2. Representation of Spatial Objects

1
2. Representation of Spatial Objects

• Ki-Joon Han
• Department of Computer Science Engineering
• Konkuk University
• E-mail kjhan_at_db.konkuk.ac.kr
• URL http//db.konkuk.ac.kr

2
Introduction

• This chapter
• Various ways of modeling and representing
  geometric and topological information in GIS
• Euclidean space
• The space of interest Rd (d2), together with the
  Euclidean distance
• Points
• Elements of the Euclidean space
• A pair of (Cartesian) coordinates (x,y)
• Map projection
• A conversion to map geographic entities on a
  glove (hence a curved surface) onto a planar
  representation
• Embedded space(or search space)
• Region of R2 that contains the relevant objects
  and is bounded
• A sufficiently large rectangle whose edges are
  parallel to the axes of the coordinate system

3
2.1 Geographic Space Modeling2.1.1 Entity-Based
Models (1/6)

• Geographic object( entity or feature)
• (1) Description
• (2) Spatial component( spatial object or spatial
  extent)
• shape and location of the object a set of
  points
• Entire set (identity, spatial object, and common
  description)
• Interpretation of space
• Depends on the semantics associated with the
  geographic territory
• E.g., territory of France
• (1) Administrative point of view several
  administrative units
• (2) Geologist view point geologic areas
• (3) Traffic control view point road network
• gt choose a new interpretation of space and
  define a new collection of
• entities(theme) describing this space

4
2.1.1 Entity-Based Models (2/6)

• Types of spatial objects
• (1) Zero-dimensional objects or points
• for representing the location of entities whose
  shape is not considered
• The area which is quite small with respect to the
  embedding space size
• E.g., cities, churches, crossings
• (2) one-dimensional objects or linear objects
• for representing networks (roads, hydrography,
  and so on)
• Polyline
• A finite set of line segments or edges, such that
  each segment endpoint is shared by exactly two
  segments
• Closed two extreme points are identical
• Simple no pairs of nonconsecutive edges
  intersect at any points
• Monotone with respect to L every line L
  orthogonal to L meets the polyline at one point
  at most

5
2.1.1 Entity-Based Models (3/6)

(a) line segment(edge)
(b) polyline
(c) non-simple polyline
(d) simple closed polyline
(f) non-monotone polyline
(e) monotone polyline
Figure 2.1 Examples of 1D objects
6
2.1.1 Entity-Based Models (4/6)

• (3) two-dimensional objects or surface objects
• for representing entities with large areas, such
  as parcel or administrative units
• Polygon
• A region of the plane bounded by a closed
  polyline, called its boundary
• Simple its boundary is a simple polyline
• Convex for any pair of points A and B in P the
  segment AB is fully included in P
• Monotone a simple polygon such that its
  boundary dP can be split into exactly two
  monotone polyline MC1 and MC2
• Region
• A set of polygons
• E.g., a country and its islands

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2.1.1 Entity-Based Models (5/6)
(a) simple polygon
(b) non-simple polygon
(c) convex polygon
(d) monotone polygon
(f) region
(e) polygon with hole
Figure 2.2 Examples of 2D objects
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2.1.1 Entity-Based Models (6/6)

• Two remarks
• (1) The choice of geometric types is arbitrary
• It depends on the future use of the collection of
  entities
• E.g., airport (scale of interest)
• Point if interested in air links
• Area if interested in the inner organization of
  the airport
• (2) The description of linear and surfacic
  objects is based on line segments
• We use only a linear approximation of entities in
  stead of higher-order polynomials in x and y
• Simplifies the design of spatial databases
• Leads to efficient ways of modeling and querying
  spatial information
• Faithful approximation of segments memory
  space

9
2.1.2 Field-Based Models

• Each point in space
• Associated one or several attribute values(e.g.,
  precipitation, temperature, and pollution),
  defined as continuous functions in x and y
• E.g., altitude above sea level
• An example of function defined over x and y,
  whose result is the value of a variable h for any
  point in the 2D space
• Comparison
• (1) Field-based models
• View space as a continuous field
• (2) Entity-based models
• Identify a set of points (region, line) as an
  entity or object

10
2.2 Representation Modes

• Representation of infinite point sets of the
  Euclidean space
• (1) Tessellation mode
• By approximating the continuous space by a
  discrete one
• E.g., city
• A set of cells that cover the citys interior
• (2) Vector mode and half-plane representation
• By constructing appropriate data structures
• E.g., city
• A list of points describing the boundary of a
  polygon

11
2.2.1 Tessellation (1/5)

• Tessellation mode(discrete model, spatial
  resolution model, tiling, meshes)
• Decomposition of the plane(grid or raster) into
  disjoint cells
• (1) Fixed(regular) tessellation mode
• Use polygonal units of equal size

(a) grid squares (square cells)
(b) hexagonal cells
Figure 2.3 Regular tessellations
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2.2.1 Tessellation (2/5)

• (2) Variable(irregular) tessellation mode
• Handles units of decomposition of various sizes

(a) cadastral zones
(b) Thiessen polygons
Figure 2.4 Irregular tessellations
13
2.2.1 Tessellation (3/5)

• Raster representation
• The rectangular 2D space is partitioned into a
  finite number of elementary cells (NxM) (i.e.,
  rectangular cell pixels)
  
• A pixel has an address in the space, (x,y) where
  x N, y M
• Field-based data in tessellation mode
• represented as a function from space
• (1) Regular tessellation
• In applications that process image data coming
  from remote sensing (satellite image), such as
  weather or pollution forecast
• Domain a finite set of pixels, as a discrete
  one
• Range temperature or elevation
• (2) Irregular tessellation
• In zoning (a typical GIS function) in social,
  demographic, or economic data
• Surface modeling using triangles or
  administrative and political units

14
2.2.1 Tessellation (4/5)

• Entity-based data in tessellation mode
• A spatial object in 2D space is represented by
  the smallest (finite) subset of pixels that
  contains it
• Point as a single pixel
• Polyline, polygon, region a finite number of
  pixels
• E.g., polygon P lt5,12,13,14,17,18,19,20,21,22,26
  ,27,28,29,30,31,35,36,37,38gt

Figure 2.5 Discrete representation of polygon P.
15
2.2.1 Tessellation (5/5)

• Tessellation mode
• Approximate a spatial object by a finite number
  of cells
• Faithful object representation
• Occupy much memory space and operate more time
  consuming

16
2.2.2 Vector Mode (1/6)

• Objects
• Constructed from points and edges as primitives
  (less memory)
• Point represented by its pair of coordinates
• Linear and surface objects represented by
  structures(lists, sets, arrays) on the point
  representation
• Polygon represented by the finite set of its
  vertices
• Entity-based data in vector mode
• Polyline
• Represented by a list of points ltp1, ,pngt, each
  pi being a vertex
• Polygon
• Represented as a list of points
• Closed polyline, i.e., (pn,p1) edge of the
  polygon
• Region
• Represented as a set of polygons

17
2.2.2 Vector Mode (2/6)

• Structure notation
• tuple
• lt gt list
• set
• point xreal, yreal
• polyline ltpointgt
• polygon ltpointgt
• region polygon
• E.g., polygon P
• lt4,4,6,1,3,0,0,2,2,2gt

Figure 2.6 Vector representation of polygon P
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2.2.2 Vector Mode (3/6)

• E.g., polylines (vertex notation)
• L1 lt1,2,3gt
• L2 lt4,5,6,7,8,9,10,11,12gt non-simple
• L3 lt13,14,19gt,lt15,16,19gt,lt17,18,19gt a set
  of polylines

(a) L1
(b) L2
(c) L3
Figure 2.7 Examples of polylines
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2.2.2 Vector Mode (4/6)

• E.g., surfacic objects (polygons and regions)
• R1 lt5,6,12,10,11gt,lt6,7,8,9,10,12gt P2 and P3
• R2 lt13,14,15,16gt P4
• R3 lt1,2,3,4gt P1
• gt concise representation
• compared to the rater mode

Figure 2.8 Examples of polygons
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2.2.2 Vector Mode (5/6)

• No way to distinguish
• A simple polygon from a non-simple one
• A convex polygon from a nonconvex one
• A polygon from a polyline
• A set of adjacent polygons from a set of disjoint
  or intersecting ones
• Field-based data in vector mode
• Digital Elevation Models (DEMs)
• A digital (and thereby finite) representation of
  an abstract modeling of space
• for any natural phenomenon that is a continuous
  function of the 2D space (temperature, pressure,
  moisture, or slope)
• Based on a finite collection of sample values
  (not all points in 2D space)
• gt Values at other points are obtained by
  interpolation (e.g., TIN)

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2.2.2 Vector Mode (6/6)

• TIN (Triangulated Irregular Networks)
• Based on a triangular partition of 2D space
• The elevation value is recorded at each vertex,
  and inferred at any other point P by linear
  interpolation of the three vertices of the
  triangle that contains P

(a) point sample
(b) triangulation
(c) TIN
Figure 2.9 Progression of a triangulated
irregular network (TIN)
22
2.2.3 Half-Plane Representation (1/4)

• Spatial objects
• Defined with a single primitive namely,
  half-planes
• H half-space in the d-dimensional space Rd
• The set of points P(x1,x2,,xd) that satisfy
  a1x1a2x2adxd ad1 0
• P convex d-dimensional polytope
• The intersection of some finite number of closed
  half-spaces
• Face of P
• H n P, where H is part of the half-spaces
  defining P
• Q d-dimensional polyhedron in Rd
• The union of a finite number of polytopes
• Not necessarily convex
• Divide the space into its interior, its boundary,
  and its exterior
• Its components are not necessarily connected and
  may overlap

23
2.2.3 Half-Plane Representation (2/4)

• Convex polygon with n edges (n vertices)
• The intersection of n half-planes delimited by
  lines
• Region
• A union of convex polygons
• Line segment
• The intersection of two half-lines or rays
• Polyline
• The union of some number of line segments
• Point
• A zero-dimensional polytope
• A set of points
• A zero-dimensional polyhedron
• gt regions, polylines, and points polyhedra of
  respective dimensions 2, 1, and 0.

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2.2.3 Half-Plane Representation (3/4)

• Figure 2.10 polygon P1
• The intersection of three half-planes, H1, H2,
  and H3
• Figure 2.11 polygon P2
• The intersection of four half-planes, H4, H5, H6,
  and H7

Figure 2.10 Definition of polygon P1
Figure 2.11 Definition of polygon P2
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2.2.3 Half-Plane Representation (4/4)

• Nonconvex polygon
• Cannot be represented by a 2D polytope that is,
  cannot be built using intersections of
  half-planes only
• Represented by a polyhedron that is the union of
  its adjacent convex pieces
• Figure 2.12 polygon P
• P the geometric union of P1 and P2

Figure 2.12 Building a nonconvex polygon
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2.3 Representing the Geometry of a
Collection of Objects

• Three representations of collections of spatial
  objects
• Spaghetti model
• Network model differ in the expression
  of topological relationships
• Topological model
• Topological relationships
• Relations that are invariant under topological
  transformations
• Preserved when the spatial objects are
  translated, rotated, or scaled in the Euclidean
  plane
• Adjacent, overlapping, disjointness, and
  inclusion
• gt helpful for query evaluation

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2.3.1 Spaghetti Model

• Spaghetti Model
• The geometry of any spatial object of the
  collection is described independently of other
  objects
• No topology is stored, and all topological
  relationships must be computed on demand
• Enable the heterogeneous representation that
  would mix points, polylines, and regions with no
  restrictions
• Advantage
• Simplicity
• Easy input of new objects into the collection
• Drawback
• Lack of explicit information about the
  topological relationships
• Some representation redundancy (boundary of two
  adjacent regions)
• Tend to large data
• Risk of inconsistency

28
2.3.2 Network Model (1/3)

• Network model
• for representing networks in network
  (graph)-based applications such as transportation
  services or utility management (electricity,
  telephone, and so on)
• Topological relationships among points and
  polylines are stored
• Two new concepts
• Node
• A distinguished point that connects a list of
  arcs -gt line connectivity
• Arc
• A polyline that starts at a node and ends at a
  node
• Two types of points
• (1) Node
• Either an arc endpoint(extreme) or an isolated
  point in the plane
• (2) Regular point
• Other line and polygon vertices

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2.3.2 Network Model (2/3)

• Figure 2.13
• ni node
• a1, a2, a3, a4 arcs

Figure 2.13 A network
30
2.3.2 Network Model (3/3)

• Network types
• (1) planar network
• Each edge intersection is recorded as a node even
  though the node does not correspond to a
  geographic object (i.e., entity in real world)
• (2) nonplanar network
• Edges may cross without producing an intersection
• E.g., ground transportation with tunnels and
  passes
• Objects in the network model
• point xreal, yreal
• node point, ltarcgt
• arc node-start, node-end, ltpointgt
• polygon ltpointgt
• region polygon
• gt No information on the relationships between 2D
  objects is stored

31
2.3.3 Topological Model (1/3)

• Topological Model
• Similar to the network model, except that the
  network is planar
• Include a planar subdivision into adjacent
  polygons
• Objects in the topological model
• point xreal, yreal
• node point, ltarcgt
• arc node-start, node-end, left-poly,
  right-poly, ltpointgt
• polygon ltarcgt
• region polygon
• Each arc is shared with a neighbor polygon (no
  redundancy, because each point/line is stored
  only once)

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2.3.3 Topological Model (2/3)

• E.g., Figure 2.14
• P1 lta, b, fgt
• P2 ltc, d, e, fgt
• N1 3, 0, lta, f, egt
• f N1, N2, P1, P2, ltgt

Figure 2.14 Representation of polygons in the
topological model
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2.3.3 Topological Model (3/3)

• Advantage
• The efficient computation of topological queries
• Easy consistency maintenance and updates
• Drawback
• Some spatial objects may have no semantics in a
  real-world application
• (? planar)
• The complexity of the resulting structure may
  slow down some operations (e.g., input of a new
  object)

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2.4 Spatial Data Formats and Exchange
Standard

• Standard types
• (1) de facto (in fact) standard ?? ??
• So dominant that everybody seems to follow it
  like an authorized standard
• E.g., mile
• (2) de jure (by law) standard ?? ??
• Standard authorized by standardization
  organization such as ISO
• E.g., meter
• DXF format
• As a data transfer format between CAD software
  data and become a popular GIS data format
• Comparison criterion between spatial data formats
• The richness of their underlying spatial model

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2.4.1 Overview of Current Spatial Data
Formats (1/2)

• Official organization
• (1) National Institutes of standards
• ANSI (American National Standards Institute)
• AFNOR (Association Française de Normalisation)
• BSI (British Standard Institute)
• (2) International Organization
• ISO (International Organization for
  Standardization)
• DCWIG (Digital Geographic Information Working
  Group)
• de facto standards
• DXF for CAD/CAM applications (from AutoCAD)
• DIGEST for military applications within many
  NATO countries (by DCWIG)
• SDTS used by many U.S. national agencies (by
  USGS)

36
2.4.1 Overview of Current Spatial Data
Formats (2/2)

• Proprietary formats
• France EDIGéO format (by AFNOR)
• Germany ALK/ATKIS format
• Switzerland INTERLIS format
• United kingdom NTF format
• Canada SAIF format
• Korea NGI format
• gt allow the description and transfer of
  raster and vector data
• gt OGCs GML
• Standards for discrete (raster) representation
• GIF (Graphics Interchange Format)
• JPEG (Joint Photographic Experts Group)
• TIFF (Tagged Image File Format) most widely
  used for spatial data
• CGM (Computer Graphic Metafile)
• ASRP (Arc Standard Raster Product)

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2.4.2 The TIGER/Line Data Format (1/7)

• TIGER (Topologically Integrated Geographic
  Encoding and Referencing)
• The name for the system and digital database
  developed at the U.S. Census Bureau
• A practical implementation of the topological
  data model
• TIGER/Line file
• A database of geographic entities such as roads,
  railroads, rivers, lakes, political boundaries,
  and census statistical boundaries
• Contains the location in latitude and longitude,
  the name, the type of feature (object), address
  ranges for most streets, geographic relationships
  to other features, and other related information
• Topological structure of the TIGER database
• Define the location and relationships of streets,
  rivers, railroads, and other features to each
  other and to the numerous geographic entities
• Designed to ensure no duplications of these
  features and areas

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2.4.2 The TIGER/Line Data Format (2/7)

• Spatial objects
• Belong to the Geometry and Topology (GT) class of
  SDTS
• Embody both geometry (coordinate locations and
  shape) and topology
• All spatial objects are mixed in a single layer
  that includes roads, hydrography, railroads,
  boundary lines, and miscellaneous features
• Introduce many spatial objects that do not
  correspond to geographic objects

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2.4.2 The TIGER/Line Data Format (3/7)

• Spatial object types
• (1) Node
• 0-D object that is a topological junction of two
  or more links or chains, or an endpoint of a link
  or chain
• (2) Entity point
• A point used for identifying the location of
  point features such as towers, buildings, or
  places
• (3) Chain (arc)
• A simple polyline described by a start node, an
  end node, and a list of intermediate points
  called shape points
• Intersect each other only at nodes
• Complete chain
• Explicitly references left and right polygons and
  start and end nodes
• Network chain
• Do not reference left and right polygons (above
  network model)
• (4) GT-polygon
• An area described by the list of complete chains
  that form its boundary

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2.4.2 The TIGER/Line Data Format (4/7)

• An example of TIGER object
• GT-polygon 3
• Polygon interior point for computing distances

Figure 2.15 TIGER objects (after TIGER
documentation)
41
2.4.2 The TIGER/Line Data Format (5/7)

• Extracting the representation of spatial objects
  from the TIGER files
• (1) Record type 1
• TLID(TIGER/Line ID), and its start and end nodes
• (2) Record type 2
• The shape points of the chains up to 10 points

Table 2.1 Examples of record type 1 (chains)
Table 2.2 Examples of record type 2 (shape points)
42
2.4.2 The TIGER/Line Data Format (6/7)

• Construction of GT-polygons
• (1) Record type I
• Give the identifiers of the left and right
  polygons (CENID, POLYID)
• (2) Record type 7
• Contains all landmarks, together with their point
  coordinates and some descriptive attributes

Table 2.3 Examples of record type I (link
chains/polygons)
Table 2.4 Examples of record type 7 (landmarks)
43
2.4.2 The TIGER/Line Data Format (7/7)

• Major types of features
• (1) Line features
• Roads
• Railroads
• Hydrography
• Transportation features, power lines, and
  pipelines
• Boundaries
• (2) Landmark features
• Point landmarks (e.g., schools and churches)
• Area landmarks (e.g., parks and cemeteries)
• Office buildings and factories
• (3) Polygon features
• Census statistical areas
• School districts
• Voting districts
• Administrative divisions states, counties,
  county subdivisions
• Blocks

44
2.4.3 Recent Standardization Initiatives (1/5)

• Standardization of exchange formats and spatial
  data models
• for improving interoperability between GISs
• OpenGIS Consortium (OGC)
• Created in 1994 to foster communication among
  GISs in order to ensure interoperability
• Dedicated to the creation and management of an
  industry-wide architecture for interoperable
  geoprocessing
• Technical goals of OGC
• (1) a universal spatio-temporal data and
  process model OGC data model
• Main entity feature which has a type and a
  geometry
• (2) a specification for each of the major
  database languages to implement the OGC data
  model
• (3) a specification for each of the major
  distributed computing environments (DCEs) to
  implement the OGC process model

45
2.4.3 Recent Standardization Initiatives (2/5)

• OGCs technical activities
• (1) the development of an abstract
  specification
• Create and document a conceptual model sufficient
  to allow for the creation of implementation
  specifications
• Essential model
• Establish the conceptual linkage of the software
  to the real world
• Abstract model
• Define the eventual software system in an
  implementation-neutral manner
• (2) the development of an implementation
  specification
• Technical specifications implementing the
  abstract requirements CORBA, DCOM, Java
• (3) the specification revision process
• gt formalism for all models UML

46
2.4.3 Recent Standardization Initiatives (3/5)

• ISO Technical Committee 211 (TC/211) Geographic
  Information/Geomatics
• Global standardization issues related GIS
• Preparing a family of geographic information
  standards in cooperation with other ISO technical
  committees
• Study the standards to specify methods, tools,
  and services for data management and transfer
  between different users, systems, and locations
• Groups of TC/211 committee
• Working group 1 Framework and reference model
• Working group 2 Geospatial data models and
  operators
• Working group 3 Geospatial data administration
• Working group 4 Geospatial services
• Working group 5 Profiles and functional
  standards
• gt since 1997, OGC and ISO seek to converge
  toward a common solution
• (i.e., interoperability in geospatial
  data processing)

47
2.4.3 Recent Standardization Initiatives (4/5)

• Open Geospatial Datastore Interface (OGDI)
• To offer a solution that leverages and
  accelerates standardization efforts
• API that resides between an application and
  various geodata products in order to provide
  standardized geospatial access methods
• A client/server architecture for delivering
  spatial data over the Internet
• To implement a simple feature interface for Java
  in OGDI as soon as OpenGIS issues the
  specifications
• Geodata integration needs of OGDI
• The distribution of geodata products via the
  Internet/Intranet
• Access to data in native format
• The adjustment of coordinate systems and
  cartographic projections
• The retrieval of geometric and alphanumeric data

48
2.4.3 Recent Standardization Initiatives (5/5)

• Current map server
• Usually transfer GIF, JPEG, etc.
• Use HTTP, which is based on a stateless
  connection
• OGDI server
• Use GLPT (Geographic Library Transfer Protocol)
• new Internet protocol for the transfer of
  geospatial data
• A statefull replacement for HTTP