Animation=

Interpolations

2

Interpolations/Régularisations

• Functions de bases– X(u) = B0(u) X0 + B1(u) X1 + B2(u) X2 + …

X0

X1 X2

X3

X(u)

3

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

4

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)

5

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

Les 12 principes de l’Animation ~1930, Studios Disney

The Illusion of Life

12

Squash & Stretch

Déformations– Pseudo physique- <<motion blur>>- Conservation du volume

13

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 »

15

Rigging

Préparation des personnages

Skelettes, articulations, etc. etc.

16

Follow-through et Overlapping

Enchaînement des actions

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Slow in & Slow out

Fluidité des mouvements

(Continuité C2 des déplacements)

18

Secondary actions

Donner de la richesse– Hiérarchie/articulations– En retard

19

Arcs

• Courbes naturelles• Non-synchronicité des mouvements

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Timing (1/2)

• Masse

• Taille

21

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