Review from Thursday. Computer Animation II. Grid acceleration. Debugging. Computer-Assisted Animation. Final project

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1 Computer Animtion II Orienttion interpoltion Dynmis Some slides ourtesy of Leonrd MMilln nd Jon Popoi Reiew from Thursdy Interpoltion Splines Artiulted odies Forwrd kinemtis Inerse Kinemtis Optimiztion Grdient desent Following the steepest slope Lgrngin multipliers Turn optimiztion with onstrints into no onstrints Leture Fll 2002 Leture 11 Slide Fll 2003 Deugging Deug ll su-prts s you write them Print s muh informtion s possile Use simple test ses Grid elertion Deug ll these steps: Sphere rsteriztion Ry initiliztion Mrhing Ojet insertion Aelertion When stuk, use step-y-step deugging Leture 11 Slide Fll 2003 Leture 11 Slide Fll 2003 Finl projet First rinstorming session on Thursdy Groups of three Proposl due Mondy 10/27 A ouple of pges Gols Progression Appointment with stff Computer-Assisted Animtion Keyfrming utomte the inetweening good ontrol less tedious reting good nimtion still requires onsiderle skill nd tlent Proedurl nimtion desries the motion lgorithmilly express nimtion s funtion of smll numer of prmeteres Exmple: lok with seond, minute nd hour hnds hnds should rotte together express the lok motions in terms of seonds rile the lok is nimted y rying the seonds prmeter Exmple 2: A ouning ll As(sin(ωt+θ 0 *e -kt ACM 1987 Priniples of trditionl nimtion pplied to 3D omputer nimtion Leture 11 Slide Fll 2003 Leture 11 Slide Fll

2 Computer-Assisted Animtion Physilly Bsed Animtion Assign physil properties to ojets (msses, fores, inertil properties Simulte physis y soling equtions Relisti ut diffiult to ontrol Oeriew Interpoltion of rottions, quternions Euler ngles Quternions Motion Cpture Cptures style, sutle nunes nd relism You must osere someone do something ACM 1988 Spetime Constrints Dynmis Prtiles Rigid ody x ( t ( t Deformle ojets Leture 11 Slide Fll 2003 Leture 11 Slide Fll 2003 Interpolting Orienttions in 3-D Rottion mtries Gien rottion mtries M i nd time t i, find M(t suh tht M(t i =M i. z y u n x M = ux x nx uy y ny uz z nz Flwed Solution Interpolte eh entry independently Exmple: M 0 is identity nd M 1 is 90 o round x-xis Interpolte ( 0 1 0, = Is the result rottion mtrix? NO: For exmple, RR T does not equl identity. This interpoltion does not presere rigidity (ngles nd lengths Leture 11 Slide Fll 2003 Leture 11 Slide Fll D rottions How mny degrees of freedom for 3D orienttions? 3 degrees of freedom: e.g. diretion of rottion nd ngle Euler Angles An Euler ngle is rottion out single xis. Any orienttion n e desried omposing three rottion round eh oordinte xis. Roll, pith nd yw Or 3 Euler ngles Leture 11 Slide Fll 2003 Leture 11 Slide Fll

3 Interpolting Euler Angles Giml lok Nturl orienttion representtion: 3 ngles for 3 degrees of freedom Unnturl interpoltion: A rottion of 90 o first round Z nd then round Y = 120 o round (1, 1, 1. But 30 o round Z then Y differs from 40 o round (1, 1, 1. = Giml lok: two or more xis lign resulting in loss of rottion degrees of freedom. Leture 11 Slide Fll 2003 Leture 11 Slide Fll 2003 Demo By Soeit oid from Euler ngles in the rel world Apollo inertil mesurement unit To preent lok, they hd to dd fourth Giml! See lso Leture 11 Slide Fll 2003 Leture 11 Slide Fll 2003 Rep: Euler ngles 3 ngles long 3 xis Poor interpoltion, lok But used in flight simultion, et. euse nturl Oeriew Interpoltion of rottions, quternions Euler ngles Quternions Dynmis Prtiles Rigid ody x ( t ( t Deformle ojets Leture 11 Slide Fll 2003 Leture 11 Slide Fll

4 Solution: Quternion Interpoltion Interpolte orienttion on the unit sphere By nlogy: 1-, 2-, 3-DOF rottions s onstrined points on 1, 2, 3-spheres 1D sphere nd omplex plne Interpolte orienttion in 2D 1 ngle But messy euse modulo 2π Use interpoltion in (omplex 2D plne Orienttion = omplex rgument of the numer θ 1 θ 0 Leture 11 Slide Fll 2003 Leture 11 Slide Fll 2003 Veloity issue & slerp liner interpoltion (lerp in the plne interpoltes the stright line etween the two orienttions nd not their spheril distne. lerp ( q0, q1, t = q( t = q0 ( 1 t + q1t => The interpolted motion does not he uniform eloity: it my speed up too muh: t q t q q q 1 0 keyfrmes lerp slerp Solution: Spheril liner interpoltion (slerp: interpolte long the r lines y dding sine term. ( t 0 1 ( q q (( t ω + q ( tω sin( ω q0 sin 1 1sin slerp q, q, = q( t =, where ω = os Leture 11 Slide Fll ngle orienttion Emed 2-sphere in 3D 2 ngles Messy euse modulo 2π nd pole Use liner interpoltion in 3D spe Orienttion = projetion onto the sphere Sme eloity orretion Leture 11 Slide Fll ngles Use the sme priniple interpolte in higher-dimensionl spe Projet k to unit sphere Proly need the 3-sphere emedded in 4D More omplex, hrder to isulize Use the so-lled Quternions Due to Hmilton (1843; lso Shoemke, Siggrph '85 Quternions Quternions re unit etors on 3-sphere (in 4D Right-hnd rottion of θ rdins out is θ q = {os(θ/2; sin(θ/2} Often noted (s, Wht if we use? Wht is the quternion of Identity rottion? Is there extly one quternion per rottion? Wht is the inerse q -1 of quternion q {,,, d}? Leture 11 Slide Fll 2003 Leture 11 Slide Fll

5 Quternions Quternions re unit etors on 3-sphere (in 4D Right-hnd rottion of θ rdins out is θ q = {os(θ/2; sin(θ/2} Often noted (s, Also rottion of θ round - Wht is the quternion of Identity rottion? qi = {1; 0; 0; 0} Is there extly one quternion per rottion? No, q={os(θ/2; sin(θ/2} is the sme rottion s -q={os((θ+2π/2; sin((θ+2π/2} Antipodl on the quternion sphere Wht is the inerse q -1 of quternion q {,,, d}? {, -, -, -d} Quternion Alger Two generl quternions re multiplied y speil rule: qq = ss, s r + s r + ( 2 1 Snity hek : {os(α/2; sin(α/2} {os(β/2; sin(β/2} Leture 11 Slide Fll 2003 Leture 11 Slide Fll 2003 Quternion Alger Two generl quternions re multiplied y speil rule: qq = ss, s r + s r + ( 2 1 Snity hek : {os(α/2; sin(α/2} {os(β/2; sin(β/2} {os(α/2os(β/2 - sin(α/2. sin(β/2}, os(β/2 sin(α/2 + os(α/2sin(β/2 + } {os(α/2os(β/2 - sin(α/2 sin(β/2, (os(β/2 sin(α/2 + os(α/2 sin(β/2} {os((α+β/2, sin((α+β/2 } Quternion rep 1 (wke up 4D representtion of orienttion q = {os(θ/2; sin(θ/2} Inerse is q -1 =(s, - Multiplition rule qq = ss ( r, s r + s r Consistent with rottion omposition θ How do we pply rottions? How do we interpolte? Leture 11 Slide Fll 2003 Leture 11 Slide Fll 2003 Quternion Alger Two generl quternions re multiplied y speil rule: qq = ss ( r 1 2, s r + s r To rotte 3D point/etor p y q, ompute q {0; p} q -1 p= (x,y,z q={ os(θ/2, 0,0,sin(θ/2 } = {, 0,0,s} q {0,p}= {, 0, 0, s} {0, x, y, z} = {.0- zs, p+0(0,0,s+ (0,0,s p} = {-zs, p + (-sy,sx,0 } q {0,p} q -1 = {-zs, p + (-sy,sx,0 } {, 0,0,-s} = {-zs-(p+(-sy,sx,0.(0,0,-s, -zs(0,0,-s+(p+(-sy, sx,0+ ( p + (-sy,sx,0 x (0,0,-s } = {0, (0,0,zs p+(-sy, sx,0+(-sy, sx, 0+(s 2 x, s 2 y, 0} = {0, ( 2 x-2sy-s 2 x, 2 y+2sx-s 2 y, zs 2+ s 2 } = {0, x os(θ-y sin(θ, x sin(θ+y os(θ, z } Leture 11 Slide Fll 2003 Quternion Alger Two generl quternions re multiplied y speil rule: qq = ss, s r + s r + To rotte 3D point/etor p y q, ompute q {0; p} q -1 Quternions re ssoitie: (q1 q2 q3 = q1 (q2 q3 But not ommuttie: q1 q2 q2 q1 ( 2 1 Leture 11 Slide Fll

6 Quternion Interpoltion (eloity The only prolem with liner interpoltion (lerp of quternions is tht it interpoltes the stright line (the sent etween the two quternions nd not their spheril distne. As result, the interpolted motion does not he smooth eloity: it my speed up too muh in some setions: t q t q q q 1 0 keyfrmes lerp slerp Spheril liner interpoltion (slerp remoes this prolem y interpolting long the r lines insted of the sent lines. q0 sin( ( 1 t ω + q1sin( tω slerp ( q0, q1, t = q( t =, sin( ω 1 where ω = os q q 0 1 Leture 11 Slide Fll 2003 Quternion Interpoltion Higher-order interpoltions: must sty on sphere See Shoemke pper for: Mtrix equilent of omposition Detils of higher-order interpoltion More of underlying theory Prolems No notion of fored diretion (e.g. up for mer No notion of multiple rottions, needs more key points Leture 11 Slide Fll 2003 Demo From Rmmoorthi nd Brr Siggrph 97 Demo From Rmmoorthi nd Brr Siggrph 97 Leture 11 Slide Fll 2003 Leture 11 Slide Fll 2003 Quternions Cn lso e defined like omplex numers +i+j+dk Multiplition rules i 2 =j 2 =k 2 =-1 ij=k=-ji jk=i=-kj ki=j=-ik Fun:Juli Sets in Quternion spe Mndelrot set: Z n+1 =Z n2 +Z 0 Juli set Z n+1 =Z n2 +C Do the sme with Quternions! Rendered y Skl (Psl Mssimino See lso Leture 11 Slide Fll 2003 Leture 11 Slide Fll

7 Fun:Juli Sets in Quternion spe Juli set Z n+1 =Z n2 +C Do the sme with Quternions! Rendered y Skl (Psl Mssimino This is 4D, so we need the time dimension s well Rep: quternions 3 ngles represented in 4D q = {os(θ/2; sin(θ/2} Weird multiplition rules qq = ss, s r + s r + ( 2 1 θ Good interpoltion using slerp q( t slerp Leture 11 Slide Fll 2003 Leture 11 Slide Fll 2003 Oeriew Interpoltion of rottions, quternions Brek: moie time Pixr For the Bird Euler ngles Quternions Dynmis Prtiles Rigid ody x ( t ( t Deformle ojets Leture 11 Slide Fll 2003 Leture 11 Slide Fll