Spatial Information Systems SIS

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Spatial Information Systems (SIS) COMP
30110 Plane Subdivisions
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Overlayed sets of entities

• If we consider overlayed sets of entities only
  disjoint and meet relations are possible between
  two polygons
• Overlayed sets of entities correspond to plane
  graphs in which we consider not only nodes (also
  called vertices) and edges but also the polygons
  (also called faces) bounded by closed cycles of
  edges

n1
e1 (n1,n2) e2
n8
n2
n9
f1
f2
n3
n10
n7
n4
n5
n6
n11
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Definitions

• Graph G pair (V,E) with V set of vertices and E
  set of pairs of vertices. Edges can be drawn in a
  given space by representing each vertex as a
  point and each edge as a (not necessarily
  straight) line segment joining two points
• Planar graph graph that can be drawn in the
  Euclidean plane in such a way that its edges do
  not intersect each other, except at their
  endpoints
• The embedding of a planar graph in the Euclidean
  plane is called a plane graph
• A planar graph can be drawn in several different
  ways corresponding to different locations of the
  vertices (i.e. obtaining different plane graphs)

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Examples
Planar graph and three possible embeddings in
the plane
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Definitions

• A straight-line plane graph is a connected plane
  graph where every edge is a straight line segment
• A straigth line plane graph defines a partition
  of the plane into a collection of simply
  connected polygonal regions (i.e. regions without
  holes) called faces
• Such a partition is called a plane subdivision

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Plane subdivisions in GIS examples

• Thematic maps
• Example
• land use
• vegetation layer
• How do we represent them?

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Plane subdivisions

• Studied in the field of Computational Geometry
  Preparata and Shamos 1985
• Representations defined in this context have been
  used in GIS
• Entities in a plane subdivision vertices, edges
  and faces
• Euler formula in a plane subdivision,
• n e f 1 n vertices
• e edges
• f (internal) faces
• It can also be shown that e and f are both
  linear in the number n of vertices. Therefore the
  space complexity for a plane subdivision is O(n)

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Relations in a plane subdivision

• With three sets of entities, we can define nine
  ordered relations
• Vertex-based
• Edge-based
• Face-based

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Vertex-based relations

• VE (vertex-edge) a vertex P is associated with
  the list of edges having P as an endpoint (edge
  incident in P), sorted in counter-clockwise order
• VV (vertex-vertex) a vertex P is associated with
  the list of the endpoints (different from P) of
  the edges having P as an endpoint, sorted in
  counter-clockwise order
• VF (vertex-face) a vertex P is associated with
  the list of the faces having P as a vertex,
  sorted in counter-clockwise order

VE(P) e1,e2,e3 VV(P) P1,P2,P3 VF(P)
f1,f2,f3
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Face-based relations

• FE (face-edge) face f is associated with the
  list of edges on its boundary, sorted in
  counter-clockwise order
• FV (face-vertex) face f is associated with the
  list of vertices on its boundary, sorted in
  counter-clockwise order
• FF (face-face) face f is associated with the
  list of faces adjacent to f along an edge, sorted
  in counter-clockwise order

FE(f) e1,e2,e3,e4 FV(f) P1,P2,P3,P4 FF(f)
f1,f2,f3,f4
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Edge-based relations

• EV (edge-vertex) an edge e is associated with
  the pair of its endpoints
• EF (edge-face) an edge e is associated with the
  pair of faces having e on their boundary
• if EV(e)(Pi , Pj), then EF(e)(fi , fj ), where
  fi and fj lie on the left and on the right with
  respect to edge e oriented from Pi to Pj

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Edge-based relations (cont.d)

• EE (edge-edge) an edge e is associated with a
  pair of edges, each incident in one endpoint of e
• if EV(e)(Pi , Pj), then EE(e)(ei , ej ), where
  ei is the first edge encountered after e moving
  counter-clockwise around Pi and ej is the first
  edge encountered after e moving counter-clockwise
  around Pj
• Note that we do not consider all edges incident
  in endpoints of e but only 2 of them!!

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Plane subdivisions particular cases

• Some plane subdivision have particular properties
• Examples include
• - Triangular plane subdivisions (triangulations)
• - Voronoi diagrams
• Both these types of subdivisions have been
  studied in computational geometry and widely used
  in GIS

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Triangulations

• Plane subdivisions with triangular faces
• Commonly used as a basis for digital terrain
  models based on a given set of sample points
  (more later)
• In particular, Delaunay triangulations have very
  good properties

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Delaunay Triangulations

• Intuitively given a set V of points, among all
  the triangulations that can be generated with the
  points of V, the Delaunay triangulation is the
  one in which triangles are as much equiangular as
  possible
• In other words, Delaunay triangulations tend to
  avoid long and thin triangles important for
  numerical problems

t
P
Does P lie inside t or on its boundary?
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Empty circle property

• Let t be a triangulation of a set of points V a
  triangle t of t is said to satisfy the empty
  circle property if the circle circumscribing t
  does not contain any points of V in its interior.
  t is called a Delaunay triangle

t does not satisfy the empty circle property
t satisfies the empty circle property
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Delaunay Triangulations (cont.d)

• A triangulation t of a set of points V is a
  Delaunay triangulation if each triangle of t
  satisfies the empty circle property
• Another definition
• A triangulation t of V is Delaunay triangulation
  if each internal edge e is locally optimal (i.e.,
  by exchanging it with the other diagonal e of
  the quadrilateral composed of the two triangles
  sharing e, the minimum internal angle becomes
  smaller)
• The Delaunay triangulation of V is unique if
• no four points of V are cocircular

P2
e
e
P1
P3
P4
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Voronoi Diagrams

• Given a set V of points in the plane, the Voronoi
  Diagram for V is the partition of the plane into
  polygons such that each polygon contains one
  point p of V and is composed of all points in the
  plane that are closer to p than to any other
  point of V.

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Voronoi Diagrams (cont.d)

• Property the straight-line dual of the Voronoi
  diagram of V is a triangulation of V
• Dual obtained by replacing each polygon with a
  point and each point with a polygon. Connect all
  pairs of points contained in Voronoi cells that
  share an edge

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Voronoi Diagrams (cont.d)

• Voronoi diagrams are used as underlying
  structures to solve proximity problems (queries)
• Nearest neighbour (what is the point of V nearest
  to P?)
• K-nearest neighbours (what are the k points of V
  nearest to P?)
• Etc.

P