Transcript Document

Spatial Information Systems (SIS)
COMP 30110
Plane Subdivisions
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
f1
n3
n7
n6
n5
n4
n9
f2
n10
n11
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)
Examples
Planar graph and
three possible embeddings in the plane
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2
4
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2
3
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2
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2
4
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3
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
Plane subdivisions in GIS: examples
• Thematic maps:
• Example:
• land use
• vegetation layer
residential
barley
wheat
rural
oats corn
industrial
• How do we represent them?
forest
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)
Relations in a plane subdivision
• With three sets of entities, we can define nine ordered relations:
• Vertex-based
EE
• Edge-based
• Face-based
E
EF
EV
VE
VV
V
FE
VF
FV
F
FF
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
P2
e2
f3
f2
P
P1
e1
VE(P) = [e1,e2,e3]
f1
VV(P) = [P1,P2,P3]
e3
P3
VF(P) = [f1,f2,f3]
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
f1
P2 e1 P1
f2
e2
P3
f
e3
f3
e4
P4
FE(f) = [e1,e2,e3,e4]
f4
FV(f) = [P1,P2,P3,P4]
FF(f) = [f1,f2,f3,f4]
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
Pj
fi
Pi
e
fj
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 counterclockwise around Pj
ei
Pi
Pj
fi
e
ej
fj
• Note that we do not consider all edges incident in endpoints of e
but only 2 of them!!
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
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
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
Does P lie inside t or on its boundary?
P
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
P2
P2
t
t’
P1
P3
P3
P4
t does not satisfy the empty
circle property
P4
t’ satisfies the empty
circle property
P1
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)
P2
e’
P3
•
The Delaunay triangulation of V is unique if
no four points of V are cocircular
P4
e
P1
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.
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
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