AND GEOMETRY

VORONOI DIAGRAM IN THE LAGUERRE GEOMETRY AND

ITS

APPLICATIONS*

HIROSHI IMAIf, MASAO IRIf ^AND KAZUO MUROTA"

Abstract. Weextendtheconcept of Voronoidiagraminthe ordinaryEuclideangeometry fornpoints totheone intheLaguerregeometryfor n circles inthe plane, where thedistancebetween a circleanda pointis defined by thelengthofthe tangentline, andshow that thereis an O(nlogn)algorithm^{for this} extended case. The Voronoi diagram in the Laguerregeometry may be applied to solving effectivelya numberofgeometrical problemssuch asthoseofdetermining whether^{or not a}pointbelongsto theunion of n circles, of finding the connected components ofn circles,andoffinding thecontourof theunionof n circles. Asinthe casewithordinaryVoronoidiagrams, thealgorithms .proposed here for those problems are optimaltowithin a constantfactor.Someextensions of^theproblem andthealgorithm fromdifferent viewpointsarealsosuggested.

Keywords. Voronoidiagram, computational geometry,Laguerregeometry, computational complexity, divide-and-conquer, Gershgorin’s theorem

Introduction. The Voronoi diagramfor a set of n points inthe Euclideanplane is one ofthemostinterestingand usefulsubjectsin computationalgeometry. Shamos and

Hoey [15]

presentedan algorithm which constructs the Voronoi diagram ⁱⁿ the Euclidean plane in

O(n

log

n)

^time by using the divide-and-conquer technique, and showed many useful applications. Since then, variousgeneralizations of the Voronoi diagram have been considered.

Hwang [6]

^and

Lee

and

Wong [10]

considered the Voronoidiagramsfora setofnpointsunder the Ll-metric, and the

L1-

and Lo-metrics, respectively, andgave

O(n

log

n)

algorithmstocompute them.

Lee

and Drysdale

[9]

studied theVoronoi diagrams fora set of n objects^{such asline} segments ^orcircles, where the distance between a point and an object is defined as the least Euclidean distance from the point to anypoint of the object, and therefore the edges of these Voronoidiagramsare nolongersimple straightlinesegments butmaycontainfragments of parabolic ^orhyperbolic curves. They gave an

O(n(log n) 2)

algorithm toconstruct these diagrams,and Kirkpatrick

[7]

^{reduced its}complexity ^to

O(n

log

n).

Here

weextend the concept of usualVoronoidiagramⁱⁿthe Euclideangeometry for npointstotheonein the

Laguerre

geometry forn circles in theplane,where the distancefrom apointtoacircle is definedby thelengthof thetangentline. Then the edges ofthese extended diagramsaresimple straight line segments whichare easyto manipulate.

We

show that thereis an

O(n

log

n)

algorithm ^{for this}extendedcase.

In

spite of the unusual distance employed here, the Voronoi diagram in the

Laguerre

geometry can beappliedtosolving efficientlyanumber ofgeometric problems concerningcircles.

By

usingthisextendedVoronoidiagram,theproblemofdetermining whetheror not apointbelongstotheunionofgivenn circles canbe solved in

O(log n)

time and

O(n)

space ^with

O(n

log

n)

preprocessing.

We

can also solve the problem offindingthe connected components ofgiven n circlesin

O(n

log

n)

time, which can be applied to a problem in numerical analysis, namely, estimating the region where the eigenvalues of a given ^{matrix lie}

[4].

^{The problem of finding the contourof the}

unionof n circles can also be solved in

O(n

log

n)

time,which canbe appliedtoimage processingand computergraphics.

As

in the caseof theproblemsconnectedwiththe

*Receivedby the editors December 15, 1981,andinfinal revised formAugust20, 1983.

"

^Department of Mathematical Engineering and Instrumentation Physics, Faculty of Engineering,

University ofTokyo,Tokyo,Japan 113.

93

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ordinaryVoronoi

diagram,

the methodsproposed^hereareoptimalto within a constant factor.

Some

further generalizations ofthe problems and the algorithmsfrom different viewpoints arealso suggested.

1.

Laguerre geometry.

Consider the three-dimensional real vector

space R

³

where the distance d(P, Q) betweentwopoints

P= (xl,

yl,

Zl)

and

O (x2,

y2,

z2)

^is

defined by

d2(P,

Q)

(Xl- x2) +(y- y)-(z- z) . ^In

^the

^Laguerre

^geometry

^[1],

a point

(x,

y,

z)

in this space

R

³ ismade to correspond to a directed circle in the Euclidean planewith center

(x, y)

and radius

Izl,

^{the circle}being endowed with the directionofrevolutioncorrespondingtothesignof z. Then the distance between two points in

R

³ corresponds to the length ofthe common tangent of the corresponding twocircles. Hereafter we consider the planewith distanceso defined.

Note

here that, solongasthedistancedL(Ci,

P)

betweenacircle

Ci

^C(Q;

r)

with center

and radius

r

^{and a point}

^{P (x, y)}

^{is concerned,} the direction of the circle has no meaningsince the distance

dL(C, P)

is expressed^as

(1) d2(C,, P) (x- x,) +(y- y,)Z-

r,,2

d(C,

P)

being the length of thetangent segment from

P

to

C

^if

^P

^{is outside of}

C.

Note

that, accordingasapoint

P

lies inthe interiorof,onthe periphery of,orin the exteriorof circle Ci,

d2(C, P)

^isnegative, zero,orpositive, respectively. The locus of the pointsequidistant from twocircles

C

^and

C

^{is a straightline, called}the radical axis of

C

^and

Cj,

which is perpendicularto the lineconnectingthetwocenters of

C

and

Cj.

Iftwo circles intersect,their radicalaxis is theline connectingthetwo points of intersection.Typicaltypes ofradicalaxes are illustrated inFig.1.If the^three,centers

(a) (b} (c}

(Ci

:Cj) (Cj’Ck)

(Ck,C

(d)

FIG. 1. Radical axesandradicalcenters.

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of three circles Ci,

Cj

and

Ck

^{are not on a line, the} three radical axesamong Ci,

Q

and

Ck

^meetata point, whichiscalled the radical center of Ci,Cjand

Ck ^(see

^Fig.1

^(d)).

2. Definitionofthe Voronoidiagraminthe

Laguerre geometry. Suppose

ncircles

Ci C ^{Q; r)}

^Q

^{(x, y))}

^aregivenin

^the

^{plane,wherethe}distance between^acircle

C

^{and apoint}

^P

^{isdefinedby dL(Ci,}

^P)

^{as in 1.Then}the Voronoipolygon

V(C)

for circle

C

^{is definedby}

(2) V(C) {pR21d2L(C,P)<-_d2L(C,P)}.

Note

that theinequality

d2c(C, P)

^_-<

d2c(C, ^P)

^{determines a}half-plane ^so that V(Ci) isconvex.

However,

note also that

V(C)

maybe

empty

^andthat

C

^maynot intersect its polygon

V(C)

when circle

C

^is contained in the union of the other circles. The Voronoi polygonsfor n circles

Ci

^(i--1,...,

n)

partition the wholeplane, whichwe

shall

^{referto as} the Voronoidiagram inthe

Laguerre

geometry

(see

Fig.

2). A

corner ofaVoronoi polygon is called a Voronoipoint, and a

boundary

edge ofthe Voronoi polygon^iscalleda Voronoiedge. Furthermore,acirclewhose correspondingVoronoi polygon ^is nonempty

(empty)

is referredto as a substantial (trivial) circle. In Fig. 2, circle

C3

^{is trivial}and all the others are substantial.

It

isalsoseenthat,inFig. 2, circle

C2

^hasnointersectionwith

V(C2). A

circle that intersects thecorrespondingVoronoi polygon is said to be proper, and a circle whichis not properis called improper. The following is immediate fromthe above definitions.

/v(ce /

7)

FIG.2. Voronoidiagram in theLaguerregeometry.

LEMMA

1. (i)

A

trivial circle is necessarilyimproper.

(ii)

An

improper circleis contained intheunion

of

^{thepropercircles.}

Obviously, if ri 0 for all i, the Voronoi diagramin the

Laguerre

geometry reduces tothatin

the

ordinaryEuclidean geometry.

In

^a Voronoi diagram in the

Laguerre

geometry, ^a Voronoi edge ^is

(part of)

a radical axis and aVoronoi pointis a radical center. ^Sincethe diagram ^isplanar, and Euler’s formula

[5]

still holds, wehave

LEMMA

2. There are

O(n)

Voronoi edgesandpoints in the ^Voronoi diagram in the

Laguerre

geometry

for

^ncircles.

In

the case of the Voronoi diagram in the ordinary Euclidean geometry for n points

P

^(i--1,...,

^n),

the Voronoi polygon

V(P)

^is unbounded iff point

P

^{is on}

the boundary of the convex hull of the n points Pi, but,

for

the Voronoi diagram in

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the

Laguerre

geometry ^{for n circles}

C

with center Q, this statement needs some modification, asinLemma3 below.

In

Fig. 3,thecenterQ2 of

C2

^{lies ontheboundary}

of the convex hull of the centers, but

V(C2)

isempty.

/"

^v(c3

FIG.3. Relationsbetween the convex hull and Voronoipolygons.

LEMMA 3.

In

the Voronoidiagram^inthe

Laguerre

geometry, theVoronoipolygon

V(

Ci) is nonempty andunbounded

if

^thecenterQi

of

^thecircle

Ci

is at a corner

of

^the

convex hull

of

^{the centers Qi,.} Q,. Furthermore,

if

^thecenter

^Qj of

^{acircle Cj is on}

the

boundary of

^{thisconvexhull butnot atacorner, itsVoronoipolygon}

^V( C)

^iseither

unboundedorempty.

If

^thecenterQk

of

^acircle

Ck

^{is notontheboundary}

of

this convex hull, its Voronoipolygon

V( Ck)

is eitherboundedorempty.

Proof.

^Consider the Voronoi diagram in the

Laguerre

geometry ^for n circles Ci(Qi;ri) (Qi=(x,

y);

i=1,...,

n),

wherewe canassume

y

yi (i

])

^{without loss}

of generality. First recall

(cf. (1), (2))

thatapoint

P=(x, y)

belongs to

V(C1)

iff

d2(C1, P) --< d(C, P),

1,

,

n,

(3)

where

(xi-

x)x + (Yi- Y)Y <=

Ri, i=l,...,n,

2_

x- ^{y2 + r21)/2.}

Ri (xZi + y-

r

Next,

note that thecenter

Oa

^of

Ca

^{lies on the} boundary (including the

corners)

of the convex hull of

{ Oli

^1,.

, n}

^itf

(4)

:l(a,

fl)( (0, 0)): a(x,-xa)+(y-yl)<=O,

i=l,...,n,

since all the centers

O

^(i=2,...,

ⁿ⁾

^{lie on one side with respect to a line passing}

through

(xa, Yl).

Suppose

that

V(Ca)

and

(Xo, Yo)

^E

V(Ca).

Then,

V(Ca) f)

isunbounded itt ahalf line startingfrom

(Xo, Yo)

^is contained in

V(Ca),

i.e.,

=l(a,/3)( (0, 0)), VM(>0): (x, y)=(xo+Ma, yo+Mfl)

satisfies

(3),

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whichiseasily seentobeequivalent^to

(4)

above,sothat

V(C1)

^#

)

^isunbounded iff thecenter Q1 of

C1

^liesonthe boundaryofthe convex hull.

When the center Q1 liesat a corner of the convex hull,there exist two distinct pairs of

(a,/3),

say,

(al,/31)

and

(a2,/32)

such that

(4)

holds and that the vectors (xi-xl,

Yi-Yl)

(i=2,...,

n)

^can be represented as linear combinations of

(al,/31)

and

(a2,/32)

^withnonpositive coefficients one ofwhich isstrictly negative.The assertion that

V(C1)#

easilyfollows from the fact that

(3)

holds for

(x, y)=(Ma, Mfl)

^with

asufficientlylarge

M(>0),

where

(a,/3) (al +

a2,/31

+/32).

3. Construction of the Voronoi diagram in the

Laguerre

geometry.

We

shall show that the Voronoi diagram in the

Laguerre

geometry ^can be constructed in

O(n

log

n)

time.Thealgorithm isbasedonthe divide-and-conquer technique,which isvery much like the oneproposed initiallyby Shamosand

Hoey [15]

ⁱⁿconstructing the Voronoi diagram in the ordinary Euclidean geometryfor n points, but which is different insome essential points.

We

shallbrieflyreviewShamosand

Hoey’s

algorithm first, and thenexplain the difference.

Shamos and

Hoey’s

algorithmworksasfollows.

For

agivensetS

{P1,

P2,"

, Pn}

of n distinct points, we sort them lexicographically by ^their

(x,

y)-coordinates ^with the x-coordinate asthe first key.Then, renumbering the indicesof thepoints in that order,wedivideSinto twosubsets

L {P1,

P2," ",

Pin

^and

^R {P[n/2]+l,"

^",

^P,}.

We

recursivelyconstruct theVoronoidiagrams

V(L)

and

V(R)

for points in L and

R,

respectively,and

merge V(L)

and

V(R).

Ifwe canmerge

V(L)

and

V(R)

ⁱⁿ

O(n)

time, theVoronoi diagram

V(S)

^canbe computed in

O(n

log

n)

^time.

By

^virtue of the manner of partitioning S into

L

and

R,

there exists a unique unicursal polygonalline, called the dividing (polygonal) line, such thateverypoint to the left[right] of^{it iscloserto} somepointin

L [R]

than toanypoint in

R [L]. Once

this dividing ^{line is}found, wecan obtain the diagram

V(S)

ⁱⁿ

O(n)

time simply by discarding that part of Voronoiedges in

V(L)

^and

V(R)

which lies, respectively, ^to the right and tothe left of thedividing line.

Hence,

the main problem in merging

V(L)

and

V(R)

is to find the dividing polygonal line in

O(n)

time, which is actually possible by virtue of the following properties

(Lemmas

^{4 and}

5)

^ofthe dividingline.

LEMMA

4. The dividinglineiscomposed

of

^{tworaysextendingtoinfinityandsome}

finite

line segments. Each element

(a

rayor a

segment)

iscontainedin theintersection

of

^{V(Pi) in}

^V(L)

^and

^V(P)

ⁱⁿ

^V(R) for

^somepair

of Pi ^L

^and

P ^R

^{and isthe}

perpendicularbisector

of Pi

^and

P.

^Iq

LEMMA

5. Each

of

^{thetworaysisthe}perpendicularbisector

of

^apair

of

consecutive pointson theboundary

of ^CH(S),

^{theconvex hull}

of

^points

of ^S,

^{such that one is inL}

and theotherinR.

Lemma

4 impliesthat, given ^aray,wecanfindthe dividinglinein

O(n)

^time by tracingitfrom theraytothe otherbymeansofaspecialscanning scheme, i.e.,bythe clockwise and counterclockwise scanning scheme

[9]. Lemma

5, on the other hand, enables us to find a ray in

O(n)

^time from

CH(S),

which, in turn, can be found in

O(n)

time from

CH(L)

and

ell(R) [14], [15].

Most

of the above ideas for theEuclidean Voronoidiagram,with suitable modifica- tions, can be carried overtoobtainan efficientalgorithmforconstructingthe Voronoi diagramin the

Laguerre

geometry for n circles C(Qi;

r)

^asfollows.

The firstproblem ^ishowtopartition thesetof givencircles into twosubsets.

We

partitionthesetSofncircles

C

into two setsLand

R

withrespecttothecoordinates ofthe centers

Q

of

C.

^{That is,} we sort centers Q (i= 1,...,

n)

lexicographically by

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their

(x,

y)-coordinates with the x-coordinate as the first key and divide them into two subsets.Then,the locus of points equidistant (inthe

Laguerre geometry)

from

L

and

R,

^which

we

call thedividingline

(see

Fig.

4),

enjoys the same propertyasinthe Euclideancase, asstated below.

I

FIG.4. Merging the Voronoi diagramsintheLaguerregeometry.

LEMMA

6. The dividing polygonal line is unicursal, consisting

of

^{two rays and}

several

finite

line segments.

Every

point tothe

left

^[right]

of

^{thispolygonallineis closer}

(in thesense

of

^the

Laguerre geometry)

tosome circlein

L [R ]

thantoanycirclein

R ILl.

Proof. ^By

^rotating clockwise the axes, ifnecessary, by a sufficiently small angle, wecan assume that

x

# xj (i# j). Then there exists noVoronoi edge parallel to the x-axis.

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Itsufficestoprovethat, forany t, thereexistsoneandonly^oneintersectionpoint

P (s, t)

ofthedividingline withthe line y t,i.e,, thedividinglineismonotoneand henceunicursal.

By

the assumption that xi# xj (i# j), there exists at least one such point

P= (s, t),

since the point

(-oo, t)

is nearer to

L

than to

R

whereas the point

(+oo, t)

is nearer to

R

than toL.

For such a point

P=(s, t)

let Ci(Qi;ri) be the circle in

L

that is nearest to the point

P

and

Cj(Q; r)

^{the circle in R that} is nearest to

P.

Since

x ^< x,

^{we see} by elementarycalculationthat, forsome e

>

0,

(5) (s+

e,

t) V(C)

^and

(s-

e,

t)

V(Ci).

Suppose

that therewere more than oneintersectionpoint, say,

P1-" ^{(S1, t),} P2-"

(s2, t),. , Pk ^(Sk, t) (Sl <

S2 <"

<

Sk; k>-

2). It

follows from

(5)

that the points

(s, t)

^{with s} Sl

+

e

< s2)

arenearerto

R

thanto

L,

whereas the points^{with s} s2-e

(> sl)

are nearerto

L

thantoR. Therefore,thereexistsone andonlyoneintersection point

P (s, t)

ofthedividing^linewiththe line y t. The

Lemma

then followsbythe continuity arguments. ^[3

Itshould be noted that thepropertyof theaboveLemma6 doesnotholdfor the Voronoi diagram^{for line}segments, i.e., that theremay

appear

an "L-island" inthe R-region and viceversa, whichmakes the problem quite complicated

[9].

Thesecondproblemis to tracethedividinglinefromagiven

ray

totheotherray in linear time. Since a statementsimilar to Lemma4 holds for the Voronoi diagram inthe

Laguerre

geometry, we cansimplyutilize theordinary ^{clockwiseand counter-} clockwise scanning scheme by taking advantageofthe factthatthe Voronoiedgesare straightlines.

The last problem is to find a ray in

O(n)

time. The ray ^is found just as in the ordinary Voronoidiagramfrom the convex hull of thecenters

(cf.

Lemma

5),

provided that thenew hull edge ^{is not} degenerate (i.e.,not collinear).

In

the degenerate

case,

however, the property of

Lemma

5, as it stands, does not necessarily hold, and something moreis needed.

For

example,consider thecase shown inFig.

5(i),

where one ofthenew hull edges^isdegenerate.

Let

be the line of thenew degeneratehull edgeof the convex hull of thecenters.

Even

ifQ4and

Q5

are the closest pair ofcenters on such that

C4

^e

^L

^and

C5

^e

^R,

the radicalaxisof

C4

^and

C5

^doesnot

^appear

^{in the}

Voronoi diagram (Fig. 5(ii)).

In

place of

Lemma

5,we have the following

Lemma

7 inthe

Laguerre

geometry.

LEMMA

7. Consider the line

of

^thenewhull edge (inthedegeneratecase,

edges)

of

^the convex hull

of

^{thecenters Q1, Q,.}

^Let L

^and

R

^besets

of

^{circles in}

^L

^and

R,

respectively, with theircenters on

I.

Let

C, LI ^L

^and

C, R _ ^R

^{be twocircles}

which have the corresponding Voronoiedge

e*

in the Voronoi diagram

V(Lt U R)

in the

Laguerre

geometry

for

^{the subset}

Lt ^I..J Rt of

^{circles. Then,}

^e*,

^{which is the radical}

axis

of C,

^{and Cj,, isa ray}

of

the dividinglineinmerging

V(L)

^and

V(R).

Proof.

^{From the}

^Lemma

^{3, itis obvious that the twocircles} corresponding ^{to a}

ray ot V(L t.J R)

havetheir centers ontheboundary

ot

theconvex hull of Q1,"

, Q,.

Therefore, theedge

e*

isthe only^candidate forthe ray

ot

the dividingline. ^[3

In

order to find the ray of the dividingline in

O(n)

time, we find the Voronoi edge

e*

inthe diagram

V(LI R)

forcirclesin

L

¹³

R

in linear time inthe tollowing way.

Note

that the Voronoiedges of

V(L t.J Rt)

are allparallel.

First, we construct the diagrams

V(L)

and

V(R)

for circles in

L

^{and in}

^Rt,

respectively,from the diagrams

V(L)

and

V(R),

whichcanbe done inlinear time as tollows. Considering ^apart of thediagram

V(L)

far fromthe line l, we see thattwo circles in

L

^{share a Voronoi edgein}

^V(L)

^{iff they share a Voronoi edge in}

^V(L).

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C₇

C₃

C6

C 7

C₃

C6

V(C

I)

^v(C

2)

^V(C

3)

^V(C

6)

^V(C

7)

FIG. 5. Finding a ray in a degenerate case. (i) Degenerate new hull edge (Lt={C1,C2,Ca,C4}, Rt={Cs,C6,C7}).(ii) V(LU R). (iii) V(L) andV(R).(iv) V(Lt)andV(Rt).

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V(RZ)

V(C

5)

V(C

1)

^V(C

2)

^V(C

3)

V(C

6)

^V(C

7)

e*

c

7

V(C

4)

V(L)

FIG.5.(cont.)

Hence,

thediagram

V(Lt)

canbe constructedsimplyby picking^outthe Voronoiedges

(rays)

of pairs

o

circles in

Lt

ⁱⁿ the diagram

V(L). (In

the example of Fig. 5, ^the

V(Lt)

shown in Fig. 5(iv) by broken lines can be obtained by extending that part (consistingof parallel lines) of

V(L)

which is fardown to the bottom inFig. 5(iii).)

A

similarconstruction is valid

or

the diagram

V(R).

Next,

we canfind

e*

from

V(L)

and

V(Rt)

in lineartime as follows. Since all the Voronoiedges

in

^bothdiagrams

V(Lt)

and

V(Rt)

are perpendicular^to l, wecan

merge

the diagrams

V(Lt)

and

V(Rt)

to obtain

V(Lt U RI)

in linear time in a way similartothatinwhichwemergetwosorted listsintoasinglesorted list.

In

themerged diagram ^of

V(Lt),

and

V(Rt),

each region between two neighbouring edges ^{is the} intersection of two Voronoi regions, one in

V(L)

and theother in

V(R). For

each region of the merged diagram, with which is associated a pair (Ci^E

L, CjE R)

^of circles, we examinewhether or not there exists a point equidistant (in ^the

Laguerre geometry) rom Ci and Cj

^withinthe region; ithere exists one, the radical axis

o Ci

and

C

^istheray

^{e*. (In}

^theexample

^ot

Fig. 5(iv),theray

e*,

lyingin theintersection of

V(C3)

in

V(Lt) and V(C6)

in

V(R),

is equidistant from

C3

^and

^6"6.)

^{Since the}

number of thoseregionsin thatdiagram is

O(n),

wecan find

e*,

which isthe ray of the dividing line, in

O(n)

time.

V(LtU Rt)

is ready to obtain from

V(Lt), V(Rt)

and

e*.

Thus,ithas been shown thatthe Voronoidiagram in the

Laguerre

geometry^for n circlescan beconstructed in

O(n log n) time.

4. Applications.

Problem 1. Given n circles in the plane, determine whether a given point

P

is contained in their unionor not.

Oncewehave constructedtheVoronoidiagramin the

Laguerre

geometryforthe given n circles Ci(i=1,...,

n),

we have only to find the Voronoi polygon

V(Cj)

containing

P

and

check

if

P

lies in Cj. If

P

is not in

C,

^{then for any circle}

^C, d(C,

P)->_

d2(Cj, P)>

0,and therefore

P

is notinany circle. Sincewecanconstruct

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the Voronoidiagram in the

Laguerre

geometryin

O(n

log

n)

time, andlocateapoint in a polygonal subdivision of the plane in

O(log n)

time and

O(n)

storage, using

O(n

log

n)

preprocessing

[8], [12],

we cansolve thisproblemcompletely in

O(log n)

time and

O(n)

storagewith

O(n

log

n)

preprocessing.

Problem 2. Partition thesetof n circlesintothe connected components. That is, findtheconnectedcomponents of the intersectiongraphofthe n circles,i.e.thegraph whose vertices are the circles and which hasanedgebetweentwoverticesiff thecircles corresponding tothem intersect in the plane.

This problem arisesinnumerical analysiswhen we estimate theeigenvaluesofa matrixbymeans ofGershgorin’stheorem

[4].

Thoughthe intersectiongraph^canhave

O(n 2)

edges, we can solve this problem in

O(n

log

n)

^{time as} follows withthe help ofthe Voronoidiagraminthe

Laguerre

geometry.

Since animproper circle is contained in theunion ofthe propercircles

(Lemma 1)

and does not affect the connectedness of the other circles, we first consider only propercircles. Forthe connectedness of propercircles, we have:

LEMMA

8.

For

anypair

of

^{propercirclesC and}

^C’

^{inthesameconnectedcomponent,}

thereexists a sequence C C1, C2,"

, Ck ^C’ of

^{propercirclessuchthateverypair}

of

consecutivecircles intersect eachothersothatthey havethe correspondingVoronoiedge.

Proof.

^Considerthe connected component

St

^{whichconsistsof}propercircles and contains

C

and

C’.

Since the union of circles in $I is a connected region and ^is partitioned into

CiN

V(Ci)

(Ci.S1)[i.e., U{CilfiSI}-

U{CiN

V(Ci)ICSI}],

^we

cantake apath within thisconnected region from apoint in C

n V(C)

toa point in

C’N V(C’).

Considering

.a ^{sequence C} CI,

C2,’",

Ck ^C’

^{of circles}

in

the order in which this path

passes through C n V(C)

(Ci

Sz),

we cansee that every pair of consecutive circles in thissequence intersect each other so thatthey have the corre- spondingVoronoi edge. ^[]

We

construct a subgraph G of the intersection graph of the n circles which is guaranteed

by Lemma

8 tocarry ^{the same} information as the intersection graph so far as theconnected components of the propercircles are concerned.

For

each pair of

proper

circles (Ci,

Cj)

having acommonVoronoiedge, weputanedge connecting

Ci

^and

Cj

ⁱⁿ

G

ifthe two circles

C

^and

C

^{haveanonempty}intersection in theplane.

The graph G can be constructed in

O(n)

time since there exist only

O(n)

Voronoi edges.Furthermore, theconnectedcomponents ofGcaneasilybefoundin

O(n)

time.

In

ordertofind which components theimproper circles belongto, we firstmake a listof

all

the impropercircles,amongwhich the trivial circlesarefoundin thecourse of the construction ofthe diagram and thesubstantialbut impropercircles arefound

by

scanning allthe Voronoiedges.

Next,

for each impropercircleCi, wefind a

proper

circlethat intersects

Ci

by locating thecenter Qi of

Ci

in the diagram; if Q

V(C),

then

C

^{is a proper circle} that contains Q, i.e., intersects

C.

^{The set of centers of}

improper circles can be located in the diagram in

O(n

log

n)

time by means of the

simple

algorithmwhichmakes use ofabalancedtree

[11].

^Thus,thetotal timetofind

the

partition of n circles into the connectedcomponents ^is

O(n

log

n).

Thisalgorithm is optimal towithin a constantfactor.

In

fact,we have

LEMMA

9.

Any

algorithm which

finds

the partition

of

^{n circles into theconnected}

componentsmakesatleast

II(

nlog

n)

comparisons under the linear decisiontreemodel Areferee has kindly informed the authors thatthislemma holdstrue notonly under thelinear decision treemodel but also under the more precise algebraic computationtreemodel, basedontherecentresult byBen-Or(see M. Ben-Or, Lowerboundsfor^algebraiccomputation trees,Proc. 15thACM Symposiumon Theory of Computing,Boston,1983,pp. 80-86).

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Proof.

^Thisfollowsimmediatelyfrom the factthat the element-uniqueness prob- lem, i.e., to determine whethergiven, n real numbers are distinct, reduces in ^linear time to the connected-component problem, where the lower bound

ot f(n

log

n)

is known for the element-uniqueness problem under the above model of computation

[3]. D

Problem 3. Find the contour ofthe unionof n given circles in theplane.

This kindofproblemissometimesencountered inimageprocessingandcomputer graphics. First, we construct the Voronoi diagram in the

Laguerre

geometry for n circles and then collect that part of the periphery of each circle

Ci

^{which lies in the}

Voronoi polygon V(Ci) ^{for i=}1,...,n. The validity of this algorithm is obvious.

Concerningthe number of circulararcson the .contour, we have thefollowing.

LEMMA

10. The number

of

^{circulararcson thecontour is}

^O(n).

Proof. ^To

^{distinctpairs of}consecutivearcsof the contour, therecorresponddistinct Voronoi edges (i.e.,radical

axes),

the. number

o

which is

O(n).

This algorithmis optimal for thecontourproblem..

In

fact,we have

LEMMA

11. The complexity

of

finding the contour

of

^{the n circles in theplane is}

f(n

log

n)

under the decision treemodel.

Proof. ^We

^showthat sortingnreal numbersXl,x2,"

,

x,reducesto thisproblem in

O(n)

^time.First, find

x.

^min(xi)and

^x*

^=max

^(x),

^andletR

x*-x.>-O.

^Then,

consider n circles with centers

(x, 0)

and radii

R (see

Fig.

6).

The contour of the union of thesecirclesconsistsof circular arcs, and the order of arcs,accordingtowhich the contourcan be tracedunicursally, gives ^usthe sorted list of n numbers.

FIG. 6. Reductionof^sortingtofinding the^contourof^theunionof^circles.

5. Discussion. Consider the Voronoi diagram in the

Laguerre

geometry for n circles C(Qi;ri) (Qi=(x, yi); i=1,...

,n).

This diagram will remain invariant if

2 2

ri.(i

^1,..’,

n)

arereplacedsimultaneouslyby

r-R

^{withsomeconstant}

^R;

^inother

words,thisdiagramcan be regardedastheVoronoi diagramfor n points Q

(x, y)

inthe plane where,^withsomeconstant

R,

a distance d(Q,

P)

between Q andapoint

P (x, y)

is defined by

d2(Q,,P)=(x-x)2+(y- yi)2-r

2

+R.

On

the other hand, the two-dimensional ^section (with z

=0)

of the Voronoi diagram in the three-dimensional Euclidean

space

for n points

P ^(x,

^{y,zi) (i}

1,...,

n)

^is a kind of Voronoi diagram for n points Qi

(x, y)

(i 1,...,

n),

which we will callthe section diagram

(or,

the generalized ^Dirichlet tessellation

[13]),

^with

the distance d(Q,

P)

^{between Q and a point}

P= (x, y)

defined by

d2(Q,,P) (x x,)2+(y- y)2+

z²

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2 2

Hence,

by setting ^r

R-

z with sufficiently large constant

R,

the algorithm we presented here can be applied to the construction in

O(n

log

n)

^time of the section with the plane z =0 of the Voronoi diagram for n points in the three-dimensional Euclidean space.

More

generally, we can consider the section of the Voronoi diagram in the k-dimensional

space

with the distance d(P,

Pj)

^{between two points,}

P x

^and

Pj x ^{R k,}

^definedby

d

2 ^{P,, P) (x,-} x ^G

^x,

x

where

G

is a k

x

k symmetricmatrix

[13], [16]. We

canapplythealgorithm presented here to such section diagrams even if

G

is not positive definite

(for

example, G

=

diag[1,-1,-1]).

Here,

itshould be noted that the Voronoi diagramin the

Laguerre

geometryitself is thesectionwiththeplane z 00t the Voronoi diagram

or

npoints

P ^(x,

^y,

^z)

in three-dimensional

space

where the square of distance between two points

(xl,

Yl,

Zl)

^and

(x2,

Y2,

Z2)

is defined by

(x1-x2)2+(y1-Y2)2-(zl-z2) 2.

Nevertheless,itwould be worth whiletoconsiderthe Voronoi diagramⁱⁿthe

Laguerre

geometryinconnection withthecirclessince,then,theVoronoiedgesand the Voronoi pointshave the geometrical andphysical meaningsof radicalaxes andradicalcenters, respectively.

Condutlingremarks.

We

have shown that the Voronoidiagramin the

Laguerre

geometrycanbeconstructed in

O(n

log

n)

time, andisusefulforgeometric problems concerningcircles.

Brown [2]

consideredatechniqueofinversionwhich isalso useful forgeometrical problemsfor circles.

In act,

it can beappliedtotheproblemstreated in the presentpaper.

However,

ourapproach^isintrinsicin theplaneand would be of interestin itself.

We

havealsodiscussed the relationbetween the Voronoidiagramin the

Laguerre

geometry and the two-dimensional section

o

the Voronoi diagram in the three-dimensional Euclidean space.

Aeknowletlgment. The authors thank the referees for many helpful comments, without which the paper mighthave been less readable.

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[3] D. P. DOBKINANDR. J. LIPTON, Onthe complexityofcomputations under varyingsetsofprimitives, J.Comput.SystemSci., 18(1979), pp. 86-91.

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Optimal searchinplanarsubdivision,thisJournal, 12(1983),pp. 28-35.

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[11] O.T.LEEANDC. C. YANG,Locationof^{multiplepointsin a}planar subdivision, Inform.Proc. Letters, 9(1979),pp. ^190-193.

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[12] R.J. LIPTONANDR.E.TARJAN,Applicationsofplanarseparatortheorem,Proc.^18th

IEEE

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[13] R. E. MILES, The random divisionof^{space,Suppl.Adv.Appl.}Prob.(1972), pp.243-266.

[14] F. P.PREPARATAANDS.J. HONG, Convexhullsoffinite^setsof^{pointsintwoor}three dimensions, Comm. ACM,20(1977), pp.87-93.

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Downloaded 09/01/14 to 143.107.45.106. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php