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

ut
u0
u1
kt
k1
k0
E
2
t
E
v0
l1
lt
vt
l
v1
(a) Two deforming edges
1 2
E & E
1
t
0
ut
kt
E
2
t
E
vt
( l
lt
t kt)
nt
(b) Projected distance
1 between E and E
Figure4: EEtestfilter:ToperformaEEtestbetweenthetwo
edges E 1 and E 2 (definedby u0, v0and k0, l0at t = 0, u1, v1
and k1, l1at t = 1),weneedtocheckthecoplanarityconditions
oftheseverticesbyfindingall t(t ∈ [0, 1])wheretheprojected
distancebetween ltandthetriangledefinedby kt, ut,and vtis
equaltozero,i.e., (lt − kt) · nt = 0.
Soweobtainthesufficientconditionfornon-coplanar: Ifallthe
fourscalarvaluesarenegative/positive,thevertices Ptwillnever
beontheplanedefinedbythethreevertices(at, bt,and ct)ofthe
deformingtriangle Tt.
Thegeometricinterpretationofsymbol Aand Bisthattheyrepresenttheprojecteddistancesat
t = 0and t = 1,respectively.From
Equation(6),weconcludethatitisnotsufficienttoonlycheckthe
signsof Aand B,thevaluesofothersymbols C-Falsoaffectthe
coplanaritycondition.
3.3.1 ExtensiontoEEtests
Inordertoperformanelementarytestbetweentwoedges E 1 and
E 2 (definedby u0, v0and k0, l0at t = 0, u1, v1and k1, l1at t =
1),weneedtocheckthecoplanarityofthesefourpoints(Fig.4).
Byreplacing Pt, at, bt, ctwith lt, kt, ut, vt,wecansymmetrically
deduceanon-coplanartheoremforEEtests:
Non-coplanarTheoremforEEtests: Fortwodeformingedges
E 1 and E 2 definedbythestartandendpositionsduringtheinterval
[0, 1],thesepositionsarelinearlyinterpolatedintheinterval
withrespecttothetimevariable, t.Ifthefollowingfourscalarvalues:
A ′ , B ′ , 2∗C′ +F ′
,and 3 2∗D′ +E ′
havethesamesign, E 3
1 and
E 2 willnotbecoplanarduringtheinterval:
A ′ = (l0 − k0) · n ′ 0, B ′ = (l1 − k1) · n ′ 1
C ′ = (l0 − k0) · ˆn ′ , D ′ = (l1 − k1) · ˆn ′
E ′ = (l0 − k0) · n ′ 1, F ′ = (l1 − k1) · n ′ 0
And: n ′ 0 = (u0 − k0) × (v0 − k0), n ′ 1 = (u1 − k1) × (v1 − k1)
ˆn ′ = (n′ 0 +n′ 1 −( vu− v k)×( vv− v k))
2
vk = k1 − k0, vu = u1 − u0, vv = v1 − v0.
3.4 CCDAlgorithm
We use the non-planarity filter defined above to perform
coplanarity-basedculling.WeuseEquations(1)-(3)tocalculatethe
valuesofthefourcontrolverticesdefinedinthoseequationsfora
primitivepair.Ifthesevariableshavethesamesign,thispairwill
beidentifiedasafalsepositive.
1
t
t
nt
2
t
BVH Traversal
Inter-object testing
Triangle based
bounding volume
culling
Intra-object testing
Continuous normal
cone
PR-Trianges
Orphan set
Primitive based
bounding volume
culling
Deforming Non-
Penetration Filter
Elementary tests
Figure5: CCDalgorithmwithdeformingnon-penetrationfilter:Thedeformingnon-penetrationfilteriscomplementarytotheoverallCCDalgorithmandcanbeperformedrightbeforetheelementarytests.
(a) Use of deforming filter (b) Only use of bounding volume tests
Figure6: EdgesinvolvedbyEEtestswithdifferentculling
methods:Thisfigureshowsedges(highlightedingreenandblue
respectively)usedbyEEtestsduringaspecificframeoftheCloth
benchmark(Fig.7(d))withcoplanarity-basedcullingandboundingvolumebasedcullingrespectively.Comparingtoonlyboundingvolumebasedculling,thenumberofEEelementarytestsare
reducedby 80%withthedeformingnon-penetrationfilter.
Thedeformingnon-penetrationfiltercanbecombinedwithhierarchicalrepresentations(Fig.5).
DuringthetraversalofBVH,we
performdifferenthigh-levelcullingmethodstoremoveredundant
pairwisetestsorprimitivepairsthatarenotincloseproximityto
oneanother.Next,thedeformingnon-penetrationfilterisusedas
partoflow-levelcullingtofurtherremovethefalsepositives.
Fig.6comparestheeffectsofboundingvolumebasedcullingand
coplanarity-basedculling.ThegreenlinesegmentsinFig.6(a)and
bluelinesegmentsinFig.6(b)describethedistributionofedgesinvolvedinEEtestswithdeformingnon-penetrationfilterandboundingvolumebasedcullingrespectively.Asshowninthefigure,comparedtothenumberoftestsintheboundingvolumebasedculling,
weobservemorethan 80%reductioninthenumberofelementary
testsonthisscene.
4 ImplementationandPerformance
Inthissection,wedescribeourimplementationandhighlightthe
performanceofouralgorithmonseveralbenchmarks.