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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.