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Animation=
Interpolations
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Interpolations/Régularisations
• Functions de bases– X(u) = B0(u) X0 + B1(u) X1 + B2(u) X2 + …
X0
X1 X2
X3
X(u)
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Splines Cubiques
• Bi polynôme degrée 3 :Bi(u) = bi3 u3 + bi2u2 + bi1u1 + bi0u0
= (u3 u2 u1 1) (bi3 bi2 bi1 bi0)
• Écriture matricielle :
X = (u3 u2 u1 1) [bij] (X0 X1 X2 X3)t
= [u] Mspline [Xj] t
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Continuité C0, C1 et C2 (1/2)
• R2, R3
Xj = (xj yj zj) Xj = (tj xj)
• Q1 : (P0 P1 P2 P3)
Q2 : (P1 P2 P3 P4)
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Continuité C0, C1 et C2 (2/2)
B-Splines• C0 C1 C2
B0(1) = 0 B’0(1) = 0 B’’0(1) = 0
B1(1) = B0(0) B’1(1) = B’0(0) B’’1(1) = B’’0(0)
B2(1) = B1(0) B’2(1) = B’1(0) B’’2(1) = B’’1(0)
B3(1) = B2(0) B’3(1) = B’2(0) B’’3(1) = B’’2(0)
0 = B3(0) 0 = B’3(0) 0 = B’’3(0)
• B0(0) + B1(0) + B2(0) + B3(0) = 1
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Bases
• B-Splines
P(u) = (u3 u2 u1 1) 1/6 ( ) (P0 P1 P2 P3)t
• C-Splines, -Splines, Catmul-Rom, etc. etc.
• NURBS : “Non-Uniform Rational B-Spline”[X(u)/W(u), Y(u)/W(u), Z(u)/W(u) ]
cercle
-1 3 -3 1 3 -6 3 0-3 0 3 0 1 4 1 0
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Conversions
P (u) = [u] Mbézier [Pbézier]
= [u] MB-spline [PB-spline]
[Pbézier] = (Mbézier -1 MB-spline) [PB-spline]
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Bézier & de Casteljau[@Renault, 72 et @Peugeot, fin 1960]
• Bi,3(u) = Ci3 ui (1-u)3-i [Polynômes de Berstein]
• P(0) = P0
P(1) = P3
P’(0) = P1-P0
P’(1) = P3-P2
Recollement C1
Construction, et reccursion.
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Tangentes & Périodicité
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Courbe de vitesse
• Contrôle de la vitesse
• Ease-in, ease-out
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Les 12 principes de l’Animation ~1930, Studios Disney
The Illusion of Life
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Squash & Stretch
Déformations– Pseudo physique- <<motion blur>>- Conservation du volume
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Anticipation
Guide l’attention,annonce l’action…
…prépare un gag mouvement C2…
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Staging
[Mise en scène]
Atmosphère, éclairage, …
« Present the idea so that it is unmistakably clear »
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Rigging
Préparation des personnages
Skelettes, articulations, etc. etc.
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Follow-through et Overlapping
Enchaînement des actions
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Slow in & Slow out
Fluidité des mouvements
(Continuité C2 des déplacements)
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Secondary actions
Donner de la richesse– Hiérarchie/articulations– En retard
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Arcs
• Courbes naturelles• Non-synchronicité des mouvements
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Timing (1/2)
• Masse
• Taille
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Timing (2/2)
[John Lasseter, Pixar, 1987]0 hit by a tremendous force.
1 hit by a brick, rolling pin, frying pan.
2 nervous tic, muscle spasm, an uncontrollable twitch.
3 dodging a brick, rolling pin, frying pan.
4 giving a crisp order, "Get going!" "Move it!"
5 friendly, "Over here." "Come on-hurry!"
6 sees a good looking girl, or the sports car he has always wanted
7 tries to get a better look at something.
8 searches for the peanut butter on the kitchen shelf.
9 appraises, considering thoughtfully.
10 stretches a sore muscle.
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Exaggeration