COT 5520 Computational Geometry

1
Planar straight line graph A planar straight
line graph (PSLG) is a planar embedding of a
planar graph G (V, E) with 1. each vertex v ?
V mapped to a distinct point in the plane, 2. and
each edge e ? E mapped to a segment between
the points for the endpoint vertices of the
edge such that no two segments intersect, except
at their endpoints.
edge (14) vertex (10) face (6)
Observe that PSLG is defined by mapping a
mathematical object (planar graph) to a geometric
object (PSLG). That mapping introduces the
notion of coordinates or location, which was not
present in the graph (despite its planarity). We
will see later that PSLGs will be useful
objects. For now we focus on a data structure to
represent a PSLG.
2
Doubly connected edge list (DCEL) The DCEL data
structure represents a PSLG. It has one entry
for each edge e in the PSLG. Each entry has 6
fields V1 Origin of the edge V2 Terminus
(destination) of the edge implies an
orientation F1 Face to the left of edge, relative
to V1V2 orientation F2 Face to the right of edge,
relative to V1V2 orientation P1 Index of entry
for first edge encountered after edge V1V2, when
proceeding counterclockwise around V1 P2 Index of
entry for first edge after edge V1V2, when
proceeding counterclockwise around V2
v3
e3
f4
f2
e5
v2
e1
v1
e4
f1
e2
f3
v4
Example (both PSLG and DCEL are partial)
3
2
2
3
F2
13
8
9
4
12
F6
11
1
3
10
F3
F1
F4
6
7
6
9
5
F5
1
4
7
8
5
Edge V1 V2 LeftF RightF PredE
NextE --------------------------------------------
----- 1 1 2 F6 F1 7
13 2 2 3 F6 F2 1
4 3 3 4 F6 F3 2
5 4 3 9 F3 F2 3
12 5 4 6 F5 F3 8
11 6 6 7 F5 F4 5
10 7 1 5 F5 F6
9 8 8 4 5 F6 F5
3 7 9 1 7 F1 F5
1 6 10 7 8 F1 F4
9 12 11 6 9 F4 F3
6 4 12 9 8 F4 F2
11 13 13 2 8 F2 F1
2 10
4
Data structures If the PSLG has N vertices, M
edges and F faces then we know N - M F 2 by
Eulers formula. DCEL can be described by six
arrays V11M, V21M, LeftF1M
Right1M, PredE1M and NextE1M. Since
both the number of faces and edges are bounded
by a linear function of N, we need O(N)
storage for all these arrays. Define array
HV1N with one entry for each vertex entry
HVi denotes the minimum numbered edge that is
incident on vertex vi and is the first row or
edge index in the DCEL where vi is in V1 or V2
column. Thus for our example in the preceding
slide HV(1,1,2,3,7,5,6,10,4. Similarly,
define array HF1F where F M-N2 , with one
entry for each face HFi is the minimum
numbered edge of all the edges that make the
face HFi and is the first row or edge index in
the DCEL where Fii is in LeftF or RightF
column. For our example, HF(1,2,3,6,5,1). Both
HV and HF can be filled in O(N) time each by
scanning DCEL.
5
DCEL operations Procedure EdgesIncident use a
DCEL to report the edges incident to a vertex vj
in a PSLG. The edges incident to vj are given as
indexes to the DCEL entries for those edges in
array A 1 3N-6 since Mlt 3N-6.
1 procedure EdgesIncident(j) 2 begin 3 a
HVj / Get first DCEL entry for vj, a is
index. / 4 a0 a / Save starting index.
/ 5 A1 a 6 i 2 / i is index for A
/ 7 if (V1a j) then / vertex j is
the origin / 8 a PredEa / Go on to next
incident edge. / 9 else / vertex j is the
destination vertex of edge a / 10 a
NextEa / Go on to next incident edge.
/ 11 endif 12 while (a ? a0) do / Back to
starting edge? / 13 Ai a 14 if (V1a
j) then 15 a PredEa / Go on to next
incident edge. / 16 else 17 a NextEa /
Go on to next incident edge. / 18 endif 19 i
i 1 20 endwhile 21 end
V2a
NextEa
a
V1a
PredEa
6

• If we make the following changes, it will become
  a procedure
• Faceincidentj giving all the edges that
  constitute a face
• 3 a HFj
• And sustitute HV by HF and V1 by F1.
• A will have a size A1N.
• Both EdgesIncident and Faceincident requires
  time proportional
• to the number of incident edges reported.
• How does that relate to N, the number of vertices
  in the PSLG?
• We have these facts about planar graphs (and thus
  PSLGs)
• (1) v - e f 2 Eulers formula
• (2) e ? 3v - 6
• (3) f ? 2/3e