Paper - Geometric Algorithms for Modeling, Motion, and Animation ...

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anddescribethecoplanarity-basedcullingmethodandthedeformingnon-penetrationfilterinSection3.WepresenttheoverallCCD algorithmanditsimplementationinSection4. Wecompareour approachwithprioralgorithmsinSection5. 2 RelatedWork ManyefficientalgorithmshavebeendesignedforCCDbetween rigidmodelsanddeformablemodels[Govindarajuetal.2005;HutterandFuhrmann2007;Zhangetal.2007;Curtisetal.2008;Tang etal.2009a;Tangetal.2009b]. Someoftheseapproacheslinearlyinterpolatebetweentheverticesofthemodelandcomputethefirsttime-of-contactbasedonhierarchicalcullingandperformingelementarytestsbetweenthetrianglepairs.Basedonlinearly interpolatingmotionofthevertices,theelementarytestsreduce tosolvingcubicequations. However,thesetestscanbeexpensiveandalsosensitivetotolerance[BrochuandBridson2009].OtherCCDformulationsuseadifferentformulationoftheinterpolatingmotion[Redonetal.2002;Redonetal.June,2004;Kimand Rossignac2003].Mostofthesemethodsreducetocomputingroots ofpolynomialfunctionsandcanbeacceleratedusingBéizerclipping[SederbergandNishita1990],recursivesubdivision [Taubin 1994],oreigenvaluecomputations[ManochaandDemmel1995]. MostoftherecentworkonCCDalgorithmshasbeenondesigninghigh-levelandlow-levelcullingalgorithmsthatcanreducethe numberofelementarytestsbetweentheprimitives. High-levelculling: Manyhigh-levelmethodsuseBVHstocull awaythenon-overlappingprimitivesandtheycanusetightfittingboundingvolumessuchask-DOPsorOBBstoobtainhigher cullingefficiency.Othercullingmethodscheckforself-collisions basedonsurfacenormalsandcurvature[VolinoandThalmann 1994; Provot1997; Mezgeretal.2003]. Recently,Tangetal. [2009a]extendedtheseideastoCCDandpresentedcontinuousnormalcones(CNCs). Low-levelculling: HutterandFuhrmann [2007]usedbounding volumes(k-DOPs)oftheprimitivestoavoidperformingelementarytestsbetweendifferentfeatures.Curtisetal.[2008]andWong andBaciu[2006]usedmaskingschemestoremovetheredundant elementarytests.However,neitheradjacenttrianglesnortheirfeaturescanbeculledbyboundingvolumes.Govindarajuetal.[2005]eliminatesomeoftheelementarytestsassociatedwithadjacenttrianglesbasedonsomeofthetestsbetweenthenon-adjacentprimitives. Tangetal.[2009a]proposedtheconceptoforphansetsto eliminatealmostalltheredundantelementarytestsbetweenadjacenttriangles.Theideahasbeenfurtherextendedtoproceduralrepresentationtriangles(PR-Triangles)toremoveallredundant elementarytestsbetweennon-adjacenttriangles. Althoughthese methodscanlowerthenumberofelementarytestsandfalsepositives,thecurrentCCDalgorithmscanstillresultinahighnumber offalsepositives(e.g., 95%ormore). 3 DeformingNon-PenetrationFilter Inthissection,weintroducethenotationusedandpresentour cullingalgorithmthatreducesthenumberoffalsepositives. 3.1 Notations Weusefollowingnotationsintherestofthepaper: • SV (X)isthesweptvolumeofadeformingprimitive X alongthelinearinterpolatingpathbetweenthevertices. • BV (X)istheboundingvolumeofadeformingprimitive X. BV (T ) T0 T0 SV (T ) P0 T1 (a) Bounding volume of a deforming triangle P1 BV (P) T1 (c) Bounding volume test BV (T ) n0 BV (P) T0 P0 n1 P0 1 P SV (P) (b) Bounding volume of a deforming vertex T1 (d) Coplanarity test Figure2: DeformingFilter:Foradeformingtriangle Tanda deformingvertex Pdefinedby T0,T1and P0, P1respectively,the boundingvolumetest(c)becomesquiteconservative. Thecoplanaritytest(d)checkswhetherapenetrationbetweentheprimitives duringthetimeinterval. Ifthevertexisalwaysonthesameside ofthetriangleduringtheentiretimeinterval,thenthefourvertices associatedwiththatelementarytestcannotbecoplanarduringthe timeinterval [0, 1]andtherefore,nocollisionoccurs. • T0, T1and Ttrepresenttheinstancesofadeformingtriangle Tat t = 0, t = 1,andarbitrary t ∈ [0, 1],respectively. • a0, b0,and c0arethethreeverticesof T0. a1, b1,and c1are theverticesof T1. at, bt,and ctaretheverticesof Tt. • n0, n1,and ntarethenormalvectorsof T0, T1,and Tt. • P0, P1and Ptaretheinstancesofadeformingvertex Pat t = 0, t = 1,andarbitrary t ∈ [0, 1],respectively. • Operator‘∗’,‘·’,and‘×’denotemultipleoftwoscalevalues, dotproductoftwovectors,andcrossproductoftwovectors, respectively. 3.2 Motivation Giventhelinearinterpolatingmotionbetweenthevertices,theCCD testbetweenatrianglepaircanbereducedtotwotypesofelementarytests:6VFtestsand9EEtests.Eachelementarytestcanbe furtherbrokendownintotwoparts:coplanaritytestandinsidetest. BoththeVFtestsandEEtestsinvolvetheuseoffourdeforming vertices,andanecessaryconditionforacollisionisthatthesefour verticesbecoplanar.Provot[1997]showedthecoplanaritytestof fourverticescanbereducedtofindingrootsofacubicequation. Insteadofsolvingthecubicequation,wewilldeduceasufficient conditionthatthesefourverticesarenon-coplanarduringthetime interval. Byusingthiscondition,manyelementarytestscanbe culledandwedonotneedtosolvethecubicequations. Weuseboundingvolumesforeachprimitive(edge,face,vertex), andperformboundingvolumebasedcullingbeforetheelementary tests.However,theboundingvolumetestscanbecomequitecon- P1
a0 a1 P0 n1 T0 T1 P t b0 Tt b1 n0 P1 nt c1 c0 (a) Deforming triangle ( T) & deforming vertex ( P) at Pt ( P a ) n t t nt (b) Projected distance between Pt and t T Figure3:VFtestfilter:ToperformaVFtestbetweenadeforming triangle(definedby a0, b0, c0at t = 0,and a1, b1, c1at t = 1) andamovingvertex(definedby P0at t = 0and P1at t = 1),we needtocheckforcoplanaritybetweenthevertexandthetriangleby findingat(t ∈ [0, 1])whentheprojecteddistancealongthenormal vectorofthetriangleisequaltozero,i.e., (Pt − at) · nt = 0. servativeforprimitivelevelCCDtests.Fortheexampleshownin Fig.2, BV (T)and BV (P)aretheboundingvolumesofadeformingtriangle T andadeformingvertex P,respectively. Inorder toperformculling, BV (T)and BV (P)mustcontaintheswept volumesofdeformingprimitives,i.e., SV (T)and SV (P),respectively(Fig.2(a-b)). TheVFpair {T, P }willbetestedforexact CCDtest,sincetheirboundingvolumesoverlapduringthetime interval [0, 1](Fig.2(c)). Wemakeuseofthefollowingobservation:ifthevertex Pisalways onthesamesideofthetriangle Talongtheentire [0, 1]trajectory, thispairshouldbeclassifiedasafalsepositivesinceitcannotbe coplanar(Fig.2(d)).Basedonthisobservation,wederiveasufficientconditiontocheckwhether Ptwillalwaysbeonthesameside of Ttduringthetimeinterval [0, 1].Ifapairofprimitivessatisfies thiscondition,thenwedon’tneedtoperformtheexacttestinterms ofsolvingacubicequation. TheideacanbesimilarlyextendedfortheEEtests:ifthereisno crossingbetweenthetwodeformingedgesalongthecontinuous trajectory,theycannotbecoplanarduringthetimeinterval. 3.3 CoplanarityTest Inordertocheckthecoplanarityofavertex Ptandatriangle Tt, weneedtocalculatetheprojecteddistancebetweenthemalongthe directionof nt,asshowninFig.3(b).Ifthisdistancebecomeszero atanytimeinterval,thanthefourverticesofthetwoprimitivesare classifiedascoplanar. Non-coplanarTheoremforVFtests: Foratriangle Ttanda vertex Ptdefinedbythestartandendpositionsduringtheinterval [0, 1],thesepositionsarelinearlyinterpolatedintheintervalwith respecttothetimevariable, t.Ifthefollowingfourscalarvalues: A, B, 2∗C+F ,and 3 2∗D+E havethesamesign, Ttand Ptwillnot 3 becoplanarduringtheinterval: A = (P0 − a0) · n0, B = (P1 − a1) · n1 (1) C = (P0 − a0) · ˆn, D = (P1 − a1) · ˆn (2) E = (P0 − a0) · n1, F = (P1 − a1) · n0 (3) t And: n0 = (b0 − a0) × (c0 − a0), n1 = (b1 − a1) × (c1 − a1) ˆn = (n0 + n1 − (vb − va) × (vc − va)) 2 va = a1 − a0, vb = b1 − b0, vc = c1 − c0. Proof. Thenormalvector ntofthedeformingtriangleattime tcan berepresentedasfollowing: nt = n0 ∗ B 2 0(t) + ˆn ∗ B 2 1(t) + n1 ∗ B 2 2(t) (4) where B 2 i (t)isthe i th basisfunctionoftheBernsteinpolynomials ofdegree 2. Wedefine: α = B 2 0(t) = (1 − t) 2 , β = B 2 1(t) = 2 ∗ t ∗ (1 − t), and γ = B 2 1(t) = t 2 .ThenEquation(4)becomes: nt = n0 ∗ α + ˆn ∗ β + n1 ∗ γ Forthemovingvertex Pt = P0 ∗ (1 − t) + P1 ∗ tandavertexof thedeformingtriangle at = a0 ∗ (1 − t) + a1 ∗ t,theirprojected distantalong ntis: (Pt − at) · nt = ((P0 − a0) ∗ (1 − t) + (P1 − a1) ∗ t) · nt = ((P0 − a0) ∗ (1 − t) + (P1 − a1) ∗ t) ·(n0 ∗ α + ˆn ∗ β + n1 ∗ γ) = (P0 − a0) · n0 ∗ (1 − t) ∗ α + (P0 − a0) · ˆn ∗ (1 − t) ∗ β + (P0 − a0) · n1 ∗ (1 − t) ∗ γ + (P1 − a1) · n1 ∗ t ∗ γ + (P1 − a1) · ˆn ∗ t ∗ β + (P1 − a1) · n0 ∗ t ∗ α Substitute α, β,and γ,wehave: (Pt − at) · nt = (P0 − a0) · n0 ∗ (1 − t) 3 + (P0 − a0) · ˆn ∗ 2 ∗ (1 − t) 2 ∗ t + (P0 − a0) · n1 ∗ (1 − t) ∗ t 2 + (P1 − a1) · n1 ∗ t 3 + (P1 − a1) · ˆn ∗ 2 ∗ t 2 ∗ (1 − t) + (P1 − a1) · n0 ∗ t ∗ (1 − t) 2 UsingthesymbolsdefinedbyEquations(1)-(3),Equation(5)becomes: (Pt − at) · nt = A ∗ (1 − t) 3 + C ∗ 2 ∗ (1 − t) 2 ∗ t + E ∗ (1 − t) ∗ t 2 + B ∗ t 3 + D ∗ 2 ∗ t 2 ∗ (1 − t) + F ∗ t ∗ (1 − t) 2 = A ∗ B 3 0(t) + + 2 ∗ D + E 3 2 ∗ C + F 3 ∗ B 3 1(t) (5) ∗ B 3 2(t) + B ∗ B 3 3(t) (6) where B 3 i (t)isthe i th basisfunctionoftheBernsteinpolynomials ofdegree 3. Byusingtheconvexhullpropertyassociatedwithcontrolpointsof theBernsteinbasis,therangeoftheprojecteddistancebetween Pt and Ttisboundedbythecontrolvertices.Inourcase,thesecontrol verticesarethefourscalarvalues: A, B, 2∗C+F 3 ,and 2∗D+E 3 .
Page 1: FastContinuousCollisionDetectionusi
Page 5 and 6: 4.1 Implementation Wehaveimplemente
Page 7: Acknowledgments WewouldliketothankJ

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