Internat. J. Math. & Math. Sci.
VOL. 14 NO. 4 (1991) 817-820
817
A NON-UNIQUENESS THEOREM IN THE THEORY OF VORONOI SETS
M. JONES
Department of Pure Mathematics
Queen’s University
Belfast BT7 INN
Northern Ireland
(Received March 14, 1990)
It is shown that two distinct, bounded, open subsets of
ABSTRACT.
2
may possess the
same Voro1oi set.
,KEY WORDS AND PHRASES.
51M15, 51KO5.
1980 AMS SUBJECT CLASSIFICATION CODES.
INTRODUCTION
1.
Let
be a finite collection of non-empty, bounded, open and simply
2
I
i
n and D. n D.
which satisfy D
D # D
].
O
i
i
{Di}o<i_<n
connected subsets of IR
DO,
,
n
I < i
<
-<- n.
and connected subset of IR
domain D
2
Vor() of
,
For any (x,y)
\
u
i=l
Di,
is a non-empty, bounded, open
Di,
n
u D..
i=O
i E i
n.)
(Loosely speaking,
is a
The the following definition of
is taken from [I].
define Near(x,y) as the set of points in
("Closest to" is, of course,
closest to (x,y).
defi,ed in terms of ordinary Euclidean distance in the
is closed, Near(x,y) is always non-empty.
Since
The Vul%
O
with boundary
containing "obstacles"
O
the Voronoi diagram
plane.)
D
Then if we define
d
Vor() of
{(x,y)
is then defined to be the set of points
Near(x,y) contains more than one point}.
Vor() is used in [I] in connection with motion planning problems.
Clearly given the sets {Di} Vor() is unique. However, here we take the
from a given
opposite point of view and consider the construction of the sets {D.}
I
Voronoi diagram.
A preliminary question that one might ask is: could it be possible for two
collections {D.} and {D} to have the same Voronoi diagrams?
I
the answer is yes: for O < e
I let
DOe
{(x,y)
x+y
DI
((x,y)
2 2
x +y
< (l+e)
2}
2
(i-) }.
and
It is easy to see that
818
H. JONES
Then if
value of
e DO\I,
E
Vor(e)
is the unit circle, centre the origin, whatever the
night be.
D,
A more subtle question is the following: Suppose D
then is it possible
O
for two different collections {D.} and {D} to have the same Voronoi diagram?
i
i
Informally, what we are asking is whether, given a fixed domain
to arrange two different sets of obstacles within
same Voronoi diagram.)
2.
DO,
DO,
it is possible
both of which produce tile
We show the answer is again in the affirmative.
THE EXAMPLE
Let
D
D
Then
-
D
O
{(x,y)[[x[
I
{(x,y)[
2
[y[
4}
<
4,
Ix
<
3, I < y < 3}
{(x,y)[ [x
<
3, -3 < y < -I}.
and Vor(.) (where
Do\D I
u
D2)
that Vor([) contains the line segment
<
are depicted in Figure i.
{(x,O)[ Ix
Note in particular
< I}.
/
")"-3,3)
(3,3""(
(-3,1)
D
\
(3,1)
1
/
(o o)
/
D
2
D
Figure I
Vor() is denoted by the dashed line
/
\
/
(4,4)
NON-UNIQUENESS THEOREM IN THE THEORY OF VORONOI SETS
We odify D
Let C
Vor()
1
819
and D^ as follows.
2 2
2} and put D
x +y
{(x,y)
I
Vor(’), (see Figure 2).
(-1,6)
/
DI\C
D
D2\C
2
(o,o)
Then if
DO\
’
u
D--
(1,o)
/
/
\
D
Figure 2
Vor(’) is denoted by the ashed line
To see that the Voronoi diagrams of
and
’
that it suffices to consider those points (x,y) in
are indeed the same first note
’
for which
Ixl
-< I and
E ’, Near(x,y) will be unchanged by the
the
To begin with, consider those points
2.
I
(x,y)
if
that
clear
It
is
triangle whose vertices are (-i,0), (O,O) and (-i,i).
{(-i,I)} and so (x,y) 4 ’. The same conclusion is
is such a point then Near(x,y)
IYl
<
/2 since for any other (x,y)
modifications made to D
and D
’
which lie on the straight lines joining (-I,I) to (-i,0)
and (-i,I) to (O,O), (excluding the endpoints of those lines). Next consider the
{(-i,i), (-I,-I)} and
points (x,O) where -i < x < O. For such a point Near(x,O)
consider those
Now
Vor(’).
so (x,O) E Vor(’). It is also clear that (O,O)
true for the points in
points within the sector of C which has vertices (0,0), (-i,I) and (0,/2). If
(x,y) is such a point then it is easy to see that Near(x,y) consists of the single
point obtained by projecting the straight line joining (O,0) to (x,y) until it
intersects D I. The same conclusion is true for the points on the straight line
between (0,O) and (0,/2) (excluding the endpoints of course). The results for
M. JONES
820
the remaining points in
Vor()
R’ follow immediately from the symmetry of R’.
Hence
Vor(’).
D
are not convex.
A possible weakness of this example is that the sets D and
1
The answer to the same question as that posed in I but with the additional
and {D] be convex would appear to be unknown.
hypothesis that all the sets in {D.}
1
REFERENCES
I.
O’DUNLAING,
a
disc,
tJ.
C. and YAP, C.K., A ’retraction’method for planning the motion of
of Algorithms:, 2.8 (i985), iO4-II.