a p p o r t
d e r e c h e r c h e
TH ` EMES 2 et 3
The expected number of 3D visibility events is linear
Olivier Devillers — Vida Dujmovi´c — Hazel Everett — Xavier Goaoc — Sylvain Lazard
— Hyeon-Suk Na — Sylvain Petitjean
N° 4671
D´ecembre 2002
OlivierDevillers
,Vida Dujmovic y
, Hazel Everett z
, Xavier Goaoc z
,Sylvain
Lazard z
, Hyeon-Suk Na x
, SylvainPetitjean z
Themes2et3|Genielogiciel
etcalcul symbolique |Interactionhomme-machine,
images,donnees,connaissances
ProjetsIsaet Prisme
Rapport derecherche n°4671 |Decembre2002|42pages
Abstract: Inthis paper,weshowthat,amongst nuniformlydistributed unit ballsin R 3
,
theexpectednumberofmaximalnon-occludedlinesegmentstangenttofourballsislinear,
considerablyimprovingthepreviouslyknownupperbound. Usingourtechniques weshow
alinearboundontheexpectedsizeofthevisibilitycomplex,adatastructureencodingthe
visibility information of a scene, providing evidence that the storage requirement for this
datastructureisnotnecessarilyprohibitive.
Ourresults generalizein various directions. We showthat thelinearbound ontheex-
pectednumberofmaximalnon-occludedlinesegmentsthatarenottooclosetotheboundary
of thescene and tangent to four unit balls extends to balls of various but bounded radii,
to polyhedra of bounded aspect ratio,and even tonon-fat 3Dobjectssuch aspolygonsof
bounded aspectratio. Wealsoprovethat ourresultsextendto otherdistributions such as
the Poisson distribution. Finally, weindicate howour probabilisticanalysis providesnew
insight on the expected size of other global visibility data structures, notably the aspect
graph.
Key-words: computationalgeometry,3Dvisibility,visibilitycomplex,visualevents,prob-
abilisticanalysis,expectedcomplexity
INRIA Sophia-Antipolis. Olivier.Devillers@inria.fr. http://www-sop.inria.fr/prisme/. Partiallysup-
portedbythe ISTProgrammeofthe EUasa Shared-costRTD(FETOpen) ProjectunderContractNo
IST-2000-26473(ECG-EectiveComputationalGeometryforCurvesandSurfaces).
y
SchoolofComputerScience,McGillUniversity;vida@cs.mcgill.ca. ResearchsupportedbyFCAR.
z
LORIA - INRIA Lorraine, CNRS, Univ. Nancy 2. feverett, goaoc, lazard, petitjeag@loria.fr.
http://www.loria.fr/feverett,goaoc,lazard, petitjeag/. Researchsupportedbythe McGill-ISAcollabo-
rativeINRIAproject.
x
HongKongUniversityofScienceandTechnology. hsnaa@cs.ust.hk. Thisauthor'sresearchwasdone
duringapost-doctoraltenureatLORIA-INRIALorraine.
Resume : Nous montrons que, pour une scene de R composee de n boules de rayon 1
uniformementdistribuees,lenombremoyendesegmentsdedroitesnonoccultes,maximaux
et tangents a quatre boules est lineaire. Nous montrons que la borne lineaire s'applique
egalementalataillemoyenneducomplexedevisibilite,unestructurededonneesencodant
l'informationdevisibilitedelascene,donnantapenserquel'espacememoirenecessaireau
stockagedecettestructurededonneesn'estpasnecessairementprohibitif.
Nousgeneralisonscesresultatsdansdierentesdirections. Nousmontrons quelaborne
lineairesur le nombre moyende segments de droites non occultes,maximaux et tangents
aquatre boulesunitessuÆsammenteloignees du bord de lascenes'applique egalement a
desscenescomposeesde boulesderayondierentmaisborne,de polyedres de proportion
bornee et aussi ades objetsnon epaiscomme despolygonesde proportion bornee. Nous
prouvonsegalementquenosresultatssegeneralisentad'autresdistributions,notammentla
distribution dePoisson. Enn, nousmontronscommentnosresultatsouvrentdenouvelles
perspectivesconcernantl'etudedelataillemoyenned'autresstructuresdevisibiliteglobale
commelegraphed'aspects.
Mots-cles : geometrie algorithmique, visibilite3D, complexe de visibilite, evenements
visuels,analyseprobabiliste,complexiteenmoyenne
1 Introduction
Visibilitycomputations arecentralin computergraphicsapplications. Computingthelim-
itsoftheumbraandpenumbracastbyanarealightsource,identifying thesetof blockers
between any two polygons and determining the view from a given point are examples of
visibility queriesthat are essentialfor the realisticrendering of 3D scenes. Inglobal illu-
mination algorithms, where theowof lightin ascene is simulated accordingto thelaws
ofgeometricaloptics,visibilitycomputationsareexcessivelycostly. Infact,morethanhalf
of the overall computation time can routinely be spent on visibility queries in radiosity
simulations[12].
Oneapproachtospeedinguprenderingistostoreglobalvisibilityinformationinadata
structurewhichcanthenbeeÆcientlyqueried. Thevisibilitycomplex,apartitionoftheset
ofmaximal freelinesegments, hasbeenproposedasaunieddata structureencodingthe
visibilityinformation ofascene [21] andhasbeenused forrenderingpurposes[10]. Other
relateddatastructuresincludePellegrini'sray-shootingstructure[18],theaspectgraph[20]
andthevisualhull[13];see[7]forarecentsurvey.
One problem with these typesof data structures which may prevent their application
in practiceis their potentiallyenormoussize; the size of thevisibilitycomplex of aset of
n triangles in R 3
is (n 4
) in the worst case [10], which is prohibitive even for scenes of
relatively modest size. Worst-case examples are somewhat articial and indeed Durand,
Drettakisand Puech [8] provideempirical evidence indicatingthat these worst-caseupper
boundsarelargelypessimisticinpracticalsituations;theyobserveaquadraticgrowthrate,
albeitforrathersmallscenes. In2D,whiletheworst-casecomplexityofthevisibilitycomplex
isquadratic,experimental resultsstronglysuggestthatthesizeofthevisibilitycomplexof
asceneconsistingofscatteredtrianglesislinear[3].
Ourgoalistoprovidetheoreticalevidencetosupporttheseobservations. Tothisendwe
investigatetheexpectedsizeofthevisibilitycomplex,orequivalently,theexpectednumberof
visibilityevents,occurringinscenesinR 3
. Avisibilityeventisacombinatorialchangeinthe
viewofamovingobserver;suchaneventoccurswhentheviewingdirectionbecomestangent
tosomeobjects. Forsetsof convexobjectsingeneralpositionin R 3
,theviewing direction
canbetangenttoat mostfourobjects. Visibilityeventsthus correspondto maximalnon-
occludedline segmentstangentto at mostfourobjects; combinatoriallydierentvisibility
eventscorrespondto thefacesof thevisibilitycomplex.
Inthispaperweprovethattheexpectednumberofmaximalnon-occludedlinesegments
tangenttofourballs,amongstnuniformlydistributedunitballsinR 3
,islinear,considerably
improving the result of Durand et al. [10] who proved a bound of O(n 8=3
) for the same
model. Theintuition behindourproof is that, given aline segmenttangentto four balls,
theprobabilitythatthissegmentisnotoccludedbyanyotherballistheprobabilitythata
cylinder-likevolumeofradius1aboutthesegmentisfreefromthecentersoftheotherballs.
Thisprobabilitydecaysroughlyexponentiallyfastwith thelengthofthesegment,yielding
theresult. Usingour techniques wethen show alinearbound onthe expected size ofthe
visibility complexofn uniformlydistributed unit ballsin R 3
. A simplecomputation then
providesuswiththesameresultforthePoisson distribution.
Worst-case Expected
possiblyoccludedlinesamongstunit balls (n 4
) O(n
8
3
)[10]
freelinesamongstunitballs (n 2
)[1,?],O(n 3+
)[1] (n)[?]
freelines amongstdisjointhomotheticpolytopes (n 3
)[2] ?
freesegmentsamongstunit balls (n 2
)[1,?],O(n 4
) (n)[?]
freesegmentsamongstarbitrarysizedballs (n 3
)[5],O(n 4
) ?
visibilitycomplexofunitballs (n 2
)[?],O(n 4
) (n)[?]
Table 1. Knownbounds on the complexityof the set of lines, freelines ormaximalfree linesegments
tangentto4amongstnobjects.Theexpectedcomplexitiesarecalculatedfortheuniformdistribution.The
resultsreferencedby?areestablishedinthispaper.
Our results generalize in the following ways. We show that the linear bound on the
expected numberof visibility eventsoccurring not too close to theboundaryof the scene
also appliesto balls of various but bounded radii, to polyhedral objectsenclosed between
twoconcentricballsof xedradius,andevento non-fatobjectssuchaspolygons,enclosed
between two concentric circles of xed radius, whose centers and normals are uniformly
distributed.
Of courseobjectsingraphics scenesareseldomdistributed uniformlyoraccordingto a
Poissonpointprocess. Wechosethismodelbecauseitallowstractableproofsoftheoretical
results. Thisisimportantinacontextwheretherearefewrigorousresultseithertheoretical
orexperimental. The same model has also been used to study theaverage complexity of
rayshooting [23]and occlusionculling for2D urbanscenes[16]. Noothermodel hasbeen
widelyacceptedbythegraphics communityand, in fact,generating randomscenesusable
fortestingalgorithmsisamajorproblem.
Previousresultsonthistopicincludethosethatboundthenumberoflinesandthenum-
beroffree(i.e.,non-occluded)linesamongstdierentsetsofobjects. Theyaresummarized
in Table 1. Agarwal, Aronov and Sharir [1] showed an upper bound of O(n 3+
) on the
complexity ofthe space ofline transversals ofn balls bystudying the lowerenvelopeof a
setoffunctions. Astudyoftheupperenvelopeofthesamesetoffunctionsyieldsthesame
upper bound onthe numberof free lines tangent to four balls [5]. Agarwalet al. [1] also
showedalowerboundonthecomplexityofthespaceoflinetransversalsofnballsof(n 3
)
for arbitrarily sizedballs and (n 2
) for unit sized balls. The construction also showsthe
sameboundsforthenumberoflinestangenttofourballs. DeBerg,EverettandGuibas[2]
showeda(n 3
)lowerbound on thenumberoffreelines (andthus freesegments) tangent
tofouramongstndisjointhomotheticconvexpolyhedra. Recently,DevillersandRamos[5]
presentedasimple(n 3
)lowerboundonthenumberoffreesegmentstangentto4amongst
narbitrarilysizedballs,whichalsoholdsfornon-intersectingballs. Wealsopresentasimple
(n 2
)lowerboundonthenumberoffreesegmentstangentto4amongstnunit balls.
In the next section we carefully dene the problem and state our main results. In
Section3andSection4weprovetheexpectedupperandlowerlinearboundsonthenumber
of freesegmentstangentto four balls. In Section 5weextend this resultto the visibility
complex. Wepresentin Section6a(n 2
)worst-caselowerbound. InSection7wediscuss
extensionsofourresultstosomeothermodels. Weconcludein Section8.
2 Our model and results
Werstdescribeourobjectsandtheirdistribution. Letn2N andbeapositiveconstant.
AsamplesceneconsistsofnunitradiusballsB
1
;:::;B
n
whosecentersp
1
;:::;p
n
areinde-
pendentlychosenfromtheuniformdistributionoverauniversalballU ofradiusRcentered
at O. Since wedistribute thecenters p
i
overU, theballsB
i
mayintersect each otherand
arecontainedintheball,denotedU +
,whoseradiusisR+1andwhosecenteristhatofU.
WedenetheradiusRoftheuniversalballU tobeafunction ofnsatisfying
R 3
=n=: (1)
The constant reects the density of the balls in the sense that the expected number
of centers lying in any given solid of volume V in the universe is 3
4
V. (The model is
interesting onlyif n is asymptotically proportional to R 3
. Indeed, if n
R 3
tends to innity
when n tends to innity, then the universe gets entirelylled up with balls and visibility
eventsonlyoccurinU +
nU. Conversely,if n
R 3
tends tozerowhenntendstoinnity,then
theballsgetscattered sofarapartthat theprobabilitythatanyfour(or three)balls have
acommontangentgoesto zero.)
We now dene the visibility complex of a set of objects [21]. A free or non-occluded
segmentisalinesegmentthatdoesnotintersecttheinteriorofanyobject. Afreesegment
ismaximalifitisnotproperlycontainedinanotherone. Thus,theendpointsofamaximal
freesegmentareeither onanobjectorat innity. Wesaythattwomaximal freesegments
are similar if their endpoints lie on thesame objects(possibly at innity). The visibility
complexofacollectionofobjectsisroughlydenedasthepartitionofthespaceofmaximal
free segments into connected components of similar segments 1
. Its faces have dimension
between0 and4; when the objectsare in adequategeneralposition,ak-dimensional face
corresponds to a connectedset of similar maximal non-occluded line segmentstangent to
4 kobjects.
Inordertoboundthetotalnumberoffacesofthevisibilitycomplex,werstboundthe
numberof0-faces. Todothis,wecounttheT4-segments,whicharethefreesegmentstangent
to4ballswithendpointsontwoofthoseballs. Sincethereis aone-to-one correspondence
between 0-faces and T4-segmentswhen the objectsare in adequate generalposition, this
yieldsaboundontheexpectednumberofverticesofthevisibilitycomplex. Notethatsince
theballsarecontainedin U +
,theT4-segmentsarealsocontainedinU +
.
Ourmainresultisthefollowing.
1
Formally, weconsider the spaceof free segmentsquotiented bythe equivalencerelation that isthe
transitiveandreexiveclosureoftheinclusion. Inotherwords,twofreesegmentsareidentiediftheyare
bothcontainedinthesamemaximalfreesegment. Thisallowsthecellsofthepartitiontobeconnected.
Theorem1 TheexpectednumberofT4-segmentsamongstnuniformlydistributedunitballs
is(n).
Weextendthisresultto thehigherdimensionalfaces ofthecomplex.
Theorem2 Theexpectedsizeofthe visibilitycomplexof nuniformlydistributedunitballs
is(n).
Wealsopresentan(n 2
)worst-caselowerboundonthenumberofT4-segmentsamongst
n unit balls in R 3
(see Proposition 27). Infact the lower bound holds for the numberof
k-facesofthevisibilitycomplex,forallk between0and4.
3 The expected number of T4-segments is at most lin-
ear
Thegeneralidea behind theproof ofthe upperbound ofTheorem 1is thefollowing. For
anyorderedchoiceoffourballs,weboundfromabovetheprobabilitythatalineistangent
tothese ballsinthegivenorderandisnotoccludedin betweenitscontactpointswiththe
balls. Thenwesumtheseprobabilitiesoverallorderedquadruplesofballsandallpotential
tangentlinesto theseballs.
Foranytwopointspandq,andpositiverealnumber,letH (p;q;) denotetheunion
of allthe balls of radius centered on theline segmentpq (see Figure 1). Werst show
thatalineistangenttofourballsB
i ,B
j ,B
k andB
l
inthat orderonlyifp
j andp
k arein
H(p
i
;p
l
;2). Thus thevolumeof H( p
i
;p
l
;2)\U givesanupperbound onthe probability
thatalinetangenttothefourballs, inthegivenorder,exists.
Wenextshowthat asegmenttangenttofourballsB
i ,B
j ,B
k andB
l
in thatorder,at
pointst
i ,t
j ,t
k andt
l
,respectively,isnotoccludedifandonlyifthecentersofallremaining
ballsare outsideor ontheboundaryofH(t
i
;t
l
;1). Thevolumeof UnH(t
i
;t
l
;1)givesan
upperboundontheprobabilitythat thetangentsegmentis notoccluded. Thus, togetan
upperbound onthat probability,weneedalowerbound onthevolumeofH(t
i
;t
l
;1)\U.
Tobound theprobability that aT4-segmentexists, weintegrate overthe distance be-
tweenp
i and p
l
, and over the distance from p
i
to the boundary of the universe U. This
integralissplit intothreepartscoveringthecaseswhere
(i)B
i andB
l
arecloseto oneanother,
(ii)atleastoneofB
i andB
l
is entirelyinsidetheuniverse,
(iii)B
i andB
l
arenotcloseto oneanotherandbotharepartiallyoutsidetheuniverse.
Ineachcaseweover-estimatethevolumeofH(p
i
;p
l
;2)\U andunder-estimatethevolume
ofH(t
i
;t
l
;1)\U.
3.1 Denitions
LetN bethesetofordered4-tuples(i;j;k;l)chosenfromf1;2;:::;ngsuchthati;j;k;lare
pairwisedistinct. Inourmodel,theprobabilitythatfourcentersarecollineariszero,sowe
ti
t
j
t
k
t
l p
l
pi
B
l
Bi
(b)H(ti;t
l
;1) ti
tj
t
k
tl p
l
pi
p
j
B
l Bj
B
i 1
2
(a)H(p
i
;p
l
;2)
1
Figure1.H( p
i
;p
l
;2)andH( t
i
;t
l
;1)areshownshaded.
mayassumethatanysetoffourballsadmitsat most12realcommontangentlines[4,14].
Moreover, the real common tangent lines correspond to the real solutions of a degree 12
systemof equations. Foranyset of four balls weorder arbitrarily the12 solutionsof the
associatedsystem.
GivenfourballsB
i ,B
j ,B
k andB
l
,wedenotebyL
!
i;j;k ;l
,for!inf1;:::;12g,theevent
thatthe! th
solutionofthesystemisreal,thatthecorrespondingrealtangentlineistangent
to thefour ballsB
i , B
j , B
k and B
l
in that order,and that p
i
is notcloserthan p
l tothe
boundary ofU. WheneverL
!
i;j;k ;l
occurs,wedenotethe pointsoftangencyof that lineon
B
i ,B
j ,B
k ,B
l byt
i ,t
j , t
k ,t
l
,respectively. LetÆ
!
i;j;k ;l
betheeventthatL
!
i;j;k ;l
occursand
theline segmentt
i t
l
is not occluded. Notice that ifÆ
!
i;j;k ;l
occurs,the balls B
i
;B
j
;B
k
;B
l
deneaT4-segment,andthataT4-segmentcorrespondstoauniqueÆ
!
i;j;k ;l .
Let x
i;l
be the random variable representing the distance from p
i to p
l , and y
i (resp.
y
l
)betherandomvariabledenotingthedistancefromp
i
(resp. p
l
)totheboundaryofthe
universe.
In the sequel,a random point pdenotes a point chosen from the uniform distribution
overU.
3.2 The Proof
There isaone-to-one correspondence betweentheT4-segmentsandthe events Æ
!
i;j;k ;l that
occur. Wethus havethefollowingstraightforwardlemma.
Lemma3 TheexpectednumberofT4-segmentsamongstnuniformlydistributedunitballs
is
X
(i;j;k ;l)2N 12
X
!=1 Pr(Æ
!
i;j;k ;l ):
Webound theprobabilityPr(Æ
!
i;j;k ;l
)byintegratingoverthedistance xbetweenp
i and
p
l
,andoverthedistancey from p
i
totheboundaryof theuniverseU. Theintegralissplit
into three partscoveringthecaseswhere (i)the ballsB
i and B
l
are closeto oneanother,
(ii)p
i
isatdistanceatleast1fromtheboundaryofU,and(iii)theballsB
i andB
l
arenot
closetooneanotherandp
i
isatdistancelessthan1fromtheboundaryofU. Notethat in
thelast case, if Æ
!
i;j;k ;l
occurs,then bothball centersp
i and p
l
are within distance 1from
theboundaryofU. Twoballsareconsideredclosetooneanotheriftheircenters arecloser
thansomesuÆcientlylargeconstant;fortechnicalreasonswhichareembeddedintheproof
ofProposition29,weactuallydeneclose tomeandistanceatmost6.
Lemma4 Pr(Æ
!
i;j;k ;l )6I
x66 +I
y>1 +I
x>6;y<1 ,where
I
x66
= Z
6
x=0 Pr(Æ
!
i;j;k ;l jx
i;l
=x)Pr(x6x
i;l
<x+dx);
I
y>1
= Z
2R
x=0 Pr(Æ
!
i;j;k ;l jx
i;l
=x; y
i
>1)Pr(x6x
i;l
<x+dxjy
i
>1);
I
x>6;y<1
= Z
2R
x=6 Z
1
y=0 Pr(Æ
!
i;j;k ;l jx
i;l
=x; y
i
=y; y
l 6y
i )
Pr((x6x
i;l
<x+dx)\(y
l 6y
i )jy
i
=y)Pr(y6y
i
<y+dy):
Proof: BytheTotalProbabilityTheorem(see[17]),
Pr(Æ
!
i;j;k ;l )=
Z
2R
x=0 Pr(Æ
!
i;j;k ;l jx
i;l
=x)Pr(x6x
i;l
<x+dx):
The integral can be split at x = 6, giving I
x66
. Then applying the Total Probability
Theoremonwhatremains,weget
Z
2R
x=6 Z
R
y=0 Pr(Æ
!
i;j;k ;l jx
i;l
=x; y
i
=y)Pr((x6x
i;l
<x+dx)jy
i
=y)Pr(y6y
i
<y+dy)
(2)
which canbesplit aty=1. Thepartcorrespondingtoy between1andR isequalto
Z
2R
x=6 Z
R
y=1 Pr(Æ
!
i;j;k ;l jx
i;l
=x; y
i
=y; y
i
>1)
Pr((x6x
i;l
<x+dx)jy
i
=y; y
i
>1)Pr(y6y
i
<y+dy)
6 Z
2R
x=6 Z
R
y=0 Pr(Æ
!
i;j;k ;l
\(x6x
i;l
<x+dx)jy
i
=y; y
i
>1)Pr(y6y
i
<y+dy):
ApplyingtheTotalProbabilityTheoremagain,weget
Z
2R
x=6 Pr(Æ
!
i;j;k ;l
\(x6x
i;l
<x+dx)jy
i
>1)
whichislessthanI
y>1
. Considernowthepartof (2)fory between0and1. Ify
l
>y
i then
Æ
!
i;j;k ;l
doesnotoccur(bydenitionofL
!
i;j;k ;l
),thuswehave
Pr(Æ
!
i;j;k ;l jx
i;l
=x; y
i
=y)Pr((x6x
i;l
<x+dx)jy
i
=y)
=Pr(Æ
!
i;j;k ;l
\(x6x
i;l
<x+dx)jy
i
=y)
=Pr(Æ
!
i;j;k ;l
\(x6x
i;l
<x+dx)\(y
l 6y
i )jy
i
=y)
=Pr(Æ
!
i;j;k ;l jx
i;l
=x; y
i
=y; y
l 6y
i
)Pr((x6x
i;l
<x+dx)\(y
l 6y
i )jy
i
=y):
Thus,thepartof (2)fory between0and1isequaltoI
x>6;y<1
. 2
Letdenote anyofthefollowingevents: (x
i;l
=x), (x
i;l
=x,y
i
>1), (x
i;l
=x; y
i
=
y; y
l 6y
i
). ThenextthreelemmasareusedtoboundPr(Æ
!
i;j;k ;l
j)appearinginthethree
integralsI
x66 ,I
y>1 andI
x>6;y<1 .
Lemma5 If a line is tangent to four balls B
i
;B
j
;B
k
;B
l
in that order at t
i
;t
j
;t
k
;t
l , re-
spectively,thenp
j
;p
k 2H(p
i
;p
l
;2). Also,the segment t
i t
l
isnotoccludedifandonlyif the
interiorofH(t
i
;t
l
;1) does notcontainthe center ofany other ball.
Proof: Segmentt
i t
l
iscontainedinH(p
i
;p
l
;1). Sincet
j andt
k
belongtothat segment,t
j
andt
k
arealsoinH( p
i
;p
l
;1). Thusp
j
;p
k
arebothin H( p
i
;p
l
;2). SeeFigure1(a).
Thesegmentt
i t
l
isoccludedifandonlyifsomeballB
,6=i;j;k;l,properlyintersects
it,thatisthecenterof B
liesin theinteriorofH(t
i
;t
l
;1). SeeFigure1(b). 2
Lemma6 Pr(p2H (p
i
;p
l
;2)j)6
(3x+8)
R 3
.
Proof:
Pr(p2H (p
i
;p
l
;2)j)=
VolumeofH (p
i
;p
l
;2)\U
VolumeofU
j
6
VolumeofH (p
i
;p
l
;2)
VolumeofU j
:
Whenoccurs,x
i;l
=x andthevolumesof H(p
i
;p
l
;2) andU are 4
3
( 3x+8) and 4
3 R
3
,
respectively. Thus
Pr(p2H (p
i
;p
l
;2)j)6 3x+8
R 3
:
2
Lemma7 Pr(Æ
!
i;j;k ;l j)6
(3x+8) 2
R 6
Pr(p62H (t
i
;t
l
;1)jL
!
i;j;k ;l
; ) n 4
.
Proof: IfÆ
!
i;j;k ;l
occurs,thenL
!
i;j;k ;l
necessarilyoccurs,thus
Pr(Æ
!
i;j;k ;l
j)=Pr(Æ
!
i;j;k ;l
\L
!
i;j;k ;l
j)=Pr(L
!
i;j;k ;l
j)Pr(Æ
!
i;j;k ;l jL
!
i;j;k ;l
; ):
By Lemma 5, Pr(L
!
i;j;k ;l
j ) is bounded by the probability that p
j and p
k
belong to
H (p
i
;p
l
;2)given,andPr(Æ
!
i;j;k ;l jL
!
i;j;k ;l
)isequaltotheprobabilitythatforall6=i;j;k;l,
pointp
isoutsideH (t
i
;t
l
;1)given. Sinceallthepointsareindependentlyandidentically
drawnfrom theuniformdistributionoverU,Lemma6yields theresult. 2
WeconsiderthethreeintegralsI
x66 ,I
y>1 andI
x>6;y<1
inthefollowingsubsections,and
provethat each is bounded by O 1
n 3
. This will complete the proof of the upper bound
ofTheorem 1since, by Lemmas3and4,theexpected numberof T4-segmentsislessthan
12 n
4
(I
x66 +I
y>1 +I
x>6;y<1 ).
3.2.1 B
i
and B
l
are closeto one another
WeproveherethatI
x66 isO(
1
n 3
). WhenB
i andB
l
areclosetooneanother,theprobability
thatthereexisttwootherballs,B
j andB
k
,deningalinetangenttoB
i
;B
j
;B
k
;B
l in that
order,issmallenoughthatwedonotneedto considerocclusionsinordertogetthebound
wewant.
WerstboundthetermPr(x6x
i;l
<x+dx)appearingin theintegralI
x66 .
Lemma8 Pr(x6x
i;l
<x+dx)6 3x
2
R 3
dx.
Proof: Whenp
i
isgiven,p
l
mustbelongto asphericalshell betweentwospheresofcenter
p
i
andradiix andx+dx. TheprobabilityPr(x6x
i;l
<x+dx), ifp
i
isknown, isexactly
the volume of the part of the spherical shell inside U divided by the volume of U. The
volume of the part of the spherical shell inside U is bounded from above by the volume
of thesphericalshellwhich is4x 2
dx. Since thevolume of U is 4
3 R
3
weget theclaimed
bound. (TheexactvalueofPr(x6x
i;l
<x+dx)isactuallygivenin[15,22]buttheabove
approximateboundisenoughforourpurposes.) 2
Proposition 9 I
x66 isO(
1
n 3
).
Proof: Recallthat(seeLemma 4)
I
x66
= Z
6
x=0 Pr(Æ
!
i;j;k ;l jx
i;l
=x)Pr(x6x
i;l
<x+dx):
ByLemma 7,
Pr(Æ
!
i;j;k ;l jx
i;l
=x)6
(3x+8) 2
R 6
Pr(p62H (t
i
;t
l
;1)jx
i;l
=x; L
!
i;j;k ;l )
n 4
6
(3x+8) 2
R 6
:
ItthusfollowsfromLemma 8that
I
x66 6
Z
6
x=0
(3x+8) 2
R 6
3x
2
R 3
dx=
3
n 3
Z
6
x=0 3x
2
(3x+8) 2
dx=O
1
n 3
:
2
3.2.2 B
i
is entirely insideU
FortheintegralI
y>1
, occlusionsmustbetakeninto account. Tothis aim, webound from
belowthevolumeofH (t
i
;t
l
;1)\U in thefollowinglemma.
Lemma10 When L
!
i;j;k ;l
occurs and y
i
>1, the volume of H (t
i
;t
l
;1)\U isgreaterthan
12 x
i;l .
Proof: LetKbetheballhavingdiameterp
i t
i
. NotethatKandp
l
arebothcontainedinU
andinH (t
i
;t
l
;1). Theconvexhullof p
l
andK isthus containedin H (t
i
;t
l
;1)\U,andits
volume islargerthan halfthe volumeof theballK,
12
,plusthevolumeofaconeofapex
p
l
,ofbaseadiskwhoseboundaryisagreatcircleofK,andofheightgreaterthanx
i;l 1.
Thevolumeofthatconeisat least 1
3
2 2
(x
i;l 1)=
12 x
i;l
12
. 2
We nowbound the probability that atangent line segmentt
i t
l
is notoccludedby any
of theother n 4balls, given that the line segmentt
i t
l
exists and theball B
i
is entirely
containedinU.
Lemma11 Pr p62H (t
i
;t
l
;1)jx
i;l
=x; y
i
>1; L
!
i;j;k ;l
n 4
<55exp
x
16
.
Proof: Firstnoticethat
Pr(p62H (t
i
;t
l
;1)jx
i;l
=x; y
i
>1; L
!
i;j;k ;l
=1
Volume ofH( t
i
;t
l
;1)\U
VolumeofU
j
xi;l=x;yi>1;L
!
i;j;k ;l :
By Lemma 10, the volume of H(t
i
;t
l
;1)\U is bounded from below by
12
x. Since the
volumeofU is 4
3 R
3
,weget
Pr p62H (t
i
;t
l
;1)jx
i;l
=x;y
i
>1; L
!
i;j;k ;l
n 4
<
1 x
16R 3
n 4
:
Forany06t61,wehave(1 t)6e t
thus
(1 t) n 4
6e t(n 4)
=e tn
e 4t
6e 4
e tn
<55e tn
:
Now06x62RandR>1sinceB
i
isentirelyinside U. Thus06 x
16R 3
6 1
8R 2
61and
Pr p62H (t
i
;t
l
;1)jx
i;l
=x; y
i
>1; L
!
i;j;k ;l
n 4
<55exp
nx
16R 3
=55exp
x
16
:
2
Thefollowingpropositionnowbounds theintegralI
y>1 .
Proposition 12 I
y>1 isO(
1
n 3
).
Proof: Recallthat
I
y>1
= Z
2R
x=0 Pr(Æ
!
i;j;k ;l jx
i;l
=x; y
i
>1)Pr(x6x
i;l
<x+dxjy
i
>1):
ByLemmas7and11wehave
Pr(Æ
!
i;j;k ;l jx
i;l
=x; y
i
>1)6
(3x+8) 2
R 6
55exp
x
16
:
SimilarlyasinLemma8wehave
Pr(x6x
i;l
<x+dxjy
i
>1)6 3x
2
R 3
dx:
Thusweget
I
y>1 6
Z
2R
x=0
(3x+8) 2
R 6
55exp
x
16
3x
2
R 3
dx6
3
n 3
Z
+1
x=0 3x
2
(3x+8) 2
55exp
x
16
dx:
Changing x
16
byzwegetanintegralofthekind
Z
1
0 z
r
exp( z)dz
which isbounded byaconstantandthusI
y>1 isO
1
n 3
. 2
3.2.3 B
i
and B
l
are notclose to one anotherand B
i
ispartially outsideU
TheonlyremainingtaskistoboundtheintegralI
x>6;y<1
. Asinthepreviouscase,weneed
tobound frombelowthevolumeof H (t
i
;t
l
;1)\U. Here,however,thetangentt
i t
l canbe
entirelyoutsideU,sothebound ofLemma10doesnotapply andamoreintricateproofis
needed. Weneed to distinguish twocases depending on thedistance of segment t
i t
l from
O,thecenterofU.
Tothisaim, we introducetwonewtypesof events. Foranys2R, letF
!
i;j;k ;l
(s)(resp.
N
!
i;j;k ;l
(s)) betheeventthat L
!
i;j;k ;l
occurs and theline segmentt
i t
l
is at distancegreater
(resp. less) than R+1 s from O. For reasons that will become clear in the proof of
Lemma15,weconsider s=y 2
3
.
ThenextvelemmasareusedtoboundthersttermoftheintegralI
x>6;y<1 .
Lemma13 ForanyrandompointpinU,Pr(Æ
!
i;j;k ;l jx
i;l
=x; y
i
=y; y
l 6y
i
)isequal to
Pr
F
!
i;j;k ;l (y
2
3
)jx
i;l
=x; y
i
=y; y
l 6y
i
Pr
p62H (t
i
;t
l
;1)jx
i;l
=x; y
i
=y; y
l 6y
i
; F
!
i;j;k ;l (y
2
3
)
n 4
+Pr
N
!
i;j;k ;l (y
2
3
)jx
i;l
=x; y
i
=y; y
l 6y
i
Pr
p62H (t
i
;t
l
;1)jx
i;l
=x; y
i
=y; y
l 6y
i
; N
!
i;j;k ;l (y
2
3
)
n 4
:
Proof: Æ
!
i;j;k ;l
impliesL
!
i;j;k ;l
whichcanbesplitintoF
!
i;j;k ;l (y
2
3
),N
!
i;j;k ;l (y
2
3
),andtheevent
thatL
!
i;j;k ;l
occursandthelinesegmentt
i t
l
isat distanceexactlyR+1 y 2
3
fromO. This
latereventoccurswithprobability0,thus
Pr(Æ
!
i;j;k ;l jx
i;l
=x; y
i
=y;y
l 6y
i )=
Pr(Æ
!
i;j;k ;l
\F
!
i;j;k ;l (y
2
3
)jx
i;l
=x; y
i
=y; y
l 6y
i )
+Pr(Æ
!
i;j;k ;l
\N
!
i;j;k ;l (y
2
3
)jx
i;l
=x; y
i
=y; y
l 6y
i );
which canbeexpandedinto
Pr(F
!
i;j;k ;l (y
2
3
)jx
i;l
=x; y
i
=y; y
l 6y
i )
Pr(Æ
!
i;j;k ;l jx
i;l
=x; y
i
=y; y
l 6y
i
; F
!
i;j;k ;l (y
2
3
))
+Pr(N
!
i;j;k ;l (y
2
3
)jx
i;l
=x; y
i
=y; y
l 6y
i )
Pr(Æ
!
i;j;k ;l jx
i;l
=x; y
i
=y; y
l 6y
i
; N
!
i;j;k ;l (y
2
3
)):
WhenF
!
i;j;k ;l (y
2
3
)occurs,theprobability
Pr(Æ
!
i;j;k ;l jx
i;l
=x; y
i
=y; y
l 6y
i
; F
!
i;j;k ;l (y
2
3
))
istheprobabilitythatthetangentisnotoccluded,thatis,p
doesnotbelongtoH (t
i
;t
l
;1)
forallthen 4valuesof6=i;j;k;l. ThesameargumentholdsforN
!
i;j;k ;l (y
2
3
). Sincethe
p
areindependent,wegettheresult. 2
InordertoboundthetwotermsinLemma 13,
Pr
p62H (t
i
;t
l
;1)jx
i;l
=x; y
i
=y; y
l 6y
i
; F
!
i;j;k ;l (y
2
3
)
n 4
and
Pr
p62H (t
i
;t
l
;1)jx
i;l
=x; y
i
=y; y
l 6y
i
; N
!
i;j;k ;l (y
2
3
)
n 4
;
weneedtoboundthevolumeofH (t
i
;t
l
;1)\U from below.
Lemma14 When x
i;l
>6, y
l 6y
i 61, L
!
i;j;k ;l
occurs andsegment t
i t
l
isatdistanceless
thanR+1 s,06s61,fromthe center ofU,thenthe volumeof H (t
i
;t
l
;1)\U islarger
than 1
6 p
2 (x
i;l 5)s
p
s.
Proof: Wegiveheretheideaof theproof; fulldetails canbefoundin Appendix A. Lett
betheclosestpointonsegmentt
i t
l
from O, andD beaunitradiusdisk centeredat tin a
planecontainingO,thecenter ofU. Wedeneaquadrilateralwith verticesa;b;a 0
;b 0
such
that aanda 0
aretheclosestand thefarthest points,respectively,in D\U from O, and b
and b 0
are the points of intersection of @D and the perpendicular bisectorof segmentaa 0
(seeFigure2). LetvbeequaltoR+1minusthedistancefromOtosegmentt
i t
l
. Weprove
that the convex hull of a;b;a 0
;b 0
and p
l
, which is included in H (t
i
;t
l
;1)\U, has volume
greaterthan 1
6 p
2 (x
i;l
5)min (2 p
2;v p
v). It followsthat, for any 06s 61,ifsegment
t
i t
l
isat distancelessthanR+1 s fromO,thenv>sand thevolumeofH (t
i
;t
l
;1)\U
isgreaterthan 1
6 p
2 (x
i;l 5)s
p
s. 2
@U t
b
b 0
a D
a 0
O
v R+1 v
R
R v
Figure2. ForthesketchoftheproofofLemma14(v2(0;1)).
Lemma15 Forany random pointpinU,x>6and06y61,
Pr
p62H (t
i
;t
l
;1)jx
i;l
=x; y
i
=y; y
l 6y
i
; F
!
i;j;k ;l (y
2
3
)
n 4
<55exp
(x 5)y 2
8 p
2
and
Pr
p62H (t
i
;t
l
;1)jx
i;l
=x; y
i
=y; y
l 6y
i
; N
!
i;j;k ;l (y
2
3
)
n 4
<55exp
(x 5)y
8 p
2
:
Furthermore, ifx>6 p
Rthen
Pr
p62H (t
i
;t
l
;1)jx
i;l
=x; y
i
=y; y
l 6y
i
; N
!
i;j;k ;l (y
2
3
)
n 4
<55exp
(x 5)
8 p
2
:
Proof: Let x
i;l
= x, y
i
= y and suppose rst that event F
!
i;j;k ;l (y
2
3
) occurs. Since p
i is
at distanceR y fromO, thesegmentt
i t
l
isat distance lessthanR+1 y from O, and
thus,byLemma14,thevolumeofH(t
i
;t
l
;1)\U isgreaterthan 1
6 p
2
(x 5)y p
y,whichis
biggerthan 1
6 p
2
(x 5)y 2
since06y 61(webound y p
y from belowbyy 2
only sothat
wecanactuallycomputetheintegralI
1
intheproofofProposition20). Wenowfollowthe
proofof Lemma 11, exceptthat thevolumeof H( t
i
;t
l
;1)\U is nowbounded from below
by 1
6 p
2
(x 5)y 2
insteadof
12
x. Weget
Pr
p62H (t
i
;t
l
;1)jx
i;l
=x; y
i
=y; y
l 6y
i
; F
!
i;j;k ;l (y
2
3
)
n 4
<55exp
(x 5)y 2
8 p
2
:
WhenN
!
i;j;k ;l (y
2
3
)occurs,the segmentt
i t
l
is at distanceless thanR+1 y 2
3
from O,
andthus, byLemma14, thevolumeof H( t
i
;t
l
;1)\U isbounded frombelowby 1
6 p
2 (x
5)y 2
3 q
y 2
3
= 1
6 p
2
(x 5)y. Then, asbefore,weget
Pr
p62H (t
i
;t
l
;1)jx
i;l
=x; y
i
=y; y
l 6y
i
; N
!
i;j;k ;l (y
2
3
)
n 4
<55exp
(x 5)y
8 p
2
:
Now,ifx>6 p
R ,thelengthofthetangentt
i t
l
isat least6 p
R 2. Sincex>6,R>3
andasimplecomputationshowsthat6 p
R 2isbiggerthan2 p
2R+1whichisthelengthof
thelongestlinesegmentthatmayentirelylieinsideU +
nU. Thusdist(O;t
i t
l
)6R=R+1 s
with s =1 and, by Lemma 14, thevolume of H (t
i
;t
l
;1)\U is greater than 1
6 p
2
(x 5).
Then,asbefore,weget
Pr
p62H (t
i
;t
l
;1)jx
i;l
=x; y
i
=y; y
l 6y
i
; N
!
i;j;k ;l (y
2
3
)
n 4
<55exp
(x 5)
8 p
2
:
2
Lemma16 Pr(N
!
i;j;k ;l (y
2
3
)jx
i;l
=x; y
i
=y; y
l 6y
i )6
(3x+8) 2
R 6
.
Proof: TheeventN
!
i;j;k ;l (y
2
3
)occursonlyifL
!
i;j;k ;l
occurs. Theresultthusfollowssince,by
Lemmas5and6,Pr(L
!
i;j;k ;l jx
i;l
=x; y
i
=y; y
l 6y
i )6
(3x+8) 2
R 6
. 2
Lemma17 If y<1,then Pr
F
!
i;j;k ;l (y
2
3
)jx
i;l
=x; y
i
=y
681 2
(x+6) 2
y 2
R 6
.
Proof: A\far"tangentt
i t
l
isatdistanceatleastR+1 y 2
3
fromthecenterOofU. Sucha
segmentalsoliesinH (p
i
;p
l
;1). LetE bethepartofH (p
i
;p
l
;1)lyingoutsideofthesphere
ofradiusR+1 y 2
3
andcenterO. SeeFigure 3(a). Now, bothp
j and p
k
mustbeinthe
regioninsideU andwithindistance 1fromE. Denotethis regionbyK. Then
Pr
F
!
i;j;k ;l (y
2
3
)jx
i;l
=x; y
i
=y; y
l 6y
i
6
VolumeofK
VolumeofU
2
:
By Proposition 32, which weprovein Appendix B, the volume of K is bounded from
aboveby 12 2
(x+6)y, which yields the result. Here we givethe intuition of the proof.
RefertoFigure3. Firstnoticethat the\length"ofK isatmostx+4. SinceKisenclosed
inbetweenasphereofradiusRandoneofradiusR y 2
3
,its\height"isatmosty 2
3
. Forthe
\width",considerFigure3(b)whichshowsacross-sectionofKtakenwithaplanethrough
O and perpendicular to p
i p
l
. The \width" of K is no morethan 2 times the \width" of
E. The\height"of Ecanbeboundedbysomeconstanttimesy 2
3
;thusits\width"canbe
bounded bysomeconstanttimes q
y 2
3
=y 1
3
. Thus, intuitively,thevolumeofK is smaller
than(x+4)y 2
3
y 1
3
=(x+4)y,uptoaconstant,andtheresultfollows. 2
WenowboundthetwolasttermsoftheintegralI
x>6;y<1 .
Lemma18 Pr(y6y
i
<y+dy)6 3dy
R .
Proof: Theevent(y6y
i
<y+dy)occursonlyifp
i
liesinthesphericalshelldelimitedby
thetwospherescenteredat Oof radiiR y andR y dy whosevolumeis smallerthan
4R 2
dy. DividingbythevolumeofU provestheresult. 2
Lemma19 For66x62R andy61, wehave
Pr((x6x
i;l
<x+dx)\(y
l 6y
i )jy
i
=y)6
6xydx
R 3
:
Proof: TheprobabilityPr((x6x
i;l
<x+dx)\(y
l 6y
i )jy
i
=y)isequalto thevolume
oftheregion(shownin greyinFigure4)which istheintersectionoftheregioninbetween
thetwospherescenteredatp
i
andofradiixandx+dx,andtheregioninbetweenthetwo
spherescenteredat O and ofradiiR andR y, dividedbythevolumeofU. Weprovein
p
i
;p
l E
R y 2
3 K
y 2
3
y 1
3 x+2
pi
pl E
R+1 y 2
3
R
R y
(a)
(b)
R+1 y 2
3
R
R y
K
Figure3.ForthesketchoftheproofofLemma17.
Proposition37inAppendixCthatthevolumeofthatregionisatmost8xydx. Roughly
speaking, thevolumeboundedbythefourspheresisat most8xydx because,its\thick-
ness"isdx, its\height"is y andits\radius"isx. DividingbythevolumeofU provesthe
result. 2
R y
x
x+dx p
i
O
Figure4.FortheproofofLemma19.
WecannowboundtheintegralI
x>6;y<1
ofLemma 4.
Proposition 20 I
x>6;y<1 isO
1
n 3
.
Proof: Recallthat
I
x>6;y<1
= Z
2R
x=6 Z
1
y=0 Pr(Æ
!
i;j;k ;l jx
i;l
=x; y
i
=y; y
l 6y
i )
Pr((x6x
i;l
<x+dx)\(y
l 6y
i )jy
i
=y)Pr(y6y
i
<y+dy):
ByLemmas18and19, weget
I
x>6;y<1 6
Z
2R
x=6 Z
1
y=0 Pr(Æ
!
i;j;k ;l jx
i;l
=x; y
i
=y; y
l 6y
i )
6xydx
R 3
3dy
R :
ByLemma 13, Pr(Æ
!
i;j;k ;l jx
i;l
=x; y
i
=y; y
l 6y
i
)isequalto
Pr
F
!
i;j;k ;l (y
2
3
)jx
i;l
=x; y
i
=y; y
l 6y
i
Pr
p62H (t
i
;t
l
;1)jx
i;l
=x; y
i
=y; y
l 6y
i
; F
!
i;j;k ;l (y
2
3
)
n 4
+Pr
N
!
i;j;k ;l (y
2
3
)jx
i;l
=x; y
i
=y; y
l 6y
i
Pr
p62H (t
i
;t
l
;1)jx
i;l
=x; y
i
=y; y
l 6y
i
; N
!
i;j;k ;l (y
2
3
)
n 4
:
Wesplit theintegralatx=6 p
R . Whenx>6 p
R ,thedistancefrom O tothetangent
t
i t
l
is lessthanR (seetheproofof Lemma15), which islessthanR+1 y 2
3
forany y in
(0;1). Thus, foranyx>6 p
R andy2(0;1),
Pr
F
!
i;j;k ;l (y
2
3
)jx
i;l
=x; y
i
=y; y
l 6y
i
=0:
Itthenfollowsfrom Lemmas15,16and17thatI
x>6;y<1 6I
1 +I
2 +I
3 with
I
1
= Z
6 p
R
x=6 Z
1
y=0 81
2 (x+6)
2
y 2
R 6
55exp
(x 5)y 2
8 p
2
6xydx
R 3
3dy
R
;
I
2
= Z
6 p
R
x=6 Z
1
y=0
(3x+8) 2
R 6
55exp
(x 5)y
8 p
2
6xydx
R 3
3dy
R
;
I
3
= Z
2R
x=6 p
R Z
1
y=0
(3x+8) 2
R 6
55exp
(x 5)
8 p
2
6xydx
R 3
3dy
R :
ComputingtheseintegralswithMaple gives(assuming>0),whenR tendstoinnity,
I
1 +I
2 +I
3
=O
1
R 9
:
SinceR 3
=n=,wegetI
x>6;y<1 2O(
1
n 3
). 2
We can now conclude the proof that the expected number of T4-segments is O(n),
because, by Lemmas 3, 4, and Propositions 9, 12, and 20, the expected number of T4-
segmentsissmallerthan
X
(i;j;k ;l)2N 12
X
!=1
O
1
n 3
+O
1
n 3
+O
1
n 3
=O(n):
4 The expectednumber of T4-segments is at least linear
In this section, we provethat the expected number of T4-segmentsamongst n uniformly
distributed unit balls is (n). Todothis, webound from belowthe probability that four
given balls have a given T4-segment. The key step is to givea condition on the relative
positionsoffour unit ballsthat guaranteesthat theyhaveexactlytwelvecommontangent
lines. Weuseherethenotationasdenedin Section3.1.
Lemma21 Let e be a real number satisfying 4
p
2
3
< e <2 and let the radius R of U be
strictlygreaterthane. Thereexistsan>0suchthatfor anypointp2U,thereexistthree
balls
1 (p),
2 (p),
3
(p)of radius containedinU and satisfyingthe following conditions:
pandthe centersofthe
i
(p) formaregulartetrahedron withedges of lengthe,and
for any triple ofpoints(p
1
;p
2
;p
3 ), p
i
takenfrom
i
(p), the four unitballs centeredatp,
p
1 ,p
2 andp
3
have exactly12 distincttangentlines.
Proof: Macdonald, Pach and Theobaldproved [14, Lemma 3] that 4 unit balls centered
ontheverticesofaregulartetrahedronwithedges oflengthe, 4
p
2
3
<e<2,haveexactly
12distinct realcommontangentlines. Moreover,these12 tangent linescorrespond tothe
12 realroots of asystem of equationsof degree 12, thus each tangentline corresponds to
a simple root of that system of equations. It thus follows that for any suÆciently small
perturbationofthe4ballcenters, the4perturbed ballsstillhave12real commontangent
lines. Let>0besuch that the4ballcenterscanmovedistance in anydirection while
keeping12distinctcommontangents.
Now,foranypointp2U,consideraregulartetrahedronwithedgelengthehavingpas
avertexandsuchthattheotherverticesareatdistanceat leastfrom theboundaryofU;
forexample,wecanchoosetheotherthreeverticesonaplaneperpendiculartothesegment
Op. Let
1 (p),
2
(p), and
3
(p) be the ballsof radius centered at the vertices, distinct
from p,of that tetrahedron. Bythe previousreasoning,for anyq 2
1
(p), r2
2
(p), and
s2
3
(p),thefourunit ballscenteredatp,q, randshaveexactlytwelvetangents. 2
Now,byLemma 3,theexpectednumberofT4-segmentsis
X
(i;j;k ;l)2N 12
X
!=1 Pr(Æ
!
i;j;k ;l ):
Thusweonlyneedto bound frombelowtheprobabilitythattheeventÆ
!
i;j;k ;l
occurs.
Lemma22 Pr(Æ
!
i;j;k ;l )=
1
n 3
.
Proof: Assume that n >8 sothat the radius R = 3 p
n=of U is largerthan 2and let
T(p)betheset
1 (p)
2 (p)
3
(p)where
i
(p)andearedened asinLemma21. First,
notethat
Pr(Æ
!
i;j;k ;l
)>Pr(Æ
!
i;j;k ;l
\(p
i
;p
j
;p
k )2T(p
l ))
=Pr((p
i
;p
j
;p
k )2T(p
l
))Pr(Æ
!
i;j;k ;l j(p
i
;p
j
;p
k )2T(p
l )):
Since
1 (p
l ),
2 (p
l ),and
3 (p
l
)arethree ballsofradiusentirelycontainedin U,wehave
Pr((p
i
;p
j
;p
k )2T(p
l ))=
4
3
3
4
3 R
3 3
=
3
9
n 3
:
By Lemmas 5 and 21, the event (Æ
!
i;j;k ;l j (p
i
;p
j
;p
k
) 2 T(p
l
)) occurs if and only if the
interiorofH(t
i
;t
l
;1)\U doesnotcontainthecenterofanyball. Notethatthevolumeof
H(t
i
;t
l
;1)\U isatmostthevolumeofH(t
i
;t
l
;1),whichisatmost 4
3
+(2+e+2)since
thelengthoft
i t
l
isatmoste+2+2. Itfollowsthat
Pr(Æ
!
i;j;k ;l j(p
i
;p
j
;p
k )2T(p
l ))>
1 (
4
3
+2+e+2)
Volume(U) n
4
:
Sincee<2,weget,after someelementarycalculations,that
Pr(Æ
!
i;j;k ;l j(p
i
;p
j
;p
k )2T(p
l ))>
1
(6+2)
n
n 4
: (3)
Wethushave
Pr(Æ
!
i;j;k ;l )>
3
9
n 3
1
(6+2)
n
n 4
:
Since
1
(6+2)
n
n 4
tendstoe
(6+2)
whenntendstoinnity, weget
Pr(Æ
!
i;j;k ;l )=
1
n 3
:
2
This completes the proof of the lowerbound of Theorem 1 sincethe expected numberof
T4-segmentsamongstnuniformlydistributedunit ballsis,byLemmas3and22,
X
(i;j;k ;l)2N 12
X
!=1 Pr(Æ
!
i;j;k ;l )=
X
(i;j;k ;l)2N 12
X
!=1
1
n 3
=(n):
5 The expected size of the visibility complex is linear
InthissectionweproveTheorem2,thattheexpectedsizeofthevisibilitycomplexofaset
ofnuniformlydistributedunit ballsislinear.
We saythat the balls arein generalposition if any k-dimensional face of thevisibility
complexisaconnectedsetofmaximalfreesegmentstangenttoexactly4 kballs. Wecan
assumethattheballsareingeneralpositionsincethisoccurswithprobability1. Wegivea
boundontheexpectednumberofk-faces, fork=0;:::;4.
Lemma23 The expectednumberof0-facesis(n).
Proof: A0-faceofthevisibilitycomplexisamaximalfreeline segmenttangentto4balls.
Each maximal free line segment tangent to 4 balls contains a T4-segment and each T4-
segmentiscontainedinonemaximalfreelinesegment. Thus, byTheorem1,theexpected
numberof0-facesislinear. 2
Todealwiththefaces ofdimensionk>1,wedivide themintotwoclasses. Ak-faceis
open ifitisincident toat leastone(k 1)-face,otherwiseitisclosed. Whentheballsare
ingeneralposition,thenumberofk-facesincidenttoa(k 1)-faceisconstant. Intheproof
ofthefollowinglemmas, anyconstantcanbeused. However,for completeness,wewilluse
theexactvalues,butwithoutjustifyingthem.
Lemma24 The expectednumberof1-facesis(n).
Proof: Note that a0-face corresponds to amaximal free segment tangent to 4balls and
itisincident tothose1-facescorrespondingtofreesegmentstangentto3amongstthose 4
balls. So,a0-faceisincidenttoexactlysix1-faces,whichimpliesthatthenumberofopen
1-facesis 6timesthenumberof0-faces,andisthus(n)bythepreviouslemma.
Provingthat theexpected number ofclosed 1-facesis O(n)canbedonein awayvery
similar to theproof of theupperbound in Theorem 1. The dierenceis that weconsider
nowonlythreeballsandthusinallproofs,weforgetballB
k
. Wehavetoconsideronly n
3
triplesof balls insteadof n
4
quadruples, but weremovefrom theintegralthe probability
Pr(p
k 2H (p
i
;p
l
;2)jx
i;l
=x)6 3x+8
R 3
. Since n
R 3
=,thisamountstodividingthetermsover
which weintegrateby (3x+8)which doesnotchangethe generalshapeofthe integrals
(apolynomialmultipliedby anexponential) which areconvergent. Noticethat B
i , B
j ,B
l
and! nowdeneaset ofsegmentst
i t
l
, ratherthanjust asinglesegment. However,those
segmentsdeneaclosed1-faceonlyifnoneofthemisoccludedbyoneofthen 3remain-
ingballs. Anyparticular choiceof atangentt
i t
l
in the1-facewill givearelevantcylinder
H (t
i
;t
l
;1)to usein theproofs. 2
Lemma25 The expectednumberof2-facesis(n).
Proof: Sincea1-facehasveincident2-faces,thetightlinearbound onthenumberof 1-
facesgivesatightlinearboundonthenumberofopen2-faces. Theclosedcaseissolvedsimi-
larlytotheproofoftheupperboundinTheorem1. Wenowconsider n
2
pairsofballsB
i
;B
l
andweremovefromtheintegralstheprobabilityPr(p
j
;p
k 2H (p
i
;p
l
;2)jx
i;l
=x)6 3x+8
R 3
2
which givesanO(n)boundonthenumberofclosed2-faces. 2
Lemma26 The expectednumbersof 3-facesand4-faces are(n).
Proof: A3-face,correspondingto linestangentto aball,canonlybeclosedifn=1. The
numberofopen 3-facesislinearby thefact thatin general position a2-faceis incidentto
four3-faces. Thenumberof4-facesislinearsincea3-faceisincidenttothree4-faces. 2
6 Worst-case lower bound
We provide herea(n 2
)lowerbound on the numberof k-faces in thevisibilitycomplex.
Recall that for the case of n arbitrarily sized balls, Devillers and Ramos [5] presented a
simple(n 3
)lowerboundonthenumberoffreesegmentstangentto 4balls,which isalso
thenumberofverticesin thevisibilitycomplex. Theirlowerbound (seeFigure 5)consists
of (i) n
3
balls such that the viewfrom theoriginconsists of n
3
disjoint diskscentered on a
circle,(ii) n
3
ballssuchthat the viewfrom theoriginconsistsof n
3
disks whoseboundaries
are concentric circles intersecting (in projection) allthe disks of (i), and (iii) n
3
tiny balls
centered aroundtheoriginsuch that from anypointonthese n
3
tiny ballsthe viewofthe
ballsin(i)and(ii)istopologicallyinvariant. Notethat ndinga(n 3
)lowerboundonthe
numberoffreesegmentstangentto4balls,amongstnballsofboundedradii,istothebest
ofourknowledge,open.
Proposition 27 The number ofk-faces in the visibilitycomplex of n disjoint unitballsin
R 3
is(n 2
)for allk between 0and4.
Proof: Werstobservethat thesize ofthevisibility complexofnunit balls cantrivially
bequadraticbyhavingtheballssparselydistributedinthespacesuchthatanypairofballs
denesaclosed2-face.
Getting aquadratic numberof freelines tangentto four balls amongst aset of nunit
ballscanbedonebytakingballsB
i
centeredat(2i;0;0)for16i6 n
2
andballsB 0
j
centered
at(2j;10;0)for16j6 n
2
. Then,foranyiand j,thelinethroughthepoints(2i+1;0;1)
and (2j+1;10;1) is free and can be moved down sothat it comes into contact with the
four balls B
i , B
i+1 , B
0
j and B
0
j+1
. This argument provesthat the numberof k-faces, for
06k62,canbequadratic.
The free segment (2i;1;0)(2j;9;0) belongs to the 4-face consisting of maximum free
segmentswithendpointsonB
i andB
0
j
. Thusthere isaquadraticnumberof 4-faces. The
boundalsoappliesto3-facesbyconsideringlinestangenttoB
i
andstabbingB 0
j .
Figure5.Quadraticviewfromtheorigin[5].
Intheaboveconstruction,theballscanbepushedtogether(theywillintersect)so that
theytinside asphericaluniverseof radius 3 p
n=withoutchangingtheresult. Notealso
thattheaboveconstructioncanbeslightlyperturbedto obtainthesameresultforasetof
nunitballs,disjointornot,withno4centerscoplanar. 2
7 Generalizations
Inthissectionweprovideseveralgeneralizationsofourresults.
7.1 Poisson distribution
Consider a set of unit balls whose centers are drawn by a 3-D Poisson point process of
parameter in the universe U. By aPoisson point process of parameter in U [11], we
meanthatwegenerateX randompointsinside U sothat
Pr(X =k)=
(Volume(U) ) k
exp( Volume(U))
k!
(4)
and for any disjoint subsetsM andM 0
of U, thenumber ofthe points inside M and the
numberofpointsinsideM 0
areindependentrandomvariables. NotethatEquation(4)yields
thattheexpectednumberofpointsinsideU isVolume(U)= 4
3 n.
The following simpleargument showsthat ourresultsextend to this distribution. Let
X betherandomvariable representingthenumberof centers ofunit balls generatedby a
PoissonpointprocesswithparameterinU,andletY betherandomvariablerepresenting
thenumberofT4-segmentsamongstthoseballs. TheexpectednumberofT4-segmentsis
E(Y)= 1
X
k =0
E(YjX=k)Pr(X =k):
Theorem1givesE(YjX=k)=(k)and
Pr(X=k)= (
4
3 n)
k
exp(
4
3 n)
k!
:
Thus
E(Y)=
4
3
n exp(
4
3 n)
P
1
k =1 (
4
3 n)
k 1
(k 1)!
=(n exp(
4
3
n) exp(
4
3
n))=(n):
ThereforetheexpectednumberofT4-segmentsamongstnballswhosecentersaregenerated
byaPoissonpointprocesswithparameterin U is(n). Similarlythisboundextends to
theexpectedsizeofthevisibilitycomplex.
Wenowinvestigatevariousmodelsin which wechangetheshapeoftheuniverseorthe
natureoftheobjects.
7.2 Smooth convex universe
Our results can be generalized to the case where the universe is no longer a ball, but a
homothet ofasmoothconvexset withhomothetyfactor proportionalto 3 p
n. This canbe
achievedbyconsidering theradius ofcurvatureoftheboundaryoftheuniverse,insteadof
R ,in theproofsofthelemmasdealingwithtangentsoutsidetheuniverse.
7.3 Other objects
Letr
min andr
max
be twostrictly positivereal constants. Inthe following, we bound the
expectednumberofT4-segmentsamongstballswhoseradiivaryintheinterval[r
min
;r
max ],
amongstpolyhedra eachenclosedbetweentwoconcentricballs ofradiir
min andr
max ,and
amongstpolygonseachenclosedbetweentwoconcentriccirclesofradiir
min andr
max . The
centersoftheconcentricballsorcirclesarecalledthecentersofthepolyhedraorpolygons,
respectively. In each case a T4-segment is called outer if the centers of the twoextremal
objects it is tangent to are farther apart than 6r
max
and are both at distance less than
2r
max
fromtheboundaryofU. OtherwisetheT4-segmentiscalled inner.
Forthese models, the proof of the (n) lowerbound on the expected number of T4-
segments(Section4) generalizesdirectlybecause,forthekindofobjectsweconsider,there
alwaysexistplacementsoffourofthemsuchthat theyadmit atleastonecommontangent
linewithmultiplicityone.
7.3.1 Balls of various radii
Wehaveconsideredamodelwherealltheballshavethesameradius. Ifweallowtheradiito
varyin theinterval[r
min
;r
max
], thentheproofof thelinearupperbound ontheexpected
number of inner T4-segments generalizes almost immediately by considering the volumes
H (p
i
;p
l
;2r
max
)andH (t
i
;t
l
;r
min
)insteadofH (p
i
;p
l
;2)andH (t
i
;t
l
;1).
Section3.2.1generalizesimmediatelytoprovethattheexpectednumberofT4-segments
tangentto fourballs B
i ,B
j ,B
k and B
l
inthat ordersuchthat p
i andp
l
arecloserto one
anotherthan6r
max
isO(n). TheonlydiÆculttaskforextendingSection3.2.2istheproof
ofthefollowinganalogofLemma10.
Lemma28 Whenx
i;l
>6r
max ,y
i
>2r
max andL
!
i;j;k ;l
occurs,thevolumeofH (t
i
;t
l
;r
min )
\U isgreaterthan
24 r
2
min (x
i;l 6r
max ).
Proof: Theproof is similar to the proof of Lemma 10. Refer to Figure 6. Letm bethe
midpointofsegmentt
i t
l
andKbethesphereofdiameterr
min
centeredonthepointclying
onsegmentt
i p
i
atdistance 1
2 r
min fromt
i
. ThesphereKisentirelyinsideH (t
i
;t
l
;r
min )\U,
mliesin H (t
i
;t
l
;r
min
)andastraightforwardcomputation showsthatm isin U sincet
i is
in U at distance at least r
max
from its boundaryand t
l
is at distance at most r
max from
U. Thus H (t
i
;t
l
;r
min
)\U containstheconvexhullof K and m which contains thecone
ofapexm,ofbaseadiskwhoseboundaryisagreatcircleofK,andofheightthedistance
frommto thecenter cofK. Now
x
i;l
=jp
i p
l j6jp
i
cj+jcmj+jmt
l j+jt
l p
l j
6r
max
+jcmj+ 1
2 jt
i t
l j+r
max
62r
max
+jcmj+ 1
2 (x
i;l +2r
max ):
Thusjcmj>
1
2 x
i;l 3r
max
andthevolumeoftheconeisatleast 1
3 (
r
min
2 )
2
( 1
2 x
i;l 3r
max )=
24 r
2
min (x
i;l 6r
max
). 2
t
i
p
i
m c
t
l p
l K
Figure6.FortheproofofLemma28.
TherestofSection3.2.2generalizeseasilyforprovingthattheexpectednumberofT4-
segments tangent to four balls B
i , B
j , B
k and B
l
in that order such that p
i and p
l are
farther apart than 6r
max and p
i
is farther than 2r
max
from the boundary of U, is O(n).
HencetheexpectednumberofinnerT4-segmentsisO(n).
Ourproofcannotbeextendedtoprovidealinearupperboundontheexpectednumberof
outerT4-segments.Thisisbecause,ifballsB
i andB
l
areofradiusr
max
thenalinesegment
t
i t
l
tangent to B
i and B
l
mightbe outside U and at distance greater than r
min
from its
boundary. ThenH (t
i
;t
l
;r
min
)doesnotintersectU andwecannotboundH (t
i
;t
l
;r
min )\U
frombelowbyapositiveconstantasinLemma14,whichiscrucialfortheproofofLemma15
andthusforProposition20.
However,bynottakingintoaccounttheocclusionintheproofofProposition20,weget
thattheexpectednumberofouterT4-segmentsisO(n 2
). RefertotheproofofProposition20
andconsiderI
x>6rmax;y<2rmax
,theanalogofI
x>6;y<1
forthiscase. TheanalogsofLemmas6
and7yieldthat
Pr(Æ
!
i;j;k ;l jx
i;l
=x; y
i
=y; y
l 6y
i )6
(3xr 2
max +8r
3
max )
2
R 6
:
Lemma18still holdsandwecaneasilyprovetheanalogofLemma 19. Both resultsimply
that
I
x>6rmax;y<2rmax 6
Z
2R
x=6r
max Z
2r
max
y=0
(3xr 2
max +8r
3
max )
2
R 6
6xydx
R 3
3dy
R
2O
1
R 6
=O
1
n 2
:
Hence the expectednumber ofinner T4-segmentsisO(n)and theexpected numberof
outerT4-segmentsisO(n 2
). ThisstillimprovestheresultofDurandetal.[10]whoproved
aboundofO(n 8=3
)forthesamemodel.
In this section wehaveassumed that the spherecenters are uniformly distributed but
wehavemadenoassumption onthedistribution oftheradii ofthespheresin theinterval
[r
min
;r
max
],whicharethusassumedto beworstcase. Theadditionofsomehypothesison
theradiidistributionmayyieldbetterresultsonthenumberofouterT4-segments.
7.3.2 Polyhedraof boundedaspect ratio
Considerpolyhedra of constant complexity, each enclosed betweentwoconcentric balls of
radiir
min andr
max
whosecentersareuniformlydistributedinU. Insuchacase,asforballs
ofvariousradii,theO(n)boundontheexpectednumberofinnerT4-segmentsimmediately
appliesaswellas theO(n 2
)bound ontheexpectednumberofouterT4-segments.
