Constructing the Exact Voronoi Diagram of Arbitrary Lines in Space

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Constructing the Exact Voronoi Diagram of Arbitrary Lines in Space

Michael Hemmer, Ophir Setter, Dan Halperin

To cite this version:

Michael Hemmer, Ophir Setter, Dan Halperin. Constructing the Exact Voronoi Diagram of Arbitrary

Lines in Space. [Research Report] RR-7273, INRIA. 2010, pp.19. �inria-00480045�

a p p o r t

d e r e c h e r c h e

0 2 4 9 -6 3 9 9 IS R N IN R IA /R R -- 7 2 7 3 -- F R + E N G

Algorithms, Certification, and Cryptography

Constructing the Exact Voronoi Diagram of Arbitrary Lines in Space

with Fast Point-Location

Michael Hemmer — Ophir Setter — Dan Halperin

N° 7273

May 2010

Centre de recherche INRIA Sophia Antipolis – Méditerranée 2004, route des Lucioles, BP 93, 06902 Sophia Antipolis Cedex

with FastPoint-Loation

Mihael Hemmer

∗

, Ophir Setter

†

,Dan Halperin

†

Theme: Algorithms,Certiation,andCryptography

Algorithmis,Programming,SoftwareandArhiteture

Équipes-ProjetsGeometria

Rapportdereherhe n° 7273May201016pages

Abstrat: We introdue a new, eient, and omplete algorithm, and its

exatimplementation,to omputetheVoronoidiagram oflinesin spae. This

is a major milestonetowardsthe robust onstrution of the Voronoi diagram

of polyhedra. As we follow the exat geometri-omputation paradigm, it is

guaranteed that we always ompute the mathematially orret result. The

algorithmisompletein thesense thatitanhandleallongurations,inpar-

tiular all degenerate ones. The algorithm requires

O(n ^3+ε )

^{time and spae,}

where

n

^{isthenumberoflines. TheVoronoidiagram is}representedbyadata struture that permits answering point-loation queriesin

O(log ² n)

^expeted

time. The implementation employs the Cgal pakages for onstruting ar-

rangementsandlowerenvelopesonparametrisurfaestogetherwithadvaned

algebraitools. Supplementarymaterialandinpartiulartheprototypialode

of our implementation anbe found in the website: http://ag.s.tau.a.

il/projets/internal-projets/3d-lines-vor/projet- page.

Key-words: VoronoiDiagram,PointLoation,LowerEnvelope,ExatGeo-

metriComputing,CGAL

ThisworkhasbeensupportedinpartbytheIsraelSieneFoundation(grantno.236/06),

by the German-Israeli Foundation (grant no. 969/07), and by the Hermann Minkowski

MinervaCenterforGeometryatTelAvivUniversity.

∗

Geometria,INRIASophiaAntipolis,Frane;Mihael.Hemmersophia.inria.fr

†

ShoolofComputerSiene,Tel-AvivUniversity,Israel;[ophirset,danha℄post.tau.a.il

droites quelonques dans l'espae

ave loalisation eae

Résumé: Nousprésentonsunnouvelalgorithmeeaeetomplet,ainsique

sonimplantationexate,pouralulerlediagrammedeVoronoïdedroitesdans

l'espae. C'est une étape majeureverslaonstrution robuste du diagramme

deVoronoïdepolyèdres. Noussuivonslemodèledualul géométriqueexat,

il est dongaranti quenous alulons toujours le résultat mathématiquement

orret. L'algorithmeestompletenesensqu'ilpeuttraitertouteslesong-

urations, enpartiuliertous lesas dégénérés. L'algorithmea une omplexité

O(n ^3+ε )

^{entempsetenespae,où}

n

^{estlenombrededroites. Lediagrammede}

Voronoïestreprésentéparunestruturededonnéespermettantderépondreaux

requêtesde loalisationsdepointsave uneomplexité moyenneen

O(log ² n)

^.

L'implantation utilise les modules CGAL de onstrution d'arrangements et

d'enveloppesinférieures surdes surfaesparamétriques ainsiquedes outilsal-

gébriques avanés. Lematériel supplémentaireet en partiulierle ode proto-

type de notre implantation peuvent être trouvés sur lesite: http://ag.s.

tau.a.il/projets/internal-projets/3d-lines-vor/projet- page.

Mots-lés : DiagrammedeVoronoï,Loalisation,EnveloppeInférieure,Cal-

ulGéométriqueExat,CGAL

1 Introdution

TheVoronoidiagram(

VD

^{)isamongthemost}fundamentalstruturesinCom- putational Geometry,andisknowntobeausefultoolin avarietyof domains.

Forinstane,struturalbiology[24,35℄androbotmotionplaning[19,34℄apply

Voronoi diagrams to enode point sets keepingmaximal distane from atoms

or obstales, respetively. A related onept is the medial-axis transform [6℄,

whih is onsidered fundamental in solid modeling and applied to problems

suhasnite elementmeshing, shapemorphing,andfeature reognition. Yet,

theadaptation ofomplexthree-dimensional Voronoidiagrams in professional

tools has been very slow. Their use is hindered by the diulty of design-

ing and implementing reliablegeometri algorithms for omplexstrutures in

three-dimensionalspae.

Voronoidiagramshavebeenthesubjetofatremendousamountofresearh.

WereferthereadertothesurveybyAurenhammerandKlein[2℄ofworkpub-

lisheduptill2000. Voronoidiagramsin

R ²

^{arewellunderstoodinalmostallas-}

pets,thatis,intermsofomplexityandoptimalalgorithmsaswellasinterms

ofrobustandeientimplementations. In

R ³

^{muhlessisknown,evenforsim-}

ple objetssuh aslines, segments, orpolyhedra. Forexample, atight bound

on the ombinatorial omplexity of the

VD

^of

n

^{lines or line segments in}

R ³

isunknown; itis onjeturedthattheomplexityisnear-quadrati;theknown

lower bound is

Ω(n ² )

^{[1℄, but the best known upper bound is1}

O(n ^3+ǫ )

^[31℄.

Intheaseoflineswith axednumber

c

^oforientationstheupperbound was improvedto

O(c ⁴ n ^2+ε )

^{[25℄. A omplete analysis ofall possible}ombinatorial asesforthree arbitrarylines ispresentedbyEverettetal.[17,18℄.

Today,therearemanypublishedresultsonrobustonstrutionsofdierent

typesofVoronoidiagramsin

R ²

^{. NotonlyVoronoidiagramsofpointsareon-}

sidered,butalsoVoronoidiagramsoflinesegments[22℄,irles[15℄,ellipses[16℄,

and more [8, Ÿ2℄. In

R ³

^{, an exat} implementation of the Voronoi diagram of additively-weightedpointswasanalyzedin[7℄,butwearenotawareofanyex-

at, omplete,and implementedalgorithm that omputesVoronoidiagrams of

lines,linesegments,orpolyhedra. Nevertheless,progresshasbeenmadetoward

theexatomputationofthearrangementofquadris[5,13℄. EahVoronoiell

ofthediagramoflinesinspaeanberepresentedastheunionofellsofsuh

an arrangement. Other approahesexpliitlyaim foran exatorrobustom-

putationof theVoronoidiagram (or themedial axis)[10,26℄. However,those

approahes are not omplete. For example, Culver'salgorithm [10℄ does not

handlesingulartrisetor-urves.

Finally, Hanniel and Elber [20℄ provided an algorithm to onstrut the

Voronoi ell of bounded planes, spheres, and ylinders in

R ³

^{. Though it is}

in some aspets similar to ours, the approah is approximate, does not deal

withdegeneraies,andleavesrobustnessissuesaside.

Wepresentanexatandomplete(andthusrobust)algorithmforomputing

theVoronoidiagram ofarbitrarylines in threedimensionswith respetto the

Eulidean metri. The algorithmrequires

O(n ^3+ε )

^{time andspae, where}

n

^is

thenumberofinputlines. Thedatastrutureadmitsansweringofpoint-loation

queriesin

O(log ² n)

^{time. Weantiipate thatthe natureof thealgorithm and}

1

Aboundoftheform

O(f(n) · n ^ε )

^{meansthattheatualupperboundis}

C ε f(n) · n ^ε

^,for

any

ε > 0

^,where

C ε

isaonstantthatdependson

ε

^{,andgenerallytendstoinnityas}

ε

^goes

to

0

^.

thegeneralapproahofitsimplementationonstituteamajormilestonetowards

anexatandrobustonstrutionoftheVoronoidiagramofpolyhedrain

R ³

^.

Weutilizethe fatthat in Eulidean spae theVoronoiellanbeonsid-

eredasalowerenvelopesinetheellessentiallyhasaertainstarshapedness

property: Foranypoint

p

^{insidetheVoronoiellofaspeilinesite}

ℓ

^,theline

segment onneting

p

^{to itsprojetion}

p ℓ

^onto

ℓ

^{,is fully ontainedin theell.}

ThisobservationenablesustorepresenttheVoronoiellof

ℓ

^asaminimization diagram,whihis(oneptually)embeddedonaninnitesimallysmallylinder

around

ℓ

^{. This}observationissimilar to (butnotthesameas) thewell-known onnetionbetweenVoronoi diagramsand lowerenvelopes[14℄. Lowerdimen-

sionalellsarerepresentedseveraltimes,namelyaspartoftheboundaryofthe

VC

^{ofeahline theyare assoiatedwith. The}implementationisdevelopedin andbasedonCgal,ComputationalGeometryAlgorithmsLibrary.

2

Thepaperisorganizedasfollows. Setion2disussespreliminarysubjets,

suh as properties of bisetors and trisetors of lines in spae and the lower

envelope algorithm. Setion 3 desribes the details of the onstrution of a

Voronoiell. Setion 4disussesthepointloationalgorithm anditsanalysis.

Setion 5gives implementation details and presentspreliminary experimental

resultsthatwereobtainedwithoursoftware.

2 Preliminaries

Let

O = { s 1 , s 2 , . . . , s n }

^{be a set of objets in}

R ^d

^{, also referred to as sites.}

We follow the Voronoidiagram denition by Everett et al. [18℄: TheVoronoi

diagram

VD( O )

^{is the} subdivision of

R ^d

^{into ells, where eah ell}

VC (S)

^is

assoiated with a subset

S ⊆ O

^{, suh that every point in}

VC(S)

^{is stritly}

loser to all sites in

S

^{than to all other sites in}

O

^{and is} equidistant from all sitesin

S

^{. Theformaldenition is:}

VC(S) =

p ∈ R ^d

∀ s ∈ S, t ∈ O \ S : d(p, s) < d(p, t)

∀ s, t ∈ S : d(p, s) = d(p, t)

In theontextof this paper,

O

^{denotesa set of arbitraryrational lines in}

R ³

and

d( · , · )

^{denotestheEulideandistanefuntion. Thesetofpointsthat isof}

equaldistanetotwoorthreesitesisalledabisetorortrisetor,respetively.

where

d( · , · )

^{denotes theEulidean distanefuntion. Whentheardinally}

of

S

^{istwoorthreeitisalledbisetorortrisetor,} respetively.

2.1 Properties of Bisetors and Trisetors

WenextstatesomepropertiesofbisetorsandtrisetorsoftheVoronoidiagram

oflinesin

R ³

^{thatareusedthroughoutthispaper.} Proposition1givesproperties ofbisetors;seeFigure1forillustrations.

Proposition1 The bisetorof twolines

ℓ 1

^and

ℓ 2

ⁱⁿ three-dimensional spae iseither(a)ahyperboliparaboloid(asurfaeofalgebraidegree2),if

ℓ 1

^and

ℓ 2

areskew,(b)aplane,if

ℓ 1

^and

ℓ 2

^{areparallel,or()apairoforthogonalplanes,}

if

ℓ 1

^and

ℓ 2

^{areonurrent. In the latterase, the singularlousof thebisetor}

2

www. gal. org

(a) (b) ()

Figure1: Bisetorof:(a)twogenerilines;(b)twoparallellines;()twointersetinglines.

Thediagramswerereatedwithourimplementation(seeSetion5),and werelippedbya

sphereforonveniene.

isalinethat isperpendiular to

ℓ 1

^and

ℓ 2

^{andpassesthrough their}intersetion point.

Themain theoremofEverettetal. [17,18℄providesagoodoverviewofthe

dierentasesofthetrisetor:

Theorem2 (Everett etal.) The trisetor of three lines is either (i) a non-

singular quarti, if the three lines are pairwise skew but not all parallel to a

ommonplanenoronthe surfaeofahyperboloidofrevolution,(ii)aubiand

a line that do not interset, if the three lines are pairwise skewand lie on the

surfaeofahyperboloid ofrevolution,(iii)anodalquarti,if thethreelinesare

pairwiseskewandallparalleltoaommonplane,(iv)oneparabolaorhyperbola,

ifthereisexatlyonepairofoplanarlineswhihareparallel,(v)twoparabolas

or hyperbolas that interset, if there is exatly one pair of oplanar lines that

interset, (vi) between 0 and 4 lines, if there are two pairs of oplanar lines,

or (vii) one line, in the ase of three oplanar onurrent lines, the ommon

singularlousof the bisetors.

WeuseaorollaryoftheabovetheoreminSetion3,wherewedesribethe

onstrutionofaVoronoiellin thediagramoflines.

2.2 Lower Envelope Algorithm

Again, we regardeah three-dimensional

VC

^{asalowerenvelopewith respet}

toitslinesite

ℓ 0

^{. Thisloverenvelopeis}representedasaminimization diagram whihisoneptuallyembeddedinin the

uv

^{-parameterspaeofthesurfaeof}

aninnitesimallysmallylinderaround

ℓ 0

^{.3 Weutilizethe}divide-and-onquer algorithmforonstrutinglowerenvelopes[1℄asitisimplementedinCgal[32,

Ÿ8.5℄,whihwebrieydesribenext.

Sinethealgorithmprojetsbisetorsintotheparameterspae,allbisetors

are initiallysplit upinto

uv

^{-monotonesurfaes. Thealgorithm thensplits the}

resultingset

G

^{intotwosubsets}

G ¹

^and

G ²

^{ofroughlyequalsize,andreursively}

omputestheir minimization diagrams

M ¹

^and

M ²

^{. Inthe onquerstep, the}

twodiagrams are mergedinto one. First,the overlayof

M ¹

^and

M ²

^{is om-}

puted, whereeahfeatureislabeledwithuptotwosets oflabels

L 1

^and

L 2

^of

andidatesurfaes frombothdiagrams. Thereafter,thearrangementis further

renedsuhthat eahfeatureaneitherbelabeledwith

L 1

^,

L 2

^,or

L 1 ∪ L 2

^{. In}

3

SeeSetion3fordetailsonthe

uv

^{-parameterspaesetting.}

partiular, eahfae that is labeled withtwobisetorsis rened by theorre-

spondingprojetedtrisetorurve. Notethatthis stepanalsosplitupedges.

After theomparison of bisetorsthealgorithm removesredundantedges and

verties,whihyieldsthenaldiagram. Theomplexityoftheabovealgorithm

is

O(n ^2+ε )

^{,withtheonditionthatthebisetorsurfaesare}well-behaved.

Note that the algorithm heavily relies on arrangement operations suh as

overlay,whihareprovidedby[32,Ÿ8.1℄and[4℄. Thoughwetreatthisasablak

boxthroughoutmostofthepaper,somemoredetailsanbefoundinSetion5.

Theadditionalonstrutionsandprediatesrequiredbythelowerenvelopealgo-

rithmare: theonstrutionoftheprojetedboundaryof

uv

^{-monotonesurfaes,}

theonstrutionoftheprojetedintersetionoftwo

uv

^{-monotonesurfaes,and}

theomparisonoftwobisetorsaboveafae,edge,orvertex.

3 Computing a Voronoi Cell

Thissetiondisussestheomputationofthe

VC

^{ofoneline, referredtoasthe}

baseline anddenotedby

ℓ 0

^.

The two-dimensional pakage of Cgal has the infrastruture to ompute

envelopesoverylinders. However,fortheeienyoftheimplementationitis

importanttokeepthealgebraidegreeoftheprojetedurvesaslowaspossible.

Therefore,weprojettheurvesontwoparallelplanesthatsandwih thebase

line,whilekeepingtheprojetiondiretionnormaltotheylinder. Thisredues

themaximumdegreeofaprojetedtrisetor urvefrom sixteendownto eight.

3.1 Parametrization and Projetion

Let

F = { − → b 1 , − →

b 2 , − →

b 3 }

^{beanorthogonalbasisof}

R ³

^{whihishosensuhthat}

− → b 1

^is

thediretionofthebaseline

ℓ 0

^{. Moreover,let}

p 0

^{besomerationalpointon}

ℓ 0

^.

Now,onsider the parametrization

X (u, v, r) = p 0 + u · − →

b 1 + v · r · − →

b 2 + r · − → b 3

^.

Notethat

X (u, v, ± 1)

^{denestwoparallelplanes(}

uv

^{-planes)thatsandwih}

ℓ 0

^,

whih we all the positiveand the negativeplane, respetively. Thus a point

X (u 0 , v 0 , ± 1)

^{representsaraythatoriginatesfrompoint}

p 0 + u 0 · − →

b 1

^on

ℓ 0

^with

diretion

± (v 0 · − → b 2 + − → b 3 )

^.

Notethattheplane

H ^∗ = { x ∈ R ³ | (x − p 0 ) ^T · − → b 3 = 0 }

^{isnotoveredbythe}

parametrization. However, onean simplyglue the arrangementstogether as

longasthehosenframeisgeneri,thatis,urvesarenotallowedtotouh

H ^∗

^,

interset in

H ^∗

^{, or even be ontainedin}

H ^∗

^{. However, urvesare allowedto}

transverselyinterset

H ^∗

^,whereeahintersetiongivesrisetoasimplevertial asymptoteintheprojetion.

Inordertoavoidtheseritialases,wegeneratetheloalframebysetting

−

→ b 2

to somerandomvetorthat is orthogonalto

−

→ b 1

^{. Thoughthisframe isgeneri}

with high probability, we also hek in all relevant prediates that the frame

is indeed generi. If neessary, we restart the omputation hoosing another

random frame. We hose the standardstrategy that inreasesthe number of

randombitsused foreahiteration. Thiswayweguaranteeterminationanda

smallnumberofadditionalbitsdue totherandomization.

Wehighlightbelowseveralmajorissuesintheprojetionofatrisetor. The

projetionof abisetors' boundaryand adetailed ase analysis is deferredto

AppendixA . Werelymerelyonthegeneriframeandonthefollowingorollary

that diretlyfollowsfromTheorem 2:

Corollary 3 Thesetofpointswherethetrisetordoesnotrepresentatransver-

sal intersetion of the bisetors is a 0-dimensional set, namely, the singular

pointsof thetrisetor. Theonly exeptionisthe aseofthreeoplanaronur-

rent lines; inthis asethe trisetor isthe ommon singular lous (line) of the

three bisetors.

For atrisetor

T 0ij

^let

B 0i , B 0j , B ij ∈ Q [x 1 , x 2 , x 3 ]

^{be the three trivariate}

polynomialsoftherelevantbisetors. Nowlet

B 1

^and

B 2

^{bethetwobisetors}

of minimal degree,

d 1

^and

d 2

^, respetively. The projetionis arried outbya resultant omputation [33℄. However,sine wewish to projettowards

ℓ 0

^we

rstsubstitute

X (u, v, r)

^into

B 1

^and

B 2

^{andomputetheresultantwithrespet}

to

r

^.

res(u, v) := resultant(B 1 ( X (u, v, r), B 2 ( X (u, v, r)), r) ∈ Q[u, v] .

−

→ v 2

−

→ v 3

−

→ v 1

B 0 i

B 0 j

X (u, v,+1)

X (u, v, − 1) v →

← v ℓ 0

T 0 ij

T 0 ij

cross-section perpendicular to ℓ ₀

Thisisatmostabivariatepolynomialof

degree

2d 1 d 2

^{. Thus, in theworst ase (the}

generi ase) this is an irreduible polyno-

mial of degree 4

only

8

^{. However,due to its}

algebrai nature the approah an not im-

mediately distinguish between the positive

andthenegativeparameterspae. TheFig-

uretotherightillustrateshowtheresultant

projets

T 0ij

^{into the positiveand negative}

plane. Werstsplituptheprojetedurveinto

u

^{-monotonesegmentsusing[4℄.}

Inpartiular,urvesaresplitupatvertialasymptotes.

Inordertodeidethatanar

α

^{isonaertainplain weutilizeCorollary3,}

namelytheobservationthatinallbutoneexeption(whihishandledexpliitly)

twobisetorsmustintersettransverselyalongthetrisetorurve,whihimplies

that theymustinterhangetheirorderwhilepassingtheprojetedtrisetor.

Thisisdetetedbytworayshootsatrationalpointsrightaboveandbelow

α

^.

Let

p

^and

p

^{bethese twopoints,} respetively. Toensurethat bothpoints are hosensuientlylose,weonstrutarationalvertialline

L

^{thatintersets}

α

initsinterior,sayatpoint

p α

^{. Wehoosethepointson}

L

^{suhthattheyisolate}

thearfromallother intersetionsof

L

^with

res

^{. Nowonsiderthepathon}

L

from

p

^(or

p

^)to

p α

^.

p

^{issuientlyloseto}

α

^{sinethispathdoesnotinterset}

res

^{until itreahes}

α

^{. Inase}

α

^{isvertial,wehoose}

L

^tobehorizontal.

Furtherdetailsandin partiularhowtoguaranteethatthehosenframeis

generi,anbefoundinAppendix A.

3.2 Lower Envelope Prediates

A ore part of the envelope algorithm is the representation of minimization

diagrams as labeled arrangementsand the overlayof suharrangements. The

requiredprediates forthese operations relateto planaralgebraiurvesonly,

4

More preisely, itisa bivariatepolynomialofbi-degree atmost

(4, 4)

^{. Fora standard}

rationalparametrizationoftheylinder,wewouldobtainapolynomialofbi-degree

(8, 8)

^or

16

^intotal.

whih are provided by[4℄. However,it remains to ensurethat nointersetion

takesplaein

H ^∗

^{. This boilsdownto testingthattheleadingoeientswith}

respet to

v

^{of twonon} overlapping (o-prime) urveshave no ommon root.

Thus, weprovideaslightly modiedset ofprediatesthat additionallyensure

thisondition.

Theprediatesthatareadditionallyrequiredbytheenvelopealgorithmare

theomparisonoftwobisetorsaboveafae,anedge,oravertex,respetively;

seealsoSetion2.2. Forafae,itissuienttoseletarationalpointinsideit

and omparethesurfaes along theorrespondingray. The pointis hosenin

asimilar way to theapproah used whensorting thetrisetorsto thepositive

and negative planes. For an edge we onstrut a vertex in its interior and

omparealongtheorrespondingray. Foravertex,whihmaynothaverational

oordinates, we rst hek whether the point is on the projetedintersetion

of the two bisetors, and report equality if it is indeed the ase. Otherwise

weomparebisetorsatarationalpointsuiently loseto thevertex, where

suientlyloseisagaindeterminedbyasimilarstrategyasinsortingtrisetor

urves.

3.3 Complexity

Forthetimeomplexityandspaeomplexityanalysisweignoreadditionalosts

that mayarise due to variable bit-lengthof various implementationsadhering

totheexatomputationparadigm[36℄. Wealsoignoretheadditionalrun-time

that anresultfrom apoorhoieofageneriframe(Setion3.1),asitisnot

thegeneralase,andhasnoimpatonperformaneinexpetation.

Thebisetorsurfaes arealgebraisurfaes ofmaximumdegreeof

8

^there-

fore,thetime-omplexityof thelowerenvelopealgorithmis

O(n ^2+ε )

^{(whih is}

alsothebestknownupperbound). Thustherun-timeomplexityofomputing

theellsforall

n

^linesis

O(n ^3+ε )

^{,whihalsobounds thespaeomplexity.}

4 Fast Point Loation

Given a query point

p

^{we wish to nd the losest line to it. Consider the}

followingpoint-loationstrategy: Westartwitharandomline site

ℓ

^{. Firstwe}

projet

p

^on

ℓ

^{andloateitsimageinthe}minimizationdiagramof

ℓ

^{. Theimage}

isloatedonafeature oftheminimizationdiagramwhihislabeledwitha(in

generalnotempty)setof linesites

S

^{. Wethen omparethedistane}

d(ℓ, p)

^to

d(ℓ ^′ , p)

^foroneline

ℓ ^′ ∈ S

^{. If}

d(ℓ, p)

^{islessthanorequalto}

d(ℓ ^′ , p)

^wereport

ℓ

or

S ∪ ℓ

^,respetively. Otherwiseweontinueintheellof

ℓ ^′

^{. Thiswalkthrough}

theVoronoidiagramterminatessinethereisonlyanite numberofellsand

thedistaneof

p

^{totheurrentlinealwaysdereases. Weanloatetheimage}

of

p

^insidetheminimizationdiagraminexpeted

O(log n)

^{timebyusingpoint-}

loation based on trapezoidal deomposition [27℄. Combining this algorithm

withtheideaoflandmarks[21℄mayalreadyhavegoodperformaneinpratie.

However,thealgorithmhasaworst-timetimeomplexity

O(n log n)

^.

Weturnitintoanalgorithmwithatime-omplexity

O(log ² n)

^byombining

itwith astrategythat issimilar toskip lists. Webuild ahierarhyofVoronoi

diagrams. Thelowest layerontainsthe

VD

^{of thefull setof lines, whileeah}

other layersontainthe

VD

^ofonly

1/k

^{(random)linesof thepreedinglayer,}

where

k > 1

^{issomeonstant. Thehighestlayer(therootlayer)ontainsonlya}

onstantnumberoflines,andthenumberoflayersis

O(log n)

^{. Inordertoloate}

apoint

p

^{werstloateitin the root layerusing the walkstrategy desribed}

above. Wethenproeedtothenextlayerstartingatthelinethatwasfoundin

thepreedinglayer.

The following theorem summarizes the performane of the point-loation

struture(see[11,23℄ forsimilar analysisin2D):

Theorem4 The expetedrunningtimeofthe point-loation queryinthe hier-

arhial VDstrutureis

O(log ² n)

^.

Proof: The number of ells visited at the root layer is obviously at most

k

^{. For all otherlayers,onsider thethepath bakward,from its targetto the}

soure: foreveryelltheprobabilitythatitisalreadythesoureis

1/k

^{. Thus,}

the expeted length of a path is

P n i=1

i

k ( ^{k −} _k ¹ ) ^{i − 1} ≤ k

^{. That is, theexpeted}

running-timeis

k P ^log k n

i=1 T(k ⁱ )

^{, where}

T (m)

^{istheexpeted timespentonthe}

point loationin theminimizationdiagramof

m

^{lines. Thus weobtainanex-}

petedrunning-timeof

O(log ² n)

^intotal.

2

Weremarkthatsomespeialasesareleftoutinthisdisussionforbrevity

(e.g.,pointsthatare ontainedin

H ^∗

^{),buttheyareompletelyhandled inour}

software.

5 Implementation Details

OurimplementationisbasedonCgal,whihfollowsthegeneri-programming

paradigm[3℄. Algorithmsareformulatedandimplementedsuh that theyare

abstratfromtheatualtypes,onstrutions,andprediates. Thus, theimple-

mentation of every algorithm and data struture in Cgal is parametrized by

a so-alledtraits lass [28℄, in whih these funtionalities are dened. Inpar-

tiular,auseranemployanalgorithmwithhisowntypes,onstrutions,and

prediatesby providinghis own traitslass. This wayitispossibleto ahieve

a great amount of exibility. At the extreme, it is possible to even partially

hange the nature of an algorithm, as we do here for the three-dimensional

lowerenvelopelass[32,Ÿ8.5℄.

The ore of our implementation is the traits lass for the lower envelope

algorithm, whih also needstobeavalidtraits lassforCGAL's arrangement

pakage. Therequiredfuntionalitiesbythearrangementpakageareprovided

bythetraits lasspresentedin [4℄. Theapproahredues allonstrutionand

prediates to ylindrial algebraideompositions of the plane for oneortwo

urves.Theapproahusesadditionalresultantomputationthatprojetsinter-

setion pointsontothe

u

^{-axis, that is,}

u

-oordinatesof intersetionpointsare represented as real roots of this univariateresultant polynomial. Thereafter,

thebersabovetherootsareinvestigatedin orderto determinethearofthe

urveonwhih theintersetion takesplae.

Weessentiallywraptheabovetraitslassandadd therequiredfuntional-

itiesby theenvelopealgorithm;see also Setion 3. Inasewe detetthat the

urrentframe is notgeneri an exeption is thrown, whih is then aught by

ourprimary lassthat omputesanewframeandrestartstheomputation of

(a) (b) ()

Figure 2: Diagramsarelippedbyasphere foronveniene. (a)

VD

^of

4

^{lines, obtained}

byrotating onelinearound the

z

^{-axis. Allbisetorsmeetinthat axis. (b)}

VD

^of

4

^lines

interseting inonepoint. ()

VD

^of

4

^{lines,twolinesintersetandtheothersareparallelto}

eahofthem,respetively.

the ell. Foreah Voronoi ellwe keep aseparate instane ofthe traits lass,

whihisusedforbothplanes. Thisallowsahingofrelevantresults.

Approximationofthethree-dimensionaloordinatesofavertex,isbasedon

multi-preision oating-point interval arithmeti (MPFI) [8, Ÿ8℄. Sine this is

aertied approximation,weobtainabounding boxthat ontainsthevertex.

This ouldbe usedto easily establishtheadjaeny among lowerdimensional

ells. Forinstane, let

v

^{denote a vertex in a}minimization diagram

M

^{. The}

labelof

v

^{pointstoallother}minimizationdiagramsthatontainarepresentation of it. Let

M ^′

^{be oneof these diagrams and}

v ^′

^{be the} representationthat we wish to ndtherein. We oulduseasimilar approah to theoneusedin [13℄:

Byusingthelabels,weidentifyallpossibleandidatesin

M ^′

^{. Thissetontains}

onlyupto

8

representationsandontainsatleast

v ^′

^{. Weompute}progressively more preise bounding boxes for allandidates until only one (the one of

v ^′

⁾

overlapsthebounding boxof

v

^.

Our implementation an handle arbitrary rational lines, in partiular, it

anhandle all possible degenerateases. Figure 2depits degenerateVoronoi

diagrams. Eah mesh was generated using CGAL's pakagefor labeledmesh

domains[30℄. Theorale,whihisrequiredbythemeshgeneration,waswritten

suhthatitonlyutilizes(andtherebytests)ourpointloationstruture. Lower

dimensional features were approximated using the approah disussed above.

In order to ahieve sharp edges the proteting balls tehnique introdued by

Bolthevaetal.[9℄wasapplied. medit[29℄wasusedforthenalvisualization.

Sineweaim to eventually inorporate our ode into aCgal pakagethe

software is developed within the revision ontrol system of the projet. All

experimentswithin thissetion wherearriedoutonan internalCgal release

CGAL-3.7-I-27,whihalreadyomprisesalltheneessaryalgebraitools[4℄.

However, the trapezoidal map is urrently not available for minimization di-

agrams due to ongoinghanges in the arrangementpakage (it is antiipated

soon),whih foresus toresortto asimplerpointloationstrategiesfornow.

Finally,wepresentpreliminaryresultsobtainedwithoursoftware. Thepoint

loationstrutureasitisdisussedinSetion4leavestheratio

k

^amonglevels

N \k

^{2 4 8 16 20 24 28}

+∞

16 6.48 4.30 4.34 N/A N/A N/A N/A 3.94

36 8.09 6.33 5.33 5.67 N/A N/A N/A 5.62

64 9.77 6.42 5.73 6.07 6.00 6.12 6.83 6.63

100 9.87 7.22 6.18 6.45 6.97 6.83 7.13 7.43

Table1: Averagenumberofvisitedellsperquery,where

k

^{denotestheratioofthehierarhy}

and

N

^{the number oflines. Entriesfor}

n < 2k

^{arerepesented bythe lastolumn. Tothe}

rightisdepitedaVoronoidiagramof

5

^{parallellines.}

undetermined. Inorder to show theimpat of

k

^{wereated random instane}

of parallel lines with oeients in the range

[0, 2 ¹⁰ ]

^{.5 For eah instane, we}

reated

10

^{Voronoidiagram}hierarhies,whihwherequeriedwith

1000

^random

pointsin

[0, 2 ¹⁰ ] ³

^eah.

Table 1shows theaverage numberof visited ellsperquerydepending on

thenumberoflinesandthehosenvaluefor

k

^{. Thelastolumnshowsthepure}

walkwithoutahierarhy,whih suggestsanaveragequery timein

O( √ n)

^{, as}

onemayalso expet due to resultsin [12℄. For largerinstanes, itseemsthat

hoosing

k

^equalto

8

^isappropriate.

6 Conlusions

We havepresented anexat, omplete, and thus robust, algorithm that om-

putes the Voronoi diagram of arbitrary rational lines in

R ³

^{. The algorithm}

requires

O(n ^3+ε )

^{time and spae, where}

n

^{is the number of lines. The in-}

trodued datastruture permits to answerpoint loation queriesin

O(log ² n)

expeted time. The implemented prototype is exat and an handle all de-

generateases. Wereferto http://ag.s.tau.a.il/projets/internal-

projets/3d-lines-vor/projet- page forthe mostreentversionand sup-

plemental material.

Thealgorithmisintentionallydesignedsuhthatitavoidstediousasedis-

tintions, whih makesit implementable,maintainable and, in partiular, ex-

tensibleto otherprimitivessuhaspoints,line segments,andtriangles. Thus,

weonsiderourapproahasamajormilestonetowardstheomputationofthe

Voronoidiagramofpolyhedrainthreedimensions.

Moreover,weexpetthatitwillpavethewaytodevisingathree-dimensional

variantofthevisibility-Voronoiomplex[34℄,astruturethatenablestotrade-

o learaneand pathlength in robot motionplanning,and hasproved tobe

espeially usefulin theplane.

Ourapproahmayalsobegeneralizedtospheres(seealso[20℄)whihwould

open the doorfor innovative solutionsto entral problems in StruturalBiol-

ogy[24,35℄.

Aknowledgments: Theauthors thankS. Lazard andM. Yvineforfruitful

disussionsandMoniqueTeillaudforhertranslationof theabstratto Frenh.

5

Sinethetrapezoidalmapisnotyetavailableforenvelopes,wehadtoresorttoinstanes

thatkeeptheomplexityofaellsmall.

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pages452492.WorldSienti,2ndedition,1995.

A Projetion of Bisetors and Trisetors

Reall, from thedisussion in Setion 3that the parametrization

X (u, v, r)

^is

dened with respetto a loal frame

F = { − → b 1 , − → b 2 , − → b 3 }

^{and that the plain}

H ^∗

^,

whihisorthogonalto

−

→ b 3

^{,isnotontainedin}

X

^.

Ingeneral this is not aproblem sinea urve (trisetor)that transversely

intersets

H ^∗

^{appearsasasimplevertialasymptoteinthe}

uv

^{-parameterspae.}

Forinstane,ifaprojetedurveleavesoneplaneasavertialasymptoteatplus

innity,thenitreappearsatminusinnityontheotherplane. Ifthisistheonly

asethathappensitislearthatthereisaone-to-onorrespondeneamongthe

vertialasymptote (andthus edges) in thetwo minimization diagrams,whih

makesitpossibletogluethemtogether.

Weallaframeforwhihthisispossibleageneriframe,morepreisely:

Denition5 (GeneriFrame) Wedenetheframe

F

^{tobe generiif(i)no}

1-dimensional omponents of the trisetor (a oni or a line) 6

are ontained

in

H ^∗

^,(ii)everyintersetionofatrisetorwith

H ^∗

^isatransversalintersetion, (iii)notwourves(trisetors)intersetin

H ^∗

^,and(iv)the intersetionsof

H ^∗

with allbisetors

B 0i

^areregular.

Inthissetion weanalyze theasesthat weenounterwhile projetingbi-

setorboundariesandtrisetorurves. Inpartiular,wedisusshowwedetet

anon-generiframe.

A.1 Projetion of Bisetors Boundary

TheaseshereorrespondtotheasedistintioninProposition1.

Generi Case Let

B 0i ∈ Q[x 1 , x 2 , x 3 ]

^{be the polynomial} representing the bisetorbetween

ℓ 0

^{(thebaseline)andsomeotherline}

ℓ i

^{. Inthegeneriase}

B 0i

representsahyperboliparaboloidthat isdenein almostalldiretions ofthe

projetion. Theonlyexeptionareraysthat areperpendiularto

ℓ i

^andpoint

awayfrom

ℓ i

^{. Thesediretionsare}representedbyahorizontal(representingrays in thesamediretion)line. Thelineis haraterizedbytheleadingoeient

of

B 0i ( X (u, v, r))

^{with respetto}

r

^{. Oneah ofthe}

uv

^{-planeweinterpret this}

surfae as one or two

uv

^{-monotone surfaes. On the plane that ontains the}

above line, we split the bisetor into two

uv

^{-monotone surfaes; on the other}

planethereisexatlyone

uv

^{-monotonesurfae.}

Parallel Lines Inase

ℓ 0

^and

ℓ i

^areparallel,

B 0i

^{isof degree}

1

^sineitrep-

resents aplane. Theleading oeientof

B 0i ( X (u, v, r))

^{with respetto}

r

^is

a horizontal line that represents all rays that do not interset

B 0i

^{. That is,}

oneah ofthe

uv

^{-planesweobtainone}

uv

^{-monotonesurfae,whose projeted}

boundaryisthisline. Theproperhalfspaeanbedeterminedbyasimpleray

shoot.

6

Notethataubiannotbeontainedin

H ^∗

^.

IntersetingLines Inase

ℓ 0

^and

ℓ i

^{interset,thebisetor}

B 0i

degenerates to twoplanesthat intersetalong asingularrational line

ℓ s

^{, whih is perpen-}

diular to

ℓ 0

^and

ℓ i

^{. Let}

(u 0 , v 0 ) ∈ Q ²

^{be theparameter valuesfor that line.}

All lines

X (u 0 , v, r), v 6 = v 0

^{have a double} intersetion with

B 0i

^{. The lines}

X (u, v 0 , r), u 6 = u 0

^{donot interset}

B 0i

^{, sine theyare parallel to it. Thepa-}

rameterspaeissplitupalong

v = v 0

^and

u = u 0

^{,whihresultsin}

uv

^-monotone

surfaes,

4

^{onethepositiveand}

4

^{onthenegativeside.}

Ensuringa GeneriFrame Forallasesitholdsthattheonstrutionmay

trigger arestart of theomputation ifone of the horizontal lines is not seen,

thatis,iftheleadingoeientof

B 0i ( X (u, v, r))

^withrespetto

r

^hasdegree

0

^.

A.2 Projetion of Trisetors

Besides the exatness of the method desribed in Setion 3.1, it has the ad-

vantage that it is general and forgoesa huge ase distintion. In partiular,

wedonotfatorize thepolynomialinto itsfators thatrepresentthe dierent

omponentsmentionedinTheorem 2.

Twobisetorsareomparedabove(below)a

u

^-monotonear

c u

^,

7

onthepos-

itive(negative) planeasfollows. Forasuientlylose rationalpoint

p(u 0 , v 0 )

above(below)

c u

^{, the} orrespondingline is substituted into thetwobisetors.

Thisisatmosttwoquadratipolynomialsin

r

^{,eahhavingatmostonepositive}

and one negativeroot. If present, the positive (negative) roots from the two

bisetors are ompared. In ase that no root is present, there is no bisetor

above

p

^{andtheomparisonisnotrequired.8}

Aspeial treatmentisgivenfortheaseofthreeoplanaronurrentlines.

Thetrisetorinthisaseisasingleperpendiularlineto

ℓ 0

^{. Hene,itsproje-}

tionisjustarationalpointthatisvalidforbothsides.

Ensuring a GeneriFrame First,itishekedthatno1-dimensionalom-

ponentisontainedin

H ^∗

^,inthatase

degree(res)

^{isneessarilylessthan}

2d 1 d 2

but the degreemay alsodrop in fewother ases. Thus ifthedegree issuspi-

ious, we simply interset

B 1

^and

B 2

^with

H ^∗

^{. There is no} 1-dimensional omponentif the degreeof the

gcd

^{of thetworesulting bivariate} polynomials is onstant. Moreover, we hek that the projetion has only simple vertial

asymptotebyhekingthattheleadingoeientofthesquarefreepartof

res

issquarefree. Thisensuresthatthetrisetorintersets

H ^∗

transverselyalways.

7

Aboveistheareatotheleftoftheurvewhengoingfromitslexiographiallysmaller

toitslexiographiallylargerend.

8

Thishappensinasethebisetorisjustasimpleplane.

Contents

1 Introdution 3

2 Preliminaries 4

2.1 PropertiesofBisetorsandTrisetors . . . 4

2.2 LowerEnvelopeAlgorithm. . . 5

3 Computinga Voronoi Cell 6

3.1 ParametrizationandProjetion . . . 6

3.2 LowerEnvelopePrediates . . . 7

3.3 Complexity . . . 8

4 Fast Point Loation 8

5 ImplementationDetails 9

6 Conlusions 11

A Projetion of Bisetorsand Trisetors 14

A.1 ProjetionofBisetorsBoundary . . . 14

A.2 ProjetionofTrisetors . . . 15

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