Implicit Modeling with Skeleton Curves: Controlled Blending in Contact Situations

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Implicit Modeling with Skeleton Curves: Controlled Blending in Contact Situations

Alexis Angelidis, Pauline Jepp, Marie-Paule Cani

To cite this version:

Alexis Angelidis, Pauline Jepp, Marie-Paule Cani. Implicit Modeling with Skeleton Curves: Con-

trolled Blending in Contact Situations. International Conference on Shape Modeling and Applica-

tions (SMI’02), ACM, May 2002, Banff, Canada. pp.137-144, �10.1109/SMA.2002.1003538�. �inria-

00537493�

Controlled Blending in Contact Situations

Alexis Angelidis, Pauline Jepp

and Marie-Paule Cani

iMAGIS y

-GRAVIR, INRIA Rh^one-Alpes

655 avenue de l'Europe, 38330Montbonnot,France

[email protected] [email protected] [email protected]

Abstract

Interactiveimplicitmodellingwithcomplexskeletons

was recently improved with the introduction of primi-

tives denedat dierent LODs. These implicit primi-

tives useasubdivision curveasaskeleton. In this pa-

peran extensiontothis representation ispresented, in

order to make it suitable for the interactive modelling

and animation of soft objects in contact situations.

The rstcontribution improves the display methodfor

subdivision-basedimplicitprimitives;thisusesanadap-

tive polygonisation which locally renes, where neces-

sary,accordingtothecurrentlyselectedLOD.Thesec-

ondcontributionconsistsof anewmethodforprevent-

ingunwantedblendingwhenaskeleton-curvefoldsback

onto itself. The third introduces local deformations

where surfaces that should not blend come into con-

tact. We illustrate the benetsof this methodology by

describing two applications: the interactive modelling

ofcomplexorganicshapesincontactsituationsandthe

physically-basedanimationof suchorganic shapes.

1 Introduction

Although parametric surfaces have been more

widelyused,theblendingpropertiesofimplicitsurfaces

makethemparticularlysuitedtoorganicmodelling. In

such casessoft and branchingobjectsare moreeasily

achievedthan with their parametriccounterparts. In

particular, skeleton-basedimplicit surfacesgenerate a

smooth coating along a skeleton, which can be ani-

mated. Inthispaperwefocusonimplicitsurfacesgen-

erated by a graph of branching curves, following the

CurrentlyatDepartementofComputerScience,University

ofCalgary,Canada.

y

iMAGISisajointprojectofCNRS,INRIA,InstitutNational

methodologyrstintroducedin [6]. Theeldfunction

that generatesthe implicit surface is computed using

convolution,whichensuresbulge-freeblendingbetween

implicitprimitivesatbranchingpoints. Moreover,the

use ofsubdivision-curvesasskeleton elementsenables

thegenerationoftheimplicitsurfaceatdierentlevels

ofdetail(LODs),whichmakesthisrepresentationvery

convenientin aninteractive modelling and animation

framework.

However,makingimplicitsurfacesbehaveasorganic

shapeswhen their skeletonmovesand deforms is not

easy: body-partsgenerated bydistantpieces of curve

on the skeleton graph should not blend, but rather

squash each otherwhenthey comeinto contact. This

requirescarefullcontrolofblending,whichwasnotpro-

vided byprevioussolutions.

Thispaperextendsimplicitsubdivisioncurveprim-

itives,givingthemtheabilitytocapturetheshapeand

local deformationsoforganicobjectsin contactsitua-

tions. Therstcontribution isto improvethedisplay

of the surfaces using an adaptiveinterativepolygoni-

sation, which provides more details in highly curved

areas. Thesecondistheintroductionofalocalconvo-

lutionmethod,whichpreventsunwantedblendswhilst

preserving smoothness, even when a skeleton curve

coilsorfolds-backonitself. Lastly,weextendthePre-

cise ContactModelling (PCM)methodology[5,11]to

the subdivision-curve skeletons' framework. Two ap-

plications illustrate the benets of this methodology:

the interactivemodelling of organicshapesin contact

situations, andtheirphysically-basedanimation.

This paper is structured as follows. In Section 2,

some of the relevant background work is presented.

Section 3 presentsthe adaptiveinterativepolygonisa-

tion. Section 4introducesthenewunwantedblending

solution. The extension of PCM to the subdivision-

curveprimitivesisdescribedinSection5. Applications

2.1 Skeleton-Based Implicit Surfaces

Animplicitsurfaceisthesetofpointsforwhicheld

function f(p) takesvalue iso,so f(p) iso =0; f(p)

is ascalareld function andiso is thevalueat which

the iso-surface is drawn. An evaluation of the eld

function at any point permits the inside/outside test

whichsimpliescomputations forcontrolled blending,

collisiondetectionandraytracing.

Theeldfunctioncanbegeneratedbypoints [1,10,

15] orbymorecomplexgeometric primitives(such as

line-segmentsortriangles) that form askeleton. The

skeleton, surrounded by its implicit skin, provides an

intuitive way of controlling aspects of an object in a

modellingoranimationenvironment[8,4]. There are

two main ways to dene an implicit surface from a

skeleton:

Distance surfaces [4] calculate the eld value at a

given point P from the distance between P and the

closest point on the skeleton. Blending the contribu-

tions of severalskeleton elements isthen usually per-

formedbysummingtheireldcontributions. Although

thiseliminatescreasesonthesurfacewheretheunder-

lyingskeletalelementsjoin,italsocreatesbulges. For

example,ifaskeletoniscreatedfromapolyline,blend-

ing of the surface is desirable but not always to the

extentthatabulgeiscreatedwherethelinesjoin.

If a bulge at the join point of skeletal elements is

considredanartifact, Convolution Surfaces[3]canbe

used. Inthisrepresentation,theeldvalueatapointP

is calculatedbyintegrating allthe contributionsfrom

the dierentpointsonthe skeleton. Smoothcomplex

surfacescanbecreatedbysummingtheintegralsofin-

dividualeldcontributionsofrelativelysimpleskeletal

elements. Theintroductionofclosed-formsolutionsfor

theconvolutionintegralallowseÆcientcomputationof

convolutionsurfaces forspecicsetsofskeletonprimi-

tives,suchasline-segmentsandarcsofcircles[14].

2.2 Subdivision-curves as Skeletons

Oneofthemainweaknessesofimplicitsurfacescom-

pared to parametric ones has long been the lack of

methodsforgeneratingthematdierentLevelsofDe-

tail (LOD). Since implicit surfaces are iso-surfacesof

eldfunctions,deningsuchLODsisequivalenttogen-

eratingtheeldfunctionsat dierentresolutions.

Such a method was recently proposed in the spe-

cic case of implicit surfaces generated by graphs of

interconnected skeleton curves[6]. Subdivision curves

function itgenerates, atdierentresolutions. Ateach

resolution, the skeleton is represented by a polyline.

Closed-formconvolutionisusedforeÆcientlygenerat-

ingabulge-freeandcrease-freeimplicitsurface. Thisis

donebyusing 1

d 3

asconvolutionkernel,whichgivesan

integratedeldthatvariesastheinverseofthesquared

distancetotheskeletonelements. Interactivevisualiza-

tionis performed bygeneratingaclosedmesh around

eachskeletoncurve. Thesemesheslocallyoverlapnear

junction points where several curves meet, and their

vertices are migrated to the iso-surface. The meshes

resolutionisuniform,butcanbetunedaccordingtothe

currentLODoftheunderlyingimplicitsurface. Exam-

plesofobjetscreatedwiththismethodaredepictedin

Figure 1.

Figure 1. From [6]: an implicit object defined from a graph of subdivision curves with two different LODs.

Inthispaperwewillrelyonthesamemethodology

fordeningimplicitsurfacesfromgraphsofsubdivision

curves.TocomplementtheLODsontheeldfunction,

one of our contributions is to automatically generate

LODs onthe meshesthat samplethesurface. This is

donebylocally reningthesemeshesaccordingtothe

localerrorfromtheiso-surface,consideringitscurrent

LOD.Thisextensionmakesthemodelverywellsuited

forinteractivemodellingandanimation.

2.2.1 Controlled Blending

When parts of an implicit surface come into con-

tact, careful blending between the eld contributions

of skeletal elements is required: indeed, parts of the

surface generated by elements that are distant in the

skeletongraphshould notblend;moreover,ifthe aim

istomodelorganicshapes,ormoregenerallyanykind

of deformable bodies, object parts should not locally

intersect,butratherdeformwhentheycomeintocon-

tact.

Previousworkshaverecommended theuse ofare-

strictedblendinggraphtoavoidunwantedblendingsit-

set of skeletal elements that are authorized to blend)

introduces creases(the surfaceis not C 1

everywhere)

[12,5].

Inthecaseofskeletonsdenedasgraphsofbranch-

ing curves,the skeleton's topologywasused for auto-

maticallydening theblendinggraph[6]. Moreover,a

solution was proposed forguaranteeing C 1

continuity

everywhere. It consisted in computing the eld con-

tribution at apoint P by propagatingblending along

theskeletoncurveaslongasthenextskeletalelements

had anon-zerocontribution atthequerypoint. How-

ever,this solutionwas notfully satisfying: largefolds

that did not blend could be generated (for instance,

inFigure1thetaildoesnotblendwiththehead),but

theauthorsstressedthatblendingcouldnotbeavoided

betweensmallerfolds,asillustratedinFigure8.

Inthispaperweproposeasolutionthatfullysolves

this problem, by introducting a local convolution for-

mulation,whichwillbedescribedinSection4.

2.2.2 Contact Modelling

In thecontext of amodelling and animationenviron-

ment, situations where neither blending nor intersec-

tion are considered appropriatecall for contact mod-

elling. Theability of objects to respond to collisions

bydeforminglocallyaroundacontactsurfaceisindeed

essentialforavisuallyrealisticmodellingofcontactsit-

uations[9]. Thisrequiresthatthecontactbetweenob-

jects,orparts ofanobject,andthedeformationsthat

occurasaresultbemodelled.

ThetechniqueknownasPreciseContactModelling

(PCM) [5,11] examinestheinteractionofobjectsand

generates the resulting local deformations. The idea

isto locallymodifytheobjects'eld functions,oncea

collisionhasbeendetected,byadding a\deformation

term" to them. Sincethis termis computedfrom the

otherobject'seldfunction,itcanbeseenasadierent

way of blending the twoshapes. More precisely, two

dierentdeformationeldtermsarecomputedforeach

object(or eachpartofanobject)in contact:

Intheregionwherethetwosurfacesoriginallyin-

tersect, theeld function f

1

of the rst object is

replacedbyf

1

+(iso f

2

),whilef

2

isreplacedby

f

2

+(iso f

1

). Thismovesthetwosurfacestothe

samelocation,wheref

1

=f

2 .

The deformation is propagated around the con-

tact region in order to preserve surface smooth-

ness, and to compensate for the loss of volume

due to the rst deformation. Opalach [11] gives

an eÆcient solution for doing so: f

1

is replaced

by f +b (f ) (similarly f by f +b (f )), the

1 2

sultingnewlocalbulgearoundthecontactregion

smoothlyconnectswiththerest oftheobject.

PCM was onlyapplied to distance surfaces for which

theeldfunctionwasalmostlinearlydecreasingaround

theiso-value[5,11]. Itsapplicationtoconvolutionsur-

faces such as those of [6], where the integrated eld

function varies as theinverse of the squared distance

totheskeleton,introducesanewsetofproblems,that

will bediscussedinSection 5.

3 Adaptive Polygonisation

The categories for visualising iso-surfaces are ray-

tracing, particles, or triangulation of the surface [2].

Evenwith improvements,therstmethod istooslow

to allow interaction with the skeleton, although pro-

viding high quality images. The second produces too

rough a display of the surface. The third has the

advantage of benetting from hardware acceleration.

The method we propose is an adaptive triangulation

method which isanextension ofsamplinganddisplay

method introduced by Desbrun [5] and improved by

Cani [6], in which sample points are sent along given

raysattached to skeletal elements, and sample its el-

ement's territory, i.e. theregions where the elements

eld contribution is the highest. Inorder to produce

an adaptive sampling, we propose to attach a rough

polygonisationto afew samplingpoints,andto rene

recursivelythepolygonisationusinganerrorcriterion.

Sample points are dened in a coordinate set at-

tached to the control skeleton. As shown in 2, the

point'srayneedsanoriginontheskeletonandadirec-

tionalongwhich thepointmoves.

u t d

n

Figure 2. Sampling point. Left: a single point.

Right: triangle supported by sampling points.

3.1 Setting local meshes

A rough polygonisation that coats the skeleton is

neededforinitialisation. Itisnotobvioushowtodene

it in branchingregions. Thus, theskeletoniscutinto

itsmaximallinearpieces(seeFigure3). Closedmeshes

are generated around each of those pieces. To avoid

visual discontinuities, themeshes are made to locally

overlap at joints, as was done in [6] (see sub-section

2.2).

Figure 3. A coarse skeleton made of seven surface pieces.

Theclosedmeshisinitializedasfollows: oneachlin-

earpieceofconnectedsegments,samplingaxesaresent

aroundthesurfaceinplanessuchthat:ifthejointisan

extremity,theplaneisnormaltothesegment;elsethe

jointisofvalence2,andtheplaneisthemedialplane

betweenthetwosegments. Additionalaxesaresentat

verticesofvalence1,alignedwiththesegmenttheyare

attachedto. Notethateachaxist

i

isattachedalonga

segmentataparameteru

i

. Theroughtriangulationis

thenbuildusingsamplepoints 1

ontheaxesasvertices

(seeFigure4).

Figure 4. Exploded construction of a coarse local mesh.

3.2 Adaptive local refinement

Therenementof thecoarse triangulationwehave

dened can be done in two ways: the rst is a triv-

ial uniformsubdivision of allthecoarse triangles; the

second is an adaptive renement using a local error

metric.

Thebestmethodtochoosewhetheratriangleneeds

subdivisionornotwouldbetomeasurethevolumethat

separatesthetrianglefromthemathematicallydened

iso-surface. This would however be time consuming.

Wehavechosentomeasuretheerrorusingdirectlythe

eld function: thus, a triangle edge is subdivided if

the dierence between the eld valueat its midpoint

andtheiso-valueisoisgreaterthanaconstant(which

couldbeview-dependenttoreneareasofinterest).

1

Convergence of the sample points towards the iso-surface

is donebynding the root of (f(root+tdir) iso) 2

using

Newton-Raphsonmethod,thusnowingt,beneciatingofcoher-

encebetweenpositions,orcanbedoneusingmethodsproposed

=

f v0+v1

2

iso

Thesubdivisionof anedgethenrequirestheinsertion

ofanewaxis: itsoriginanddirectionareinterpolated

linearly between the two axes of the vertices of that

edge. Figure5illustratesthebenetsofadaptivepoly-

gonisation: curvedareasaremorenely sampledthan

atterareas.

Figure 5. Adaptive triangulation of an implicit surface with varying radius along a skeleton curve.

4 Avoiding Unwanted Blending using

Local Convolution

As mentioned previously, shape modeling and ani-

mationwith skeletondened implicitsurfaces is intu-

itive,astheshapecorrespondstoacoatingoftheskele-

ton. Thanksto the blending property of this model,

theresultingiso-surfaceis smoothandhasanorganic

aspect without any eort. Unfortunately, this prop-

ertyalsoturnsouttobeadrawback,aswhenskeleton

primitivesaretooclose,allcanblendwitheachother,

withoutanycontrolonthetopologyofthenalshape.

AsstressedinSection2,noneoftheprevioussolutions

to thisproblemwassatisfying.

Oursolutioncanbeseenasalocalconvolution,since

we use a restricted skeleton range for computing the

eld valueat aquerypoint: aparameteru

i

alongthe

skeletonis assignedto eachquerypointp

i 2IR

3

. See

Figure 6. We then dene theportion of the skeleton

to be used forcomputing f(p

i

) as a neighborhood of

constantsizeDaroundu

i

(DisexpressedinEuclidian

distance along the skeleton). The eld value is com-

putedexactlythesamewayasbefore,exceptthatonly

theusefulpartoftheskeletonisconsideredduringthis

computation.

p

j u

i

u

j

D

j D

i

p

i

Figure 6. Local convolution

The only diÆculty consists in assigning askeleton

parameteru

i

toeachpointp

i 2IR

3

. Deningu

i asthe

parameter of the closest point on the skeleton would

i

oneachskeleton-curveoftheskeleton. Thetotaleld

value near abranching point is then dened by sum-

ming thesecontributions.

Letusconsider agivenskeleton-curve. Ifthequery

pointp

i

isoneofthesamplepointsusedfordisplaying

thesurface,ndingu

i

iseasy: wejust usetheparam-

eter of the reference point on the skeleton which was

usedfordening p

i

(seeSection3).

Inthegeneralcase,weneedto deneasimilar ref-

erencepointontheskeleton. This isdonebyre-using

the set of planes dened at each vertex of the skele-

ton, and depicted in Figure 4(for vertices of valence

1, the plane is normal to the segment, and for ver-

tices of valence 2, it is the medial plane between the

twoadjacentsegments). In-betweenplanesaredened

along segmentsusinglinearinterpolation betweenthe

planesassociatedwithvertices. Finding u

i forp

i then

simplyconsistsin ndingwhichofthese planespasses

throughp

i

,and usingthecorrespondinginterpolation

parameter. Thisisdonebysolving:

8

>

>

<

>

>

: r

i

=(1 u

i )r

0 +u

i r

1

d

i

=(1 u

i )d

0 +u

i d

1

d

i (p

i r

i )=0

u

i 2[0;1]

where r

0 and r

1

are vertices that dene the current

curve-segment,and d

0 and d

1

are the normal vectors

that dene the planesat r

0 and r

1

, respectively. See

Figure7foratwodimensional illustration.

di

r0

r1

ri

d1 d0

Figure 7. Assignation of a parameter to a point in IR

for a segment of a linear skele- ton.

When several dierent planes generated by dier-

entline-segmentsofthecurvepassthroughp

i

,theone

whose parametergenerates themaximal eld value is

selected.

Thebenetsofournewcontrolledblendingmethod,

incomparisonwiththesolutionin[6],isshownonFig-

ure8. Thelocalconvolutionsolutionenablesustopre-

vent unwantedblendsbetweensmall folds,which was

impossibleusingtheprevioussolution.

5 Precise Contact Modelling

PreviousworkonPCMdealtwithdistanceimplicit

(a) (b)

Figure 8. Controlled blending: comparison between the solution in [6] (a) and our new solution based on local convolution (b).

suited tointeractivemodelingand animationenviron-

ments [11]. In thecontext ofconvolutionsurfaces [6],

the main dierence lies with the shape of the convo-

lution kernel; unlike the previous method, the eld

functionisnotlinearlydecreasing. Also,thesubdivion

structure of ourskeleton canbeused to decrease the

computation cost of collision detection. This section

will rst describe how we deal with the collision de-

tectionandprecisecontactmodeling(PCM),andthen

wewill focuson the issuesintroduced by convolution

surfaces.

The PCM process has three main steps: detect a

collision;ndthecontactsurface;andgeneratethede-

formations.

Detectingthecollisionisrstperformedusing axes

aligned bounding boxes, as in the previous method.

The bounding boxes are constructed using sample

points from the object which completely enclose the

surfaceevenafterthePCMprocess(thedeformedsur-

face has a dierent shape and perhaps volume after

contact). Eachoftheskeletalelementshavetheirown

boundingbox,asanobjectmightbeincollisionwithit-

self. Anelement'sboundingboxissizedusingitssam-

plepoints(denedasdescribedinSection3). Thesize

isnallyincreasedtocompensateforroughlysampled

curved areas,using the currentvalue of thesampling

error.

A bounding box is associated with a skeletal ele-

ment,sowhenaskeletonisrenedtheresultingbound-

ingboxesarebetterttedtothesurface.Theserened

orsub-boundingboxesallowusto usethebenetsas-

sociatedwiththelevelsofdetailforthecollisiondetec-

tion.

Samplingpointsusetheboundingboxintwostages:

rst a check is performed to detect if the bounding

boxesoftwonon-adjacentskeletalelements(orskeletal

elementsofdierentobjects)comeintocontact. Ifthis

testistruethenextstage queriesthesamplepointsof

therstobjecttodetectiftheyareinsidethebounding

boxofthesecond. Whenthesamplingpointisdetected

tobepotentiallyincontactwithanotherprimitive(ie,

aggedashavingbeenincollisionandthedeformation

term,presentedbelow,is addedtothesamplepoint.

Thedeformationtermhastworoles: todeform the

surfaces suchthat surfacesthat penetratedeach other

are in exact contact; and to add bulges to the sur-

roundingsurfacetokeepthemsmootharoundcontact

areas.

Fortwofunctionsf

i

(p)andf

j

(p)whoseiso-surfaces

are in contact, the separation limit veries f

i (p) =

f

j

(p). Thus, forsampling pointsp

i

ofi that standon

objectj's side, adding theterm iso f

j (p

i )to f

i (p

i )

makesp

i

convergetothedesiredsurface(whichisdone

symmetricallyforsamplingpointsofj). Ontheborder

ofthisrstdeformation,thesurfaceishoweverdiscon-

tinuous;thus,adeformationtermisaddedtosampling

points within a propagation area around the contact

surface.

Thebulgeareaismodeledwithadilationtermthat

is added around the objects. See for instance Fig-

ure 9. It is such that surfaces' C 1

continuity is pre-

served. Consideringobject i's deformation by object

j,thebulgefunctionb

i (f

j (p

i

))whichsurroundsobject

j is choosen as to satisfy: b

i

(iso) = 0, b 0

i

(iso) = 1,

b

i (c

j

)=0and b 0

i (c

j

)=0,where c

j

<iso controls the

end of the bulge createdby object j. (it is acoating

eld aroundj that stopsatc

j ).

Opalach[11]didnotgiveanyequationforfunction

b

i

. We are experimenting with three versions of the

function. The rst one is very simple to implement

and to use,sinceits onlyparameteristhe bulge'sex-

tentinspace, controlledbyc

j

: weuseitin animation

situations,where theuserequiresaneÆcientsolution.

The rst uses the fourth Hermite basis function.

Thisdoesnotusethehparameter,sooerslesscontrol

oftheshapeofthebulge(seeFigure9):

if f

j (p)2[c

j

;iso]

b

i (x)=x

2

(1 x)

where x=(f

j (p) c

j

)=(iso c

j )

Figure 9. Fourth Hermite basis function.

The second one is a cubic piecewise polynomial

which provides more control on the bulge's shape,

sincethelocalmaximumofthebulgecanbecontrolled

in position and height: the position of the bulge is

controlledwithaeld valuemverifyingiso<m<c

j ,

maximum.

if f

j (p)2[c

j

;m]

b

i (f

j

(p))=(3 2x)x 2

h

wherex=(f

j (p) c

j

)=(m c

j )

if f

j

(p)2[m;iso]

b

i (f

j

(p))=h+((1 3h)+(2h 1)x)x 2

wherex=(f

j

(p) m)=(iso m)

Figure 10. Height control with a piecewise polynomial.

The third piecewise function is non-polynomial. It

isanadaptationofthesecondfunction,wheretheleft

parthas been modied byacosine. The control over

thepositionofthebulgeislosttoensurethattheperi-

odslengthsareconstant: m=c

j

+(iso c

j )(1

1

4n+3 ).

Thenumberofripples isspeciedthroughn.

if f

j (p)2[c

j

;m]

b

i (f

j

(p))=(3 2x)x 2

h

cos(((2n+1)x+1))+1

2

wherex=(f

j (p) c

j

)=(m c

j )

if f

j

(p)2[m;iso]

b

i (f

j

(p))=h+((1 3h)+(2h 1)x)x 2

wherex=(f

j

(p) m)=(iso m)

Figure 11. Ripples modeled using function with local maxima.

Theheighthisconstrainedtobeatleastlowerthan

iso,otherwise,anunwantedimplicitsurfacewouldap-

pear,coatingbothsurfaces. Inthecaseofconvolution

surfaces,thisdoesnotprovideenoughconstraints: un-

likethepreviousmethod[11], theeldfunction isnot

linearlydecreasing;atsomedistancefromtheskeleton

theelddecreasestooslowlyandisalmostconstant. If

suchavalueofisoischosenthentheresultofthebulge

function maypropagateandcoattheotherobject(see

Figure13). Inpractice,weneedtochooseaniso-value

(a) (b) (c)

Figure 12. Line primitives with intersecting re- gions: (a) with fast bulge function, (b) with height controlling function, (c) with ripple function.

fj(p)=iso

(a) (b) (c)

fi(p)=iso

fj(p)+bj(fi(p))=iso f

i (p)+b

i (f

j

(p))=iso

fj(p)2[cj;iso]

Figure 13. (a) Without PCM (b) With PCM (c) object

’s iso-surface and object

bulge func- tion.

ThebulgesofFigure12wereobtainedusingthefol-

lowingcoeÆcients:

iso m c

j

h n

column(a) 2 0:4

column(b) 2 1:3 0:4 0:5

column(c) 2 0:4 0:5 2

6 Applications

6.1 Modelling organic shapes in contact situations

Figure 14(a) and (b) showapartof ahandwhere

thengerand thumb thatsquashasmall ballare de-

formedusingPCM.Figure14(c)and(d)comparethe

eect ofnotusingandusing PCM.

OurunwantedblendingsolutionisillustratedinFig-

ure 15, with the snail character. The wireframe pic-

(c) (d)

Figure 14. Hand: (a) and (b): finger and thumb grabbing a samll ball (with PCM); (c) and (d):

hand grabbing a ball, without and with PCM respectively.

Figure 15. Top: snail character. Bottom: 3 error bounds for the adaptive polygonisaton.

6.2 Animation of soft, organic shapes

Anotherapplicationofthesystemistoanimatede-

formable bodies made of skeleton curves coated with

animplicit\skin". Motionandthemaindeformations

are controlled through theanimation of the skeleton.

The implicit surfacethat coats it may locally deform

whendierentobjects,orpartsof anobjet,comeinto

contact.

Figure16showsanexampleofasuchananimation.

Here, physically-based animation has been used for

generatingthemotionoftheobject'sskeleton,whichis

modelledasachainofmassesandsprings. Theobject

falls under the action of gravity, sliding along awall.

A penalty method is used for computing collisionre-

sponsewiththewallandbetweendierentfoldsofthe

object. Sincephysically-basedmotionsaretunedusing

mationisrsttunedusingaverylowresolution. When

theuserissatised,highqualityimagesarecomputed.

Figure 16. Controlled blending in a physically- based animation framework, where a de- formable object falls under gravity.

7 Conclusion and Future Work

In this paper, we have extended the subdivision-

curveimplicitprimitiverepresentationtomakeitmore

suitablefortheinteractivemodellingandanimationof

complex, soft objects in contact situations. Blending

control is performed usingalocal convolution: onlya

portion ofthe skeleton istaken into accountfor com-

putingtheeldcontributionataquerypoint. Contact

modellingisessentialformodellingthedeformationof

non-blendingshapeswhen theycome into contact. It

usesaspeciccombinationofthedierentobjectseld

functionstogeneratecontactsurfaceswhilepreserving

smoothness. Lastly, the surfaces are displayed using

anadaptivepolygonizationscheme. This increasesef-

cienty withoutdropping quality, sinceahigher reso-

lutionisgeneratedin highlycurvedareas.

Thesystemhasbeenused to createavirtualhand

thatusescontactmodellingtomodelinteractionswith

soft objects. This hand could be used as a tool in a

virtualsculptingenvironment[7,13]. Atpresenttools

in such systemstend to belimited to theaddition or

removalof material. It would beinteresting to use a

virtualhandwithprecisecontactmodellinginorderto

emulatetheactionsofarealhandanditseectonthe

clay. Wehavealso shown that ourmodelling method

can be used in a physically-based animation frame-

work,the implicit \skin" generated with our method

being used forcoating ananimated skeleton. We are

planning to use this methodology in the context of a

pedagogic intestinal surgery simulator, aimed at sur-

by the EU Trainingand Mobility of Researchersnet-

work(TMRFMRX-CT96-0036)\PlatformforAnima-

tionandVirtualReality".

TheauthorswouldliketothankFrancoisFaureforhis

contributionsto thephysically-basedsimulation.

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