Animation = Interpolations

2 Interpolations/Régularisations Functions de bases –X(u) = B 0 (u) X 0 + B 1 (u) X 1 + B 2 (u) X 2 + … X0X0 X1X1 X2X2 X3X3 X(u)

3 Splines Cubiques B i polynôme degrée 3 : B i (u) = b i3 u 3 + b i2 u 2 + b i1 u 1 + b i0 u 0 = (u 3 u 2 u 1 1) (b i3 b i2 b i1 b i0 ) Écriture matricielle : X = (u 3 u 2 u 1 1) [b ij ] (X 0 X 1 X 2 X 3 ) t = [u] M spline [X j ] t

4 Continuité C 0, C 1 et C 2 (1/2) R 2, R 3 X j = (x j y j z j )X j = (t j x j ) Q 1 : (P 0 P 1 P 2 P 3 ) Q 2 : (P 1 P 2 P 3 P 4 )

5 Continuité C 0, C 1 et C 2 (2/2) B-Splines C 0 C 1 C 2 B 0 (1) = 0 B 0 (1) = 0 B 0 (1) = 0 B 1 (1) = B 0 (0) B 1 (1) = B 0 (0) B 1 (1) = B 0 (0) B 2 (1) = B 1 (0) B 2 (1) = B 1 (0) B 2 (1) = B 1 (0) B 3 (1) = B 2 (0) B 3 (1) = B 2 (0) B 3 (1) = B 2 (0) 0 = B 3 (0) 0 = B 3 (0) 0 = B 3 (0) B 0 (0) + B 1 (0) + B 2 (0) + B 3 (0) = 1

6 Bases B-Splines P(u) = (u 3 u 2 u 1 1) 1/6 ( ) (P 0 P 1 P 2 P 3 ) 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

7 Conversions P (u) = [u] M bézier [P bézier ] = [u] M B-spline [P B-spline ] [P bézier ] = (M bézier -1 M B-spline ) [P B-spline ]

8 Bézier & de Casteljau 72 fin 1960] B i,3 (u) = C i 3 u i (1-u) 3-i [Polynômes de Berstein] P(0) = P 0 P(1) = P 3 P(0) = P 1 -P 0 P(1) = P 3 -P 2 Recollement C 1 Construction, et reccursion.

9 Tangentes & Périodicité

10 Courbe de vitesse Contrôle de la vitesse Ease-in, ease-out

Les 12 principes de lAnimation ~1930, Studios Disney The Illusion of Life

12 Squash & Stretch Déformations –Pseudo physique - > -Conservation du volume

13 Anticipation Guide lattention, annonce laction… …prépare un gag mouvement C 2 …

14 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

17 Slow in & Slow out Fluidité des mouvements (Continuité C 2 des déplacements)

18 Secondary actions Donner de la richesse –Hiérarchie/articulations –En retard

19 Arcs Courbes naturelles Non-synchronicité des mouvements

20 Timing(1/2) Masse Taille

21 Timing(2/2) [John Lasseter, Pixar, 1987] 0hit by a tremendous force. 1hit by a brick, rolling pin, frying pan. 2nervous tic, muscle spasm, an uncontrollable twitch. 3dodging a brick, rolling pin, frying pan. 4giving a crisp order, "Get going!" "Move it!" 5friendly, "Over here." "Come on-hurry!" 6sees a good looking girl, or the sports car he has always wanted 7tries to get a better look at something. 8searches for the peanut butter on the kitchen shelf. 9appraises, considering thoughtfully. 10stretches a sore muscle.

22 Exaggeration