Using the witness method to detect rigid subsystems of geometric constraints in CAD

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Using the witness method to detect rigid subsystems of geometric constraints in CAD

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

To cite this version:

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

Using the witness method to detect rigid subsystems of geometric constraints in CAD. Symposium

on Solid and Physical Modeling, 2010, Haïfa, Israel. pp.91-100, �10.1145/1839778.1839791�. �hal-

00691703�

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subsystems of geometri onstraints in CAD

Dominique Mihelui

∗

Pasal Shrek

†

Simon E.B. Thierry

†

Christoph Fünfzig

*

Jean-David Génevaux

†

September1 st

3 rd

,2010

Abstrat

Thispaperdeals withtheresolution ofgeometri onstraintsystems

enounteredinCAD-CAM.Themainresultsarethatthewitnessmethod

an be used to detet that a onstraint system is over-onstrained and

thattheomputationofthemaximalrigidsubsystemsofasystemleads

toapowerfuldeompositionmethod.

In a rst step, we reall the theoretial framework of the witness

methodingeometri onstraint solvingand extendthis methodto gen-

erateawitness. Weshowthenthat itanbe usedto inrementallyde-

tetover-onstrainedness. Wegiveanalgorithm toeientlyidentifyall

maximalrigidpartsofageometrionstraint system. Weintroduethe

algorithmofW-deompositiontoidentifyallrigidsubsystems: itmanages

to deompose systemswhihwerenotdeomposableby lassial ombi-

natorialmethods.

Keywords:GeometriConstraintsSolving,witnessonguration,Ja-

obianmatrix,rigiditytheory,W-deomposition

1 Introdution

GeometrionstraintssolvinginComputer-AidedDesign(CAD)aimsatyield-

ingagurewhihmeetssomeinideneandmetrirequirements(e.g.distanes

betweenpointsoranglesbetweenlines),usuallyspeiedingraphialform.For-

mally,ageometrionstraintsystem(GCS)onsistsinonstraints(prediates),

unknowns (geometri entities) and parameters (metri values). Solutions are

returnedastheoordinatesofthegeometrientities. Theleftofgure1shows

anexampleofatehnialsketh, andtherightshowsapossiblesolution.

Theliteraturedesribesanumberofdierentapproahestosolvegeometri

onstraintsystems:

∗

LE2I,UMRCNRS5158,UniversitédeBourgogne

†

LSIIT,UMRCNRS7005,UniversitédeStrasbourg

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p1

p¹ p²

p² p³

p³

p⁴ p⁴

p⁵ p⁵

p⁶ p⁶

7

5

9

8 6

7 135

◦

120

◦

115

◦

Figure1: A2Dtehnialsketh(left)andapossiblesolution(right).

• ^algebrai^methods ^onsistⁱⁿ translatingthe GCSinto aset ofequations andworkingontheequationsystem,thusforgettingthegeometrialbak-

ground. Algebraimethodsanbelassiedinnumerialmethods[22℄(it-

erativeomputations onverging to anapproximate solutionfrom initial

valuesgivenby theuser)andsymbolimethods[2, 11℄(diret omputa-

tionsontheequations thesemethods areseldom usedbeauseoftheir

omplexity),

• ^geometri^methods^use^the^geometri^knowledge^to^solve^the^system:^graph-

basedmethods[6,9,22,28,29,31℄ompilethisknowledgeintoalgorithms

whih onsider only ombinatorial and onnetivity riteria, rule-based

methods[3,17℄dedueonstrutionsplansbyanexpliituseofgeometri

rules,

• ^hybrid^methods^[4,^8,^18℄âlternateâlgebraiând^geometri^phasesôfôm-

putationstousethepowerofbothapproahes.

For moredetailsongeometri onstraintsolving,see[12℄. Ageneraltrend,

both to redue omplexity and to enhane resolution power, is to deompose

theGCS into solvablesubsystems and to assembletheir solutions[4, 5, 9, 13,

15,22,28,29,31,33℄. Forinstane,onthe2Dexampleofgure1,itiseasyto

separatelysolveeahtriangle (p¹p²p⁶^,p²p³p⁴ ândp⁴p⁵p⁶⁾ând^thenâssemble

them. Foradetailedsurveyofdeompositionmethods,see[16℄.

Notiethat,ontheexampleofgure1,ifone removesoneof thetriangles,

sayp²p³p⁴^, ând^then ^tries^to ^solve^the^remaining^system,ône ^needs^toâdd în-

formationfromthesolvedsubsystem,otherwisetheremainingsystembeomes

artiulated. Thispieeofinformationisalledtheboundary[24℄. Althoughsev-

eralmethodsexisttondtherelevantinformationinspeiresolutionframe-

works [28℄, no generalalgorithm yet exists to omputethe boundary without

addingtoomuh information.

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

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• under-onstrainedifit hasan innitenumberof solutionsbeausethere arenotenoughonstraintstopindowneverygeometrientity,

• over-onstrainedifithasnosolutionbeauseofonstraintontraditions,

• well-onstrainedifithasanitepositivenumberofsolutions.

Invarianeofrigidsystemsbydisplaementsisgenerallytakenintoaountby

anhoring a point and a diretion in 2D, a point and two diretions in 3D.

Thepointandthediretionarealledareferene forthedisplaements. Other

transformationgroupsmaybeonsidered [30℄.

Alotofworkhasbeendoneaboutthedetetionofover-onstrainedness[14,

27℄orunder-onstrainedness[19,32,37℄ andmoregenerallyaboutthehara-

terization of rigidity [21, 20,30, 35℄. Yet, methods desribed in the literature

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

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

Sine a theorem list annot be exhaustive, it is impossible to develop arule-

basedor graph-based algorithm that detets geometri properties induedby

mathematialtheorems.

Inthisartile,weextendthewitnessmethod[25℄toaddressseveralproblems

ited above: how to determine the onstrainedness level of a GCS without

being triked by mathematial theorems (see for instane gure 6); how to

eientlydetetallmaximalwell-onstrainedsubsystemsofagivenGCS;how

to deompose a well-onstrained system into the set of all its minimal well-

onstrainedsubsystems.

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

unlessexpliitlymentionnedotherwise. Yet,allalgoritmsanbeextendedto3D

systemswithnearlynohangesand,mostofthetime,theonlymodiationto

bemadeforthetexttobevalidin 3Distoexhangementionsofthreedegrees

offreedom/parameterswithmentionsofsixdegreesoffreedom/parameters.

Thisartileisorganizedasfollows: setion2reallsthepriniplesofthewit-

nessmethodandgivesawaytogenerateawitness;setion3demonstratesthat

aninremental versionof the Gauss-Jordanelimination has the sameompu-

tationalostthantheoriginalversionbutallowstodetetoveronstrainedness

in allases;setion 4givesalgorithmsto eiently identify themaximalrigid

subsystems of anartiulated system; setion 5dedues from these algorithms

a method to further deompose a rigid system into rigid subsystems; nally,

setion7onludesandgivesperspetivestothiswork.

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2.1 Priniple

ThewitnessmethodomesfromideasofStruturalTopology,orRigidityThe-

ory [10℄where the questionofrigidityis studied throughthenotion offrame-

works. Aframework isatriple(V, E, p)^where(V, E)îsâ^graphândp:V →R^d

arealizationofthegraph,whihmapsthevertiesofV ^to^pointsôf^dimension d^. ^Thinking ôf^graphêdgesâs^rigid^barsândôf^vertiesâsartiulationpoints, themain goalof ombinatorial rigidityis toanswerIs (V, E, p)^rigid?, î.e. ît

admitsonlyrigidmotionsasawhole,nodeformations.

Innitesimal exion. InRigidity Theory, aninnitesimal exion is amap

q : V → R^d ^suh ^that (p(i)−p(j))·(q(i)−q(j)) = 0^, ^for ^eah (i, j) ∈E^. ^A

framework is alled innitesimally rigid, if theonlyinnitesimal exionsarise

fromthediretisometriesofR^d^,^i.e. ^thetranslationsand rotations.

Under mild assumptions onerning inidene relationships, if one frame-

work (V, E, p⁰) ^isinnitesimally rigidthen almostall frameworks (V, E, p)^are

innitesimally rigid. And theinnitesimal rigidity impliesthe rigidity of the

framework. Note thatthere areounter-examplesfor theonverse, whih on-

tainspeialinidenes.

Inotherwords,aframeworkinrigiditytheoryorrespondstotherealization

ofageometrionstraintsystemwhereallonstraintsarepoint-to-pointdistane

onstraints: suhasystemisgeneriallywell-onstraineduptodiretisometries

ifit is generiallyrigid. This was generalizedby Mihelui et al. [25, 26℄ to

metrionstraintsoverpoints,lines,et.(distanesandangles)andtoinidene

onstraints(olinearitiesin2Dand3D,oplanaritiesin3D).

InCAD whenthe designer drawsasketh, he/she hasasolutionX⁰ ^for ^a

systemF(X, Ae) = 0^,^with^some^parameter^valuesAe^read^on^the^sketh. ^Then

the goalis a solutionfor the system F(X, Aa) = 0^, ^where Aa ^are ^the ^values

givenforthedimensioning.

Witness. Let F(X, A) = 0 ^be â ônstraint ^system, ^where X âre ^the ûn-

knowns and A ^theparameters. We suppose that F(X, A) ^is dierentiable. A witness isthenasolutionX0 ^ofF(X, A) = 0^for^some^parameter^valuesAe^.

Using aTaylorexpansionfora small perturbation aroundthe solutionX0

ofF(X, Ae) = 0^,^we^have

F(X0+εv, Ae) =F(X0, Ae) +εF^′(X0, Ae)v+O(ε²)

wherev ânâlso ^be^seenâs^the înstant^veloityôf êah ôbjetînvolved ⁱⁿ ^the

systemandεîsâ^small^time^step. ^Thus,îfâninnitesimallysmallperturbation isanothersolutionofF(X, Ae)^,^we^must ^have

F^′(X⁰, Ae)v= 0

Thespaeoftheinnitesimalmotionsallowedbytheonstraintsatthewitness

isthengivenbyker(F^′(X⁰, Ae))^. ^Note^that

• ^the^matrixF^′(X⁰, Ae)îs ^knownâs^the^Jaobianôf^systemF(X, Ae) = 0

takenatpointX⁰^;

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• ^whenâllônstraintsârepoint-to-pointdistanes,theJaobianistherigid- itymatrixonsideredin RigidityTheory;

• ^for ôtherônstraints^with ^parameters^the^generiityônditionsâre ^more

ompliatedthan intheombinatorialase: aparametervalueAe^and ^a

orrespondingsolutionX⁰âre^generiîf^the^rootîsânîmpliit^funtionôf

theparametersinsomeopenneighborhoodof(X⁰, Ae)^;^for^instane,^for^a

trianglespeiedwiththreelengthparameters,thisonditionforbidsthat

onelengthisthesumoftheothers;moregenerallythisonditionimplies

thatthematrix

∂F(X, A)/∂X ∂F(X, A)/∂A

0 Id

has thesamerankin anopen neighborhood of (X⁰, Ae)^It ^remains ^that

the generi parameter values are dense in the set of parameter values

orrespondingtoarealization.

Wegivesomeexamplesfortheformulationofgenerionstraints. Forpoint,

line,planeinidenes,weassumethattheorrespondingonstraintsarespei-

edexpliitlywithoutparameters. Thisistoavoidexpressingpoint-pointini-

denesbyadistaneonstraint(P¹,x−P²,x)²+ (P¹,y−P²,y)²=d²^with^distane

parameterd= 0^. ^Fôrâ^distaneônstraint(P¹,x−P²,x)²+ (P¹,y−P²,y)²=d²^,

the parameterd = 0 îs ^not ^generi, âs^the ônstraint îs ^singular ât ^the ^solu-

tionpoint. Foran angle onstraint angle(P¹, P², P³) = θ^, î.e. P¹P²·P³P² = lP1P2lP3P2cosθ^, ^the ^parameter ^valuesθ =±π^, θ = ±π/2^, ând θ = 0 âre ^not

generi. Similarly, point-line, line-plane inidenes and line-line, plane-plane

parallelism/orthogonalityonstraintsarenotexpressedbyangleonstraintsbe-

auseitwouldintroduenon-generiangles.

Typiality. A witnessistypial ifit isrepresentativeforthe searhedsolu-

tion,i.e. it hasthesameombinatorialproperties (oinidenes,ollinearities,

oplanarities, et.). So a random solution (X0, Ae)^, {(X, A) : F(X, A) = 0}

with the speiedombinatorial properties is typial with probability 1 ^for ^a

setof witnesssolutions. Notethat systemsexistwith witnesssolutions,whih

aredierentinombinatorialproperties,and noontinuousdeformationexists

totransformoneintotheother. Foranexampleofsuhasystemsee gure14

in[16℄.

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

rank of the Jaobian F^′(X⁰, Ae) ôn â ^typial ^witness X⁰^, ând ⁱⁿ ^the âse ôf

under-onstrainedness,thestrutureoftheallowedinnitesimalmotionsanbe

deduedfromthestudyofthekernelofF^′(X⁰, Ae)^.

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

representonstraintsand olumnsrepresentoordinatesofthe unknowns. We

lassiallydenotebym^the^numberôf^rowsând^byn^the^numberôfôlumnsôf

thematrix.

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Theskethisusuallyawitnessbutduetoimpliedinidenesthismaynotbethe

ase. Inthisase,wesolvetheunder-determinedsystem{(X, A) :F(X, A) = 0}

forawitness(X⁰, Ae)^. În^thesubdivision solverpresentedin [7℄, thenonlinear monomialsx²_i ândxixj^fori < jâre^replaed^byâdditional^variablesxi,iândxi,j^,

whihareenlosedin apolytopeBD(xi, xi,i, xi,j,i<j)≥0 ^with^halfspaes^given

bythe non-negativity of relevant Bernstein polynomials (Bernstein polytope).

The quadrati onstraint system beomes a polytope S(xi, xi,i, xi,j,i<j) ≥ 0

after rewriting into the additional variables xi,i ând xi,j^. ^The^subsript D ôf BD(xi, xi,i, xi,j,i<j)≥0îndiates^that^this^polytope^dependsôn^the^domainD^.

In this way, bounds for the solutiondomain of quadrati polynomialsan be

expressedastwolinearprograms

minxi ^and maxxi

S(xi, xi,i, xi,j,i<j)≥0 BD(xi, xi,i, xi,j,i<j)≥0

Domainboundsareomputedbylinearprogramminginordertoreduethe

urrent solutiondomain D^. Îf ^the ^feasible ^set îs êmpty^, ^whih îs ^deteted^by

linearprogramming,thentheurrentdomainboxontainsnosolution. Other-

wise,weanperformasequeneof redutions andbisetions ofdomain boxes

untilthedomain boxD = [x¹, x¹]×. . .×[xn, xn] ^isδ^-small: (xi−xi)< δ ^for

alli^. ^These δ^-small^boxes^over^the^solution^set^pieewise.

The subdivision solverrequires adomain box to start thesearh. The in-

tervalsfor generi parameter valuesof onstraintsare easy to nd: angle pa-

rameterscosθ⁽cosθînsteadôfθ^toâvoidtrigonometrifuntions inthesolver) arein [−1 +ǫ,−ǫ]ôr [ǫ,1−ǫ] ^with â^small, ârbitrary ǫ^; întervals^for ^distane

parametersdân^beôbtained^from^magnitude^boundsôf^the^pointoordinates.

Findingaboundonthemagnitudeofanyroot[36℄,wouldbeneessarytoprove

thatthe systemhasnosolution. Fortheproblems here,abound onthepoint

oordinatesisknownbeforehand.

Inordertoenumerateallsolutionsofasystem,weusedmid-bisetionofthe

largestintervalin [7℄, whih minimizestheheightof theexplorationtreewhile

ylingthroughdimensions. Fortheaseofdeterminingasinglesolutionasfast

aspossible,thehoie ofthesmallestinterval(greaterorequalδ⁾^is^beneial

assettingvariablestovaluesallowingsolutionsimprovestheeetivenessofthe

domainredutionstep.

We selet the next domain box (of smallest minimum side length greater

thanδ⁾^for ^redutionând ^bisetion ât^random. În^this ^way,^we^ndâ^solution

boxontaininga randomsolution, and we take thebox enter projetedonto

thesolutionsetasawitness.

As examples, weshow two systems of dierent diulty. In gure 2, two

triangles with a ommon point p⁰ ^are ^speied ^by ^six ^side ^lengths. ^In ^the

randomsolution,thesidelengthsare alldierent. Ingure3,four pointsand

velineswith10^point-lineînidenesâre^speied^by^fourângle^parametersând

adistane parameter. Theleft partshowsasolutionwithsymmetriandnie

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p0 p1

p2 p3

p4

Figure2: Thebuttery: 2Dsystemwith5pointsand 6distane parameters

d(p⁰, p¹)^,d(p¹, p²)^,d(p², p⁰)^, d(p⁰, p³)^,d(p³, p⁴)^,d(p⁴, p⁰)^.

p

q r

c

p

q r c

Figure3:2Dsystemof4pointsand5lineswith10^point-line^inidenes,⁴^angle

parameterangle(qp, cp)^,ângle(cp, rp)^,ângle(rq, cq)^,ângle(cq, pq)ând¹^distane

parameterd(r, c)^. ^Symmetri ^solution^(left)^and ^random,^typial^witness^solu-

tion(right).

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thetrianglepoints. Intherightpart,atypialwitnesssolutionisshown,whih

wasfoundat random. Itisusedforfurther analysis.

3 Over-onstrainedness

Wealreadyshowedin setion1that thedetetionofover-onstrainednessis a

ompliatedyet essentialproblemin theeld ofgeometrionstraintssolving.

In this setion, we show that the use of the witness method leads to an

eientandrobustdetetionofredundanyin geometrionstraints.

Wealsoshowtheusefulnessofthewitnessmethodtoenhanerobustnessof

deompositionmethodsbyanaurateomputationoftheboundary.

3.1 Inremental detetion of redundany

Weshowedin[25℄ thatitispossibleto interrogateawitnessinordertodetet

whetherasetofonstraintsisdependentornot.Indeed,itispossibletoompute

the rank of the Jaobian matrix at the witness and to ompare it with the

number of onstraints. However, nding a maximal independent subset of a

dependentset isnotatrivialproblem. Working onthewitness,thenaiveidea

wouldbetotryand removeonstraintsonebyoneand,ateahstep,ompute

therankagainto determineif theonstraintisredundant withtheremaining

set. If therank of S −c êquals^the ^rankôf S^, ^then ônstraintc îs^redundant

andanberemoved. Performedthisway,theremovalofredundantonstraints

is expensive. Yet, onsidering an inremental onstrution of the geometri

onstraint system allows to identify the set of redundant onstraints with no

additionalostsinomparisonto thebasidetetionof redundany.

Indeed, onsider a geometri onstraintsystem S ^with ^no ^redundany ^be-

tweentheonstraints. ApplyingtheGauss-Jordan eliminationmethod on the

Jaobianmatrix at the witnessleads to amatrix J^′ = (IP) ^with I ^am×m

diagonal matrixand P â m×f ^matrix, f =n−m ^being ^the ^number ôf â-

tualdegrees of freedom of the system. This method hasa known omplexity

of O(min(n, m)nm)^. ^Let ûs ^nowônsider â ^systemS^′ ^with S ⊂ S^′^. Înôrder

to knowifS^′ ^is over-onstrained,oneonlyneeds to inrementallyadd thege- ometrientities andtheonstraints(bearing in mindthat aonstraintanbe

insertedonlywhen the geometrientities it onernsare all in thesystem) of

S^′− S ^to S ^and ^applying Gauss-Jordanagain. Sinethe leftmost partof the

matrixis thediagonal, thenumber ofoperations is at most2 min(m, n)f^: ^for

eahrowofI^,êah^non-zeroêlementôfP ^must^be^multipliedândâdded^to^the

newrow. Thenumberof operationsisin fat farsmaller,sinethenumberof

zeroelementsin thenewrowofthematrixishigh.

Proeeding inrementallydoesnot raise thenumber ofoperations: it only

hangestheorderoftheoperations. Indeed,thelassialGauss-Jordanelimina-

tionmethodonsistsinolumn-by-olumnoperations: foreaholumnc^,^divide

rowc ^byJc,c^,^then^substratJr,c^times^this^new^row^from^rowr^for^everyr^,^so