.
4. Case Studies Cadastral data for an urban residential community in Xuzhou, Jiangsu province, are selected as the experimental data in our case study. The experimental data includes 17,618 points, 28,487 boundary lines, 9,462 boundary faces and 1,484 buildings. These building models are imported into the software system CAUSTA [32]. The data model is implemented as a plug of the system CAUSTA. To ensure compatibility with GA-based operations, the experimental data are reorganized in CaVector data formats for the storage structure. The storage structure is subsequently assembled according to the multivector structure. With these transformations and organizations, we can manage the 3D cadastral spatial units in CAUSTA. The demonstrations are performed in the hardware environment of a Lenovo E420 notebook with a 2.4-GHz Intel Core i5-2430M CPU and 4 GB of RAM. The software environment is a Windows 7 operating system with Visual Studio 2008. Conventional 3D cadastral data models, which are based on a simplified spatial model (SSM) [18] and an object-oriented 3D GIS model (OO3D) [46], have been selected to make a comparison with the performance of a CGA-based cadastral data model. The results for memory requirements and response time in 3D visualization for the SSM, OO3D and CGA-based data models are listed in Table2. The comparison results indicate that the CGA-based data model performs more efficiently than the SSM data model concerning response time and memory occupancy. Although data storage in the CGA-based data model occupies less memory than the OO3D data model, it has a slightly longer response time than the OO3D data model.
Table 2. The memory requirements and response time in 3D visualization based on OpenGL.
CGA-Based Model OO3D Model SSS Model time(sec) 3.76 3.71 4.19 memory(kb) 905 968 1491
The geometric and algebraic representation of 3D spatial units in the CGA-based 3D cadastral data model are displayed in Figure8. The representation and organization of the spatial units differ from the representation in Euclidean space. In the CGA representation, the dimensional structures of objects are represented according to the hierarchical Grassmann structure. Figure8 shows the algebraic equations for each spatial unit, which are constructed by the outer product. Different dimensional geometry objects, such as the lines and planes shown in Figure8, can be uniformly expressed with the outer product in CGA. In Euclidean space, representations of geometry objects are primarily based on coordinates, which causes problems such as complicated expression structure, non-uniform expression form in different dimensions, and unclear geometry meaning [47]. Geometric objects are expressed in CGA, and the outer product differs from the traditional methods, which are based on Euclidean geometry. Geometric objects that are represented by the outer product not only support uniform expression among different dimensions but also propose a simpler construction structure and clearer geometric meanings for the geometric object expression of different dimensions. These advantages can be applied to represent complex 3D geometric objects, as shown in Figure9. ISPRS Int. J. Geo-Inf. 2016, 5, 20 12 of 16 ISPRS Int. J. Geo-Inf. 2016, 5, 20 12 of 16 ISPRS Int. J. Geo-Inf. 2016, 5, 20 12 of 16
FigureFigureFigure 8 8.. 8UniformUniform. Uniform representation representation representation form form forfor different different dimensionaldimensional objects objects in in CGA. CGA.
FigureFigure 9. 9.Multidimensional Multidimensional unified unified expressionexpression for a cadastral object object within within the the multivector multivector structure. structure. Figure 9. Multidimensional unified expression for a cadastral object within the multivector structure. WithWith the the unified unified expression expression form form for for geometric geometric objects of of different different dimensions, dimensions, geometr geometricic informationinformationWith the and andunified topological topological expression relations relations form of of for complex complex geometric cadastral objects objects objects of different can can be be represented representeddimensions, within within geometr the the ic informationmultivectormultivector andstructure structure topological in in CGA.CGA. relations One of of ofthe the complex 3D 3D cadastral cadastral cadastral objects objects objects was was selected can selected be to represented demonstrate to demonstrate within how howto the multivectortorepresent represent structure complex complex cadastralin cadastral CGA. One objects objects of the with with 3D the cadastral the multivector multivector objects structure, structure,was selected as as illustrated to illustrated demonstrate in in Figure Figure how 9. 9to. Compared with the separate expression of different dimensional geometric objects in Euclidean representCompared complex with the separate cadastral expression objects with of different the multivector dimensional structure, geometric as objects illustrated in Euclidean in Figure space, 9. space, all dimensional elements that comprise the 3D cadastral objects are multidimensional and Comparedall dimensional with elements the separate that comprise expression the 3Dof different cadastral objectsdimension are multidimensionalal geometric objects and in integrated Euclidean in integrated in the multivector structure. Geometric information about cadastral objects is expressed space,the multivector all dimensional structure. elements Geometric that information comprise the about 3D cadastral cadastral objects objects is are expressed multidimensional by an algebraic and by an algebraic formula with hybrid dimensional components, and topological relations among integrated in the multivector structure. Geometric information about cadastral objects is expressed by an algebraic formula with hybrid dimensional components, and topological relations among
ISPRS Int. J. Geo-Inf. 2016, 5, 20 13 of 16 formula with hybrid dimensional components, and topological relations among different dimensional components can be represented in the distinct dimensional hierarchy. The geometric information and topological relationships of a cadastral object can be expressed in the multidimensional multivector structure in a unified manner, which distinguishes our method from existing approaches based on Euclidean geometry. The multidimensional unified method for the geometry and topology provide a flexible and convincing method for topological relations analysis and geo-object updating [30,34]. Compared with existing 3D cadastral data models that are based on Euclidean geometry, the data model that is based on CGA realizes a geometrically and topologically multidimensional unified expression for 3D cadastral objects. A 3D cadastral data model that is based on CGA provides a compact and geometrically distinct method for representing cadastral objects in a cadastral database. Multidimensional construction elements are integrated in a multivector structure to express 3D cadastral objects, which enables the spatial unit to be handled as an entity in topological and spatial analysis [31,32,34]. With the advantage of the outer product in the representation of dimensional objects, cadastral objects can be directly and simply expressed. All dimensional components of 3D cadastral objects can be expressed within a multivector structure. In addition, a multivector structure can be employed to express cadastral objects and provides a hierarchical data storage form, which is helpful in organizing 3D cadastral data in a database.
5. Discussion In this paper, we introduce the theory of CGA to 3D cadastral data modeling to solve problems with spatial representations that are encountered in the extension from a 2D cadastre to a 3D cadastre. With the advantages of CGA in geometric representation, expression forms for both 2D and 3D spatial units have been united in an algebraic manner. The geometric information and topological relations of spatial units have also been integrated into the algebraic expressions. In CGA, different dimensional objects are uniformly represented using outer products, which enables a direct and intuitive extension of spatial representations from 2D structures to 3D structures. The multivector structure is subsequently employed to algebraically integrate the multidimensional geometric components for complex 3D spatial units as entities. The geometries and topological relations of spatial units are represented in a unified manner in the form of algebraic expressions. With the development of a CGA-based 3D cadastral data model, the representations of spatial units in the 3D cadastral database not only have distinct geometric meanings but also combine the representations of the geometries and the topological relations. The CGA-based 3D cadastral data model also unifies the representation of multidimensional objects in the same representation form. Therefore, direct, compact and unified representations of objects are feasible. In this paper, we primarily focus on a unified expression form between 2D spatial units and 3D spatial units and the integration of geometry and topology for spatial units. However, cadastral registration and transaction data may update daily, which indicates that the time component of cadastral data should be considered in 3D cadastre development. A method for the construction of a spatio-temporal cadastral data model will be a future research topic. The integrated updating of both the geometries and the topological relations remains complex. Although the computation ability of CGA is very powerful, we did not discuss the spatial analysis and updating of 3D cadastral objects using GA operators or algorithms. For example, the division and combination of the spatial units in the 3D cadastre can be achieved with the intersection product in CGA [31] because the geometric construction in CGA is coordinate-free, i.e., the geometries can be adaptively modified according to the relative construction structures that are related to the dimensional structures and the topological relations. The complete utilization of these advantages in CGA to develop an integrated data updating method that updates both the geometries and topological relations in the integrated framework is also an important objective of future studies. The topological relations computations are potentially more complete than the classical 9IM model for 3D triangles using the CGA model [35]. Because the 3D cadastral model is the foundation of the data representation, the construction of the analysis using ISPRS Int. J. Geo-Inf. 2016, 5, 20 14 of 16 the CGA operators is simple and direct. This topic is another potential direction for CGA-based 3D cadastre development.
6. Conclusion In this paper, we introduced a CGA-based 3D cadastral data model to solve problems such as the division in the spatial representation forms from a 2D cadastre to a 3D cadastre. In this data model, geometric constructions are produced by the outer product according to a Grassmann structure. Different dimensional objects are represented with the multivector data structure using a unified form. In this manner, the topological relations between different dimensional objects can be implicitly inherited in the object representation. The data model was tested with real-world data. The characteristics of the data model are also discussed. The test suggests that our data model can unify not only the representation of different dimensional objects but also the representation of geometric structures and topological relations. The maintenance and updating of the data in this data model become much more compact and simple. Because our data model is completely compatible and can support GA operators and analysis, multidimensional-unified 3D cadastral data analysis is feasible.
Acknowledgments: This work was supported by the Surveying and Mapping Industry research Special Funds for Public Welfare Projects (Grant No. 201512011). The authors also wish to thank the editors and reviewers for their interesting and constructive comments. Author Contributions: All six authors have contributed to the work presented in this paper. Ji-yi Zhang, Peng-cheng Yin and Gang Li constructed the overall framework for 3D cadastral data model based on CGA. Ji-yi Zhang, Hehe Gu and Hua Zhao contributed in the discussion section. Ji-yi Zhang, Peng-cheng Yin and Jian-chun Fu worked on the designing of case study and experimental data processing. All authors worked collaboratively on writing this paper. Conflicts of Interest: The authors declare no conflict of interest.
References
1. Stoter, J.; Ploeger, H.; Van Oosterom, P. 3D cadastre in the Netherlands: Developments and international applicability. Comput. Environ. Urban Syst. 2013, 40, 56–67. [CrossRef] 2. Paulsson, J. Reasons for introducing 3D property in a legal system—Illustrated by the Swedish case. Land Use Policy 2013, 33, 195–203. [CrossRef] 3. Guo, R.Z.; Li, L.; Yin, S.; Luo, P.; He, B.; Jiang, R.Z. Developing a 3D cadastre for the administration of urban land use: A case study of Shenzhen, China. Comput. Environ. Urban Syst. 2013, 40, 46–55. [CrossRef] 4. Benhamu, M. A GIS-related multi layers 3D cadastre in Israel. In Proceedings of the XXIII FIG Congress, Munich, Germany, 8–13 October 2006. 5. Aien, A.; Kalantari, M.; Rajabifard, A.; Williamson, I.; Wallace, J. Towards integration of 3D legal and physical objects in cadastral data models. Land Use Policy 2013, 35, 140–154. [CrossRef] 6. Stoter, J.; Munk Sørensen, E.; Bodum, L. 3D Registration of real property in denmark. In Proceedings of the 2004 FIG Working Week, Athens, Greece, 22–27 May 2004. 7. Zlatanova, S.; Rahmanb, A.A.; Shi, W.Z. Topological models and frameworks for 3D spatial objects. Comput. Geosci. 2004, 30, 419–428. [CrossRef] 8. Ying, S.; Guo, R.; Li, L.; Van Oosterom, P.; Stoter, J. Construction of 3D volumetric objects for a 3D cadastral system. Trans. GIS 2014.[CrossRef] 9. Tse, R.O.C.; Gold, C. A proposed connectivity-based model for a 3-D cadaster. Comput. Environ. Urban Syst. 2003, 27, 427–445. [CrossRef] 10. Aien, A.; Rajabifard, A.; Kalantari, M.; Shojaei, D. Integrating legal and physical dimensions of urban environments. ISPRS Int. J. Geo-Inf. 2015, 4, 1442–1479. [CrossRef] 11. Tack, F.; Buyuksalih, G.; Goossens, R. 3D building reconstruction based on given ground plan information and surface models extracted from space borne imagery. ISPRS J. Photogramm. Remote Sens. 2012, 67, 52–64. [CrossRef] 12. Karki, K.; Thompson, R.; McDougall, K. Development of validation rules to support digital lodgement of 3Dcadastral plans. Comput. Environ. Urban Syst. 2013, 40, 34–45. [CrossRef] ISPRS Int. J. Geo-Inf. 2016, 5, 20 15 of 16
13. Soltanieh, S.M.K. Cadastral Data Modeling—A Tool for E-Land Administration. Ph.D. Thesis, University of Melbourne, Parkville, VIC, Australia, 2008. 14. Worboys, M.; Duckham, M. GIS: A Computing Perspective; CRC Press: Boca Raton, FL, USA, 2004. 15. Van Oosterom, P. Research and development in 3D cadastres. Comput. Environ. Urban Syst. 2013, 40, 1–6. [CrossRef] 16. Yuan, L.W.; Yu, Z.Y.; Luo, W.; Yi, L.; Lü, G.N. Geometric algebra for multidimension-unified geographical information system. Adv. Appl. Clifford Algebras 2013, 23, 497–518. [CrossRef] 17. Molenaar, M. A formal data structure for 3D vector maps. In Proceedings of the first European Conference on Geographical Information Systems, Amsterdam, The Netherlands, 10–13 April 1990. 18. Zlatanova, S. 3D GIS for Urban Development. Ph.D. Thesis, ITC, Hengelosestraat, The Netherlands, 2000. 19. Coors, V. 3D-GIS in networking environments. Comput. Environ. Urban Syst. 2003, 27, 345–357. [CrossRef] 20. Li, R.X. Data structure and application issues in 3D geographical information system. Geomatic 1994, 48, 209–224. 21. Penninga, F.; Van Oosterom, P. A simplicial complex-based DBMS approach to 3D topographic data modeling. Int. J. Geogr. Inf. Sci. 2008, 22, 751–779. [CrossRef] 22. Robles-Ortega, M.D.; Ortega, L.; Feito, F.R. Design of topologically structured geo-database for interactive navigation and exploration in 3D web-based urban information systems. J. Environ. Inform. 2012, 19, 79–92. 23. Kumar, N.P. On the topological situations in geographic spaces. Ann. GIS 2014, 20, 131–137. [CrossRef] 24. Clementini, E.; Di Felice, P. A model for representing topological relations between complex geometric features in spatial databases. Inf. Sci. 1996, 90, 121–136. [CrossRef] 25. Egenhofer, M.J.; Shariff, A.R. Metric details for natural-language spatial relations. ACM Trans. Inf. Syst. 1998, 16, 295–321. [CrossRef] 26. Li, J.; Ouyang, J.; Liao, M. Representation and reasoning for topological relations between a region with broad boundaries and a simple region on the basis of RCC-8. J. Comput. Inf. Syst. 2013, 9, 5395–5402. 27. Li, H.B.; Hestenes, D.; Rockwood, A. Generalized homogeneous coordinates for computational geometry. In Geometric Computing with Clifford Algebras; Springer: Berlin, Germany, 2001; pp. 27–59. 28. Li, H.B. Invariant Algebras and Geometric Reasoning; World Scientific Publishing: Hackensack, NJ, USA, 2008. 29. Dorst, L.; Fongijne, D.; Mann, S. Geometric Algebra for Computer Science: An Object-Oriented Approach to Geometry; Morgan Kaufmann: Burlington, MA, USA, 2007. 30. Yuan, L.W.; Yu, Z.Y.; Luo, W.; Zhou, L.C.; Lü, G.N. A 3D GIS spatial data model based on conformal geometric algebra. Sci. China Earth Sci. 2011, 54, 101–112. [CrossRef] 31. Yu, Z.Y.; Luo, W.; Hu, Y.; Yuan, L.W.; Zhu, A.X.; Lü, G.N. Change detection for 3D vector data: A CGA-based Delaunay-TIN intersection approach. Int. J. Geogr. Inf. Sci. 2015.[CrossRef] 32. Yuan, L.W.; Yu, Z.Y.; Chen, S.F.; Luo, W.; Wang, Y.J.; Lü, G.N. CAUSTA: Clifford algebra-based unified spatio-temporal analysis. Trans. GIS 2010, 14, 59–83. [CrossRef] 33. Hildenbrand, D. Foundations of Geometric Algebra Computing; Springer: Heidelberg, Germany, 2013. 34. Yu, Z.Y.; Luo, W.; Yuan, L.W.; Hu, Y.; Zhu, A.X.; Lü, G.N. Geometric algebra model for geometry-oriented topological relation computation: Trans. GIS; 2015. [CrossRef] 35. Yuan, L.W.; Yu, Z.Y.; Luo, W.; Yi, L.; Lü, G.N. Multidimensional-unified topological relations computation: A hierarchical geometric algebra-based approach. Int. J. Geogr. Inf. Sci. 2014, 28, 2435–2455. [CrossRef] 36. Rosenhahn, B.; Sommer, G. Pose estimation in conformal geometric algebra part I: The stratification of mathematical spaces. J. Math. Imaging Vis. 2005, 22, 27–48. [CrossRef] 37. Hildenbrand, D. From Grassmann’s vision to geometric algebra computing. In From Past to Future: Graßmann’s Work in Context; Springer: Berlin, Germany, 2011; pp. 423–433. 38. Bayro-Corrochano, E.; Sobczyk, G. Applications of lie algebras and the algebra of incidence. In Geometric Algebra with Applications in Science and Engineering; Springer: Berlin, Germany, 2001; pp. 252–277. 39. Hitzer, E. Introduction to Clifford’s geometric algebra. J. Control Meas. Syst. Integr. 2012, 4, 1–10. 40. Hitzer, E.; Sangwine, S.J. Quaternion and Clifford Fourier Transforms and Wavelets; Springer: Basel, Switzerland, 2013. 41. Perwass, C. Geometric Algebra with Applications in Engineering; Springer: Heidelberg, Germany, 2009. 42. Hu, Y.; Luo, W.; Yu, Z.Y.; Yuan, L.W.; Lü, G.N. Geometric algebra-based modeling and analysis for multi-layer, multi-temporal geographic data. Adv. Appl. Clifford Algebras 2015.[CrossRef] ISPRS Int. J. Geo-Inf. 2016, 5, 20 16 of 16
43. Hitzer, E. Conic sections and meet intersections in geometric algebra. In Computer Algebra and Geometric Algebra with Applications; Springer: Berlin, Germany, 2005; pp. 350–362. 44. Hitzer, E.; Tachibana, K.; Buchholz, S.; Yu, I. Carrier method for the general evaluation and control of pose, molecular conformation, tracking, and the like. Adv. Appl. Clifford Algebras 2009, 19, 339–364. [CrossRef] 45. International Organization for Standardization (ISO). Geographic Information—Land Administration Domain Model (LADM), ISO 19152; International Organization for Standardization (ISO): Geneva, Switzerland, 2012. 46. Shi, W.Z.; Yang, B.S.; Li, Q.Q. An object-oriented data model for complex objects in three-dimensional geographical information systems. Int. J. Geogr. Inf. Sci. 2003, 17, 411–430. [CrossRef] 47. Yuan, L.W.; Lü, G.N.; Luo, W.; Yu, Z.Y.; Yi, L.; Sheng, Y.H. Geometric algebra method for multidimensionally-unified GIS computation. China Sci. Bull. 2012, 57, 802–811. [CrossRef]
© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons by Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).