AMS 345/CSE 355 (Fall, 2010)
Joe Mitchell
COMPUTATIONAL GEOMETRY
Homework Set # 6 – Solution Notes
(1).
[30 points]
Let
S
be a set of
n
points in the plane in general position (no three are collinear, no four are
cocircular). Let
h
denote the number of points of
S
that are vertices of the convex hull,
CH
(
S
)
.
(a).
Consider the Delaunay diagram,
D
(
S
)
, of the set
S
; since no four points are cocircular, we know that the
Delaunay diagram is a triangulation, with each face (except the face at infinity) being a triangle (and each point of
S
being a vertex of some triangle). As a function of
n
and
h
how many triangles does
D
(
S
)
have? How many Delaunay
edges are there in
D
(
S
)
?
Actually, the fact that we are dealing with a Delaunay triangulation is irrelevant for this question: the combi
natorics depend solely on the input,
T
, being a triangulation. Let
t
denote the number of triangles in
T
. There is
one face at infinity, so, when we view
T
as a planar graph, we know it has
t
+ 1 faces,
t
of which are 3sided and
one of which is
h
sided. Thus, the sum of the degrees of the faces is 3
t
+
h
, which we know is twice the number of
edges; thus, 2
e
= 3
t
+
h
. We also know from Euler’s formula that (
t
+ 1)
−
e
+
n
= 2. Putting
e
= (3
t
+
h
)
/
2 into
this, we get (
t
+ 1)
−
(3
t
+
h
)
/
2 +
n
= 2. We solve for
t
to get
t
= 2
n
−
h
−
2 for the number of triangles. Then,
e
= (3
t
+
h
)
/
2 = 3
n
−
h
−
3 is the number of edges.
(b).
Now we are interested in decomposing the convex hull of
S
into pentagons (5sided polygons,
not
necessarily
convex), such that each point of
S
is a vertex of some pentagon. Such a decomposition is called a “pentagonalization”
of
S
.
(i).
Give an example of a set
S
with

S
 ≥
5
such that
S
does
not
have a pentagonalization.
Justify briefly your
claim.
See below, left. The example has
h
= 4 points on the hull, with 1 point interior to the hull. The one interior
point must be joined to at least two hull vertices in any decomposition of the hull into polygons such that each point
of
S
is a vertex of some polygon. If the interior point is joined to two hull vertices, the hull is split into either a
pentagon and a triangle or into two quadrilaterals; in either case, we do not get a pentagonalization. If the interior
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 Spring '08
 Mitchell,J
 Delaunay triangulation, Delaunay diagram, Delaunay edges

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