Extensions of the witness method to characterize under-, over- and well-constrained geometric constraint systems

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Extensions of the witness method to characterize under-, over- and well-constrained geometric constraint systems

Simon Thierry, Pascal Schreck, Dominique Michelucci, Christoph Fünfzig, Jean-David Génevaux

To cite this version:

Simon Thierry, Pascal Schreck, Dominique Michelucci, Christoph Fünfzig, Jean-David Génevaux. Ex- tensions of the witness method to characterize under-, over- and well-constrained geometric constraint systems. Computer-Aided Design, Elsevier, 2011, 43 (10), pp.1234-1249. �10.1016/j.cad.2011.06.018�.

�hal-00691690�

over- and well-onstrained geometri onstraint systems

SimonE.B.Thierry a,∗

,PasalShrek a

,DominiqueMihelui b

,Christoph

Fünfzig b

, Jean-DavidGénevaux a

a

LSIIT,UMRCNRS7005

UniversitédeStrasbourg

b

LE21,UMRCNRS5158

UniversitédeBourgogne

Abstrat

Thispaperdesribesnewwaystotakleseveralimportantproblemsenountered

in geometri onstraint solving, in the ontext of CAD, and whih are linked

tothehandlingofunder- andover-onstrainedsystems. Itpresentsapowerful

deompositionalgorithmofsuhsystems.

Our methods are based on the witness priniple whose theoretial bak-

ground is realled in a rst step. A method to generate a witness is then

explained. Weshowthathavingawitnessanbeusedtoinrementallydetet

over-onstrainednessandthustoomputeawell-onstrainedboundarysystem.

An algorithmis introduedto hekifanhoringagiven subsetof theoordi-

natesbringsthenumberofsolutionsto anitenumber.

An algorithm to eiently identifyall maximalwell-onstrainedparts of a

geometri onstraint systemis desribed. This allows us to designa powerful

algorithmof deomposition,alled W-deomposition,whihis ableto identify allwell-onstrainedsubsystems: itmanagesto deompose systemswhih were

notdeomposablebylassiombinatorialmethods.

Keywords: GeometriConstraintsSolving,Witnessonguration,Jaobian

matrix,W-deomposition,Under-onstrainedness,Over-onstrainedness, Well-onstrainedness,Transformationgroups

1. Introdution

Geometri onstraint solving in Computer-Aided Design (CAD) aims at

yieldingagurewhihmeetssomeinideneandmetrirequirements(e.g.dis-

tanes between points or angles between lines), usually speied in graphial

∗

Correspondeneshouldbeaddressedtothisauthor

Emailaddresses: simon.thierryunistra.f r (SimonE.B.Thierry),

shrekunistra.fr(PasalShrek),dmihelu-bourgogne.fr(DominiqueMihelui),

.fuenfziggmx.de(ChristophFünfzig),jean-david.genevauxetu.u nis tra. fr

(Jean-DavidGénevaux)

(C, X, A) ^with C ^{a set of onstraints} (prediates) on a set X ^{of unknowns}

(geometrielements)withrespettoasetA^{ofparameters(metrivalues). So-}

lutionsarereturned astheoordinatesof thegeometrielements,i.e. aset of

funtionsf :X →R^{i in}idimensions. Formoreformaldenitionsofgeometri onstraintsystems,thereadermayreferto[29℄.

Example : Geometri onstraintsystem

Figure1showsalassialexampleofageometrionstraintsystem. We

give a formal statement of the system and a sketh as the user would

draw it. Therightof thegure showsa possiblesolution. Many other

solutionsexist:

• ^{ifasymmetryisappliedonasolution,ityieldsanothersolution;}

• ^{ifarotationand/ora}translationisapplied onasolution, ityields anothersolution;

• ^{ifasymmetryisapplied forinstane onpoint}p3^,withaxisp2p4^,it

yieldsanothersolution.

dist(p1, p2, k1)^,ang^_ppp(p6, p1, p2, θ1)^, dist(p², p3, k2)^,ang^_ppp(p², p3, p4, θ2)^, dist(p³, p4, k3)^,ang^_ppp(p⁴, p5, p6, θ3)^, dist(p4, p5, k4)^,dist(p5, p6, k5)^, dist(p1, p6, k6)^,withk1= 7^,k2= 5^, k3= 9^,k4= 8^,k5= 6^,k6= 7^,θ1= 135^, θ2= 120^,θ3= 115

PSfragreplaements

p1

p1 p2

p2 p3

p3

p4

p4

p5

p5

p6

p6

7

5

9

8 6

7 135

◦

120

◦

115

◦

Figure1: Formalstatement(left)ofa2Dtehnialsketh(middle)and apossiblesolution

(right).

Theliteraturedesribesanumberofdierentapproahestosolvegeometri

onstraintsystems:

• ^{algebraimethods onsistin} translatingthe GCSinto aset ofequations andworkingontheequationsystem,thusforgettingthegeometrialbak-

ground. Algebraimethods anbe lassiedin numerialmethods (iter-

ative omputations onverging to an approximate solution from initial

valuesgivenbytheuser,suhastheNewton-Raphsonortheontinuation

method[27℄)andsymbolimethods(diretomputationsontheequations

those methodsareseldomusedbeauseoftheirtimeomplexity[1℄),

• ^{geometrimethodsusethegeometriknowledgetosolvethesystem: rule-}

based methods [2, 19℄ dedue onstrution plans by an expliit use of

geometri rules,graph-based methods[7, 10,27, 33, 34,38℄ ompilethis

knowledgeintoalgorithmswhihonsideronlyombinatorialandonne-

tivityriteria,

• ^{hybridmethods[5,9,20℄alternatealgebraiandgeometriphasesofom-}

putationstousethepowerofbothapproahes.

For moredetails ongeometrionstraintsolvers,see[13℄. Ageneraltrend,

both to redue omplexity and to enhane resolution power, onsists in de-

omposing the GCS into solvable subsystems and in assembling their solu-

tions[5,10,14,17,27,33,34,38,42℄.

Example : Deomposition ofgeometri onstraintsystems

Itisimpossibletodrawdiretlyasolutionofthe2DexampleofFigure1

withonlyarulerandaompass. Butitiseasytoseparatelysolveeah

triangle(p¹p²p^6,p²p³p^4andp⁴p⁵p^{6)andthenassemblethem. Byusing}

deomposition,theresolutionpoweristhusgreater.

Notiethat,ontheexampleofFigure1,ifoneremovesoneofthetrian-

gles,sayp²p³p^{4,andthentries tosolvetheremainingsystem,oneneeds}

toaddinformationfromthesolvedsubsystem,i.e. adistaneonstraint

betweenp^2andp^{4,otherwisetheremainingsystembeomes}artiulated.

For a detailed survey of deomposition methods, see [18℄. The piee of

informationadded when removing asubsystem is alled aboundary[29℄. Al-

thoughseveralmethodsexisttondtherelevantinformationinspeiresolu-

tionframeworks[33℄,nogeneralalgorithmyetexiststo omputetheboundary

withoutaddingtoomuhinformation.

Indeed, it is important for resolution methods, espeially for graph-based

methods, that the system does not have too few or too many onstraints.

Looselyspeaking, asystemisalled

• under-onstrainedifit hasan innitenumberof solutionsbeausethere arenotenoughonstraintstopindowneverygeometrielement,

• over-onstrainedifithasnosolutionbeauseofonstraintontraditions 1

,

• well-onstrainedifithasanitepositivenumberofsolutions.

Invarianeofrigidsystemsbydiret isometriesisgenerallytakeninto aount

byanhoringapoint andadiretion in 2D, apointand twodiretions in 3D.

Thepointand thediretion arealledareferene forthediretisometriesand

onstitutewhatweallananhor ofthesystem. Othertransformationgroups

may be onsidered[37, 42℄: wesaythat asystemis G-well-onstrainedif itis well-onstrainedmodulo G^[29℄.

1

Notiethatthedenitionsofthelevelsofonstrainednessaregeneralanddonottakeinto

aountthe generiityhypothesis, furtherdisussedinsetion2. Thus,thereexistsystems

whiharesaidtobeonsistentlyover-onstrained,whentheyaregeneriallyover-onstrained

butthevaluesoftheparametersaresuhthattherearesolutions.

ThesystemofFigure1isrigid,orwell-onstrainedmodulodiretisome-

tries. Thesystemof Figure2a iswell-onstrained modulo diret isome-

tries, but ontains a subsystem, shown at Figure 2b, whih is well-

onstrained modulo similarities [37℄: one an yield a nite number of

solutionsfrom whihanysolutionanbegenerated byapplying asal-

ing,arotationand/oratranslation.

The systemof Figure2 is under-onstrainedmodulo any groupating

globally on the system: if one anhors points p^{1 and} p^{2, for instane,}

pointsp^3andp^{4maystillmovewithoutviolating}onstraints.

PSfragreplaements

p¹ p²

p³ p⁴

Figure2: Arigidsystem(left);asubsystemwell-onstrainedmodulo similarities(middle);

anartiulatedsystem(right)

Alotofworkhasbeendoneaboutthedetetionofover-onstrainedness[15,

17,32℄orunder-onstrainedness[21,41,46℄andmoregenerallyaboutthehar-

aterizationofrigidity[22,24,37,44℄. Yet,methodsdesribedintheliterature

may fail to onsider the onsequenes of mathematial theorems that are not

expliitly taken into aount in the onstrution of the resolution algorithm.

Sinea theoremlist annot beexhaustive 2

, it is impossibleto developarule-

basedorgraph-basedalgorithmwhih detetsallgeometripropertiesindued

bymathematialtheorems.

Inthisartile,weextendthewitnessmethod[30℄toaddressseveralproblems

itedabove: howtodeterminetheonstrainednesslevelofaGCSwithoutbeing

trikedby mathematialtheorems(seefor instaneFigure 11);howtobuild a

well-onstrainedboundarysystem;howtohekifapotentialanhordoesnot

make the systemover-onstrained; how to eiently detet all maximal well-

onstrainedsubsystems of agiven GCS;howto deompose awell-onstrained

systemintotheset ofallitsminimalwell-onstrainedsubsystems.

For onisenessreasons,in the rest ofthis paper,weonsider 2D systems,

unless expliitlymentioned otherwise. Yet,all algorithms anbe extended to

3Dsystemswithnearlynohanges.

This artile is organizedas follows: Setion 2 realls the priniples of the

witnessmethod andgivesawaytogenerateawitness;Setion 3introduesan

2

Morepreisely,thesetoftheoremsisreursivelyenumerable,butnotreursiveingeneral.

thatitleadstoaorretgreedyalgorithmtoomputeawell-onstrainedbound-

arysystem;Setion4showshowto deideifapotentialanhorisvalidornot;

Setion 5givesalgorithms to eientlyidentify themaximal well-onstrained

subsystemsofanartiulatedsystem(alsoalledexible system);Setion 6de-

duesfromthesealgorithmsamethodtofurtherdeomposearigidsysteminto

well-onstrained subsystems; Setion 7 disusses the robustness issues of our

algorithms;nally,Setion8onludesandgivesperspetivestothiswork.

2. The witnessmethod

2.1. Priniple

The notion of witness appears in dierent domains suh as the study of

polynomial systems through the priniple of algebrai probability one [39℄,

probabilistiproofsin geometry[6℄ortheRigidityTheory[11℄.

The ideaonsists in studying generi properties of a ontinuous olletion

ofobjetsthroughthestudy ofasingleoneof these objets: awitness. Sine

therigidityisanimportantpartofouronern,wereallherethebasisofthe

rigiditytheoryasdesribedin [11℄.

2.1.1. Frameworks andrigidity

The question of rigidity is studied through the notion of frameworks. A

framework is a triple (V, E, p) ^where (V, E) ^{is a graph and} p : V → R^{d a}

realizationofthegraph,whihmapsthevertiesofV ^{topointsofdimension}d^.

Thinkingofgraphedgesas rigidbarsandofvertiesasartiulationpoints,the

maingoalofombinatorialrigidityistoanswerIs(V, E, p)^{rigid?,i.e. arerigid}

bodymotionsallowedonlyonthewholeframework,withnoloaldeformation.

An innitesimal exion isthen amapq:V →R^{d suh that} (p(i)−p(j))· (q(i)−q(j)) = 0^,foreah(i, j)∈E^{. Aframeworkisalled}innitesimallyrigid, iftheonlyinnitesimalexionsarisefrom thediretisometries ofR^{d,i.e. the}

translationsandrotations. It is proventhat innitesimalrigidity isastronger

propertythan rigidity: a framework anberigidbut not innitesimallyrigid,

famousexamplesaregiveninFigure3(see[11℄). Counter-examplesofrigidbut

notinnitesimallyrigidframeworksarisewhentheframeworkissingular.

Figure3:Non-innitesimallyrigidframeworks. Theframeworkontheleftisrigid.

A frameworkF = (V, E, p)^{is said generi if there is a}neighborhood of F

whereallframeworkswithgraph(V, E)^arerigidifF ^{is,andnotrigidotherwise.}

theoryarestatedbythefollowingproposition:

Proposition 1. Consider agraph (V, E)^{. If there is a}realization p^{suh that}

theframework(V, E, p)^{isgeneriallyrigid,thentheframeworks} (V, E, q)^where q ^{is another} realization of the graph, is rigid for almost any q^{. On the other}

hand, if twoframeworks (V, E, p)^and (V, E, q)^{are generi, thenthey are both}

rigidorboth notrigid.

Thisproposition justiesthefatthatin 2D,theLamantheorem[24℄gives

aombinatorialharaterizationofrigidity. Alas, suhaharaterizationisan

openproblemindimension3orhigher.

From the geometri onstraintpoint of view, a framework in rigiditythe-

ory orresponds to the realization of ageometri onstraint system where all

onstraintsarepoint-to-pointdistaneonstraints: suhasystemisgenerially

well-onstrainedupto diretisometriesifit isgeneriallyrigid. This wasgen-

eralized by Mihelui et al. [30, 31℄ to metri onstraints over points, lines,

et.(distanesandangles)andtoinideneonstraints(olinearitiesin2Dand

3D,oplanaritiesin3D).

2.1.2. ExtensiontoCAD

Intheprevioussetion,wehaverealledthebasisofrigiditytheoryandthe

priniplesofwitnessinterrogationinthisontext. Intherestofthisartile,we

donothavethepoint of viewof rigidity theory andof frameworks, but fous

ongeometrionstraintsystem.

As stated above, geometri onstraint systems in CAD naturally lead to

onstraint graphs, or more generally to hypergraphs. It is then tempting to

extrapolate results of the rigidity theory, suh as Laman's theorem, into the

eld of onstraintsolving. In ourase, theonstraintsare put in agraphial

formonthesketh (seeFigure 1)whihisarealizationoftheonstraintgraph

thesamewayasinRigidityTheory. Undersomegeneriityassumptions,itisa

perfet andidatetobeawitnessfortheonstrainednessproperties.

Indeedwhen thedesigner drawsasketh, he/she hasasolutionXw ^foran

equation system F(X, Aw) = 0^{, with someparameter values} Aw ^{read on the}

sketh. ThenthegoalisasolutionforthesystemF(X, Aa) = 0^,where Aa ^are

thevaluesgivenforthedimensioning. Thisfat hasbeenusedwithin theon-

tinuationmethod with homotopy in CAD [3, 25℄ orto dene aneighborhood

relationship between gures [4℄. In fat, our purpose is lose to these prob-

lematis sine we laim that the sketh is like the searhed solution from the

onstrainednesspointofview.

IntheCADdomain,allthegeometrionstraintsanbeputundertheform

ofpolynomialequations,andF ^isaC^{∞lassfuntion. Weanthenonsidera}

TaylorexpansionofsystemF(X, Aw) = 0^,andget:

F(Xw+ε−→V , Aw) =F(Xw, Aw) +εF^′(Xw, Aw)−→V +O(ε²)

where

−

→V ^{an also be seen as the instant veloity of eah objet involved in}

thesystemand ε^{is asmall timestep. Then,} ε−→V ^{is an}innitesimalexion,or motion, if it leads from a solution to another solution of the system. Or, in

other words, the O(ε²) ^{term in the previous formula beomes in fat a} o(ε²)

term. Underthese onditions,wemusthave

F^′(Xw, Aw)−→V = 0 ⁽¹⁾

The spae of the innitesimal motions allowed by the onstraints at the

witnessisthengivenbyker(F^′(Xw, Aw))^{. Notethat}

• ^thematrix F^′(Xw, Aw) ^{isknownas theJaobianmatrixof the funtion} F(X, Aw)^takenatpointXw^;

• ^{when allonstraintsare}point-to-pointdistanes, the Jaobianmatrixis therigiditymatrixonsideredinRigidityTheory;

• ^{for other onstraintswith parametersthe generiity onditionsare alike}

thoseintheombinatorialase: aparametervalueAw^andaorresponding solutionXw^{aregeneriiftherootisanimpliitfuntionoftheparameters}

insomeopenneighborhoodof(Xw, Aw)^{;forinstane,foratrianglespe-}

ied withthreelengthparameters,thisonditionforbidsthat onelength

is the sum of the others; moregenerally this ondition implies that the

matrix

∂F(X, A)/∂X ∂F(X, A)/∂A

0 Id

has thesamerank in an openneighborhood of (Xw, Aw)^{. When all the}

equationsarepolynomials,beauseofthealgebraiprobabilityoneprini-

ple,thegeneriparametervaluesaredenseinthesetofparametervalues

orrespondingtoarealization.

Example : Generi formulationof onstraints

Forpoint,line,planeinidenes,weassumethattheorrespondingon-

straintsarespeiedexpliitlywithoutparameters. Thisisto avoidex-

pressingpoint-pointinidenesbyadistaneonstraint(p¹,x−p²,x)²+ (p¹,y−p²,y)² =d^{2 with distane parameter}d= 0^{. For adistane on-}

straint (p¹,x−p²,x)²+ (p¹,y−p²,y)² =d^{2, theparameter} d= 0 ^{is not}

generi, as the onstraint is singular at the solution point. Foran an-

gle onstraint angle(p¹, p², p³) = θ^{, i.e.} −−→p²p¹ · −−→p²p³ = lp¹p²lp³p²cosθ^,

the parameter values θ = ±π^, θ = ±π/2^{, and} θ = 0 ^{are not generi.}

Similarly,point-line,line-planeinidenesandline-line,plane-planepar-

allelism/orthogonalityonstraintsarenotexpressedbyangleonstraints

beauseitwouldintroduenon-generiangles.

Typiality. A witnessistypial ifit isrepresentativeforthe searhedsolu-

tion,i.e. ithas thesameombinatorialproperties(oinidenes, olinearities,

oplanarities,et.). [4℄assumethattheskethisawitness,andthatthesketh

(Xw, Aw)^, {(X, A) : F(X, A) = 0} ^{withthe speied} ombinatorial properties istypialwith probability1 ^{foraset ofwitnesssolutions. Notethat [18℄built}

an(artiial)ounter-example,i.e. asystemwithtwolassesofsolutions(thus

withtwokindsofwitnesses)whiharedierentinombinatorialproperties,and

noontinuousdeformationexiststotransformoneintotheother. Insuhases,

awitnessisrepresentativeofonlyonelass(itslass)ofsolutions. Thisexample

istheoneofFigure4. Suhartiialsystemsareignoredinthisartile.

PSfragreplaements

A

A

B B

C C

D D

PSfragreplaements

p¹

p² p³

6

4 3

50

Figure4: Ambiguous system: ifpoints arerequired to be

distint,thenitisrigid(leftase);otherwise,whenAandC

havethesameoordinates,pointDanbeanywhereonthe

dottedirle(exampletakenfrom[18,Figure14℄).

Figure5: A2Dsketh ofa

rigidtriangle.

We an then study the degrees of freedom of the system by studying the

rankof the JaobianmatrixF^′(Xw, Aw)^{on a typialwitness} Xw^{, and in the}

aseofunder-onstrainedness,thestrutureoftheallowedinnitesimalmotions

anbededued fromthestudyofthekernelofF^′(Xw, Aw)^.

In the rest of this paper, we onsider that rows of the Jaobian matrix

representonstraintsandolumnsrepresentvaluesoftheunknowns(oordinates

ofthegeometrielements). Welassiallydenotebym^{thenumberofrowsand}

byn^{thenumberofolumns ofthematrix.}

2.2. Pratial example

Let us now detail a step-by-step onstrution of a witness and its use to

haraterizethe degreesof freedom of the orresponding geometri onstraint

system. WeonsiderthetrivialgeometrionstraintsystemofFigure5.

ThisGCSonsistsofthreesets:

• ^unknowns: X={p¹, p², p³}^;

• parameters: A={k1, k2, α}^;

• onstraints: C={dist(p¹, p², k¹)^,dist(p², p³, k²)^,ang^_ppp(p¹, p², p³, α)}

Thedimensioninggivenbytheskethisafuntion ρ:A→R^,withρ(k¹) = 6^, ρ(k²) = 4^, ρ(α) = 50^{. If onewere to onsiderthe bottom left ornerof the}

skethastheoriginandgiveoordinatestothepointsonthesketh,theywould

notsatisfytheonstraintswiththisdimensioning.