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

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 .