1.2 Goals for Animation Control

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1 A Direc Manipulaion Inerface for 3D Compuer Animaion Sco Sona Snibbe y Brown Universiy Deparmen of Compuer Science Providence, RI 02912, USA Absrac We presen a new se of inerface echniques for visualizing and ediing animaion direcly in a single hree-dimensional scene. Moion is edied using direc-manipulaion ools which saisfy high-level goals such as reach his poin a his ime or go faser a his momen. These ools can be applied over an arbirary emporal range and mainain arbirary degrees of spaial and emporal coninuiy. We separae spaial and emporal conrol of posiion by using wo curves for each animaed objec: he moion pah which describes he 3D spaial pah along which an objec ravels, and he moion graph, a funcion describing he disance raveled along his curve over ime. Our direc-manipulaion ools are implemened using displacemen funcions, a sraighforward and scalable echnique for saisfying moion consrains by composiion of he displacemenfuncion wih he moion graph or moion pah. This paper will focus on applying displacemen funcions o posiional change. However, he echniques presened are applicable o he animaion of orienaion, color, or any oher aribue ha varies over ime. CR Descripors: I.3.7 [Compuer Graphics]: ThreeDimensionalGraphicsandRealism; I.3.6 [CompuerGraphics]: Mehodology and Techniques; I.3.5 [Compuer Graphics]: Compuaional Geomery and Objec Modeling. Addiional Keywords and Phrases: Animaion, Ineracion Techniques, Splines. 1 Inroducion Kinemaic compuer animaion is a painsaking process requiring hand adjusmen of hundreds of key posiions for every objec in an animaed scene. Mos animaion sysems provide precise conrol of moion using wo-dimensional graphs of individual parameers (e.g. x ranslaion vs. ime). Animaors mus menally inegrae his 2D informaion wih saic 3D views and occasional moion previews o mainain a clear sense of he moion which hey are creaing. This research is an aemp o develop ineracive echniques for visualizing and modifying moion direcly in a wo- or hree-dimensional scene. The principles of direc manipulaion are used o achieve he goal of fluid and naural ineracion. Our soluion uses exising keyframe and parameric echniques in combinaion wih displacemen funcions inspired by digial signal processing for real-ime direc manipulaion of spaial and emporal changes. The discussion in his paper will focus on animaing he posiion of an objec. However, he soluions presened here are adapable o he animaion of orienaion, color, or any oher aribue ha varies over ime. Expanding his research o hese areas is discussed in secion Problems in Exising Animaion Sysems Several problems found in a majoriy of commercial and research animaion sysems serve as he moivaion for his research. No all of hese problems are presen in all sysems, bu hese are curren rends in a large class of exising sysems. Animaors can compleely visualize and edi moion only in separae 2D graphs. The only means o edi an objec s ime-varying properies and visualize he value of hese properies over ime is hrough 2D graph ediors. The 3D sceneview is usedprimarily for viewing and ediing an objec a a single poin in ime. y The auhor is currenly a Adobe Sysems Incorporaed,411 Firs Avenue Souh, Seale, WA USA.

2 Ediing of moion curves is limied o single channels of moion. Moion curves are normally limied o represening a one-dimensional parameer vs. ime (e.g. x ranslaion vs. ime, y roaion vs. ime, red color componen vs. ime). Animaors mus menally inegrae all of hese channels o visualize he animaion which hey are creaing. The naural parameerizaion of splines does no advance uniformly wih respec o disance. Many sysems allow he animaor o specify he pah of an objec hrough space wih a wo- or hree-dimensional spline curve. Moion along his curve is hen described by a single funcion of u vs. ime, where u is he parameer of he spline curve. However, equal seps in u resul in unequal disances raveled along he curve. In hese sysems, a graph ha appears o indicae consan velociy will acually resul in a velociy ha varies based on he shape of he curve and he spacing of is conrol poins. The animaor is forced o cancel ou he iming induced by he spline before creaing he desired moion. The shape of a moion curve is alered o achieve iming goals. Some sysems aler he shape of a moion pah when users edi he iming of an animaion. This problem is also a resul of ying moion o he u-parameer of a spline. The acual shape of he curve mus be changed in order o aler he disance ravelled over equal ime seps. Direc manipulaion of he animaed objec is allowed only a conrol poins. When a spline curve is used as he underlying represenaion of spaial change, mos sysems only allow he animaor o change he objec a he spline conrol poins [2][14]. If he animaor wans o aler a posiion beween conrol poins, she mus eiher work indirecly, alering surrounding conrol poins and angens, or she mus add a new conrol poin. Adding conrol poins can inroduce undesired complexiy o he animaion and reduces he range over which changes have effec. Animaions wih densely spaced keyframes are difficul o modify. Mos producion qualiy animaions end up being specified by very densely spaced keyframes (10-15 keyframes/second is normal). If an animaor decides ha par of he moion should be changed, she mus individually change a wide range of conrol poins surrounding he specific change in order o blend i wih he surrounding moion here are no ools for modifying muliple keyframes simulaneously. Many animaors find i faser o re-do he animaion from scrach in his siuaion. 1.2 Goals for Animaion Conrol The following se of goals is an aemp o describe an animaion sysem which addresses he above se of problems: Creae an sysem which allows visualizaion and ediing of emporal and spaial informaion in a single 3D view. Express moion goals in erms of disance or velociy vs. ime. Mainain emporal and spaial coninuiy while ediing animaions. Allow an arbirary range over which ediing ools are applied. Develop moion conrol echniques which are naually exensible o orienaion, scale and any oher animaed parameers. Provide real-ime performance for complex scenes. As an inerface o he above goals, we require direc-manipulaion ools which correspond o he high-level goals of an animaor: Temporal ranslaion Saisfies he goal Reach his poin a his ime while mainaining he shape of he moion pah, bu changing he speed a which he objec ravels along he given pah. Spaial ranslaion Saisfies he above goal by modifying he spaial curve while mainaining eiher he duraion or velociy of he given segmen.

3 Temporal scale Changes he duraion of segmen of animaion. Saisfies he goal Make his segmen of animaion longer, shorer, or a specific duraion. Velociy modificaion Saisfies he goals Go faser, Go slower, or Reach a specific velociy a a given poin, while mainaining he shape of he spaial curve and he duraion of he emporal segmen. 2 Prior Work There is a large body of previous published research on kinemaic animaion and moion conrol. This secion only addresses hose prior models which incorporae splines o conrol spaial inerpolaion and moion conrol. Kochanek and Barels invened a spline which inerpolaes a se of spaial conrol poins and allows emporal conrol via high-level parameers a each conrol poin [6]. Their model allows visualizaion and ediing moion wihin a single view. Temporal adjusmens are made using hree parameers (ension, bias and coninuiy) whose values are se a each conrol poin. These parameers aler boh he shape of he curve and he parameric spacing around each conrol poin. The effecs of hese parameers is someimes inuiive, changing he shape and iming in a manner consisen wih many nauralisic moions. However, animaors ofen end up in a ug-of-war in which hey achieve he desired iming a he expense of he moion curve s shape, or vice-versa. Addiionally, modificaions o he shape of he curve only has an effec on he wo adjacen spline segmens Affecing a larger or smaller range involves moving addiional poins or adding more conrol poins o he curve. This mehod of animaion conrol is sill exremely popular in compuer animaion and is presen in many of oday s commercial sysems [2][14]. Sekeee and Badler [13] published a mehod for separaing emporal and spaial conrols in parameric animaion. Their work uses B-splines for graphing objec aribues. One se of curves represens posiion vs. keyframe for individual objec aribues (e.g. x, y, z). A second curve of keyframe vs. ime is used o separaely modify he iming of objec aribues. The composiion of he wo curves resuls in he final value for a given aribue. A drawback of heir sysem is ha he spaial aribues are separaed ino muliple channels and no direc manipulaion of emporal informaion is allowed. Conrol of moion involves ediing wo separae graph views (he objec aribue graph and he iming curve), hen viewing he composie spaial pah o evaluae changes. Furhermore, he effecive range of a change in any graph is limied by he number and spacing of conrol poins in he given curve and canno be arbirarily conrolled by he animaor. The Menv sysem developed a Pixar uses graphs of single parameers vs. ime for objec aribues [9]. Key values along hese curves are conneced by spline segmens. The shape and ype of hese splines can be modified in a piecewise manner, choosing he bes shape and ype for each segmen of ime. The sysem has he drawback ha each animaed parameer (e.g. x posiion, z roaion, y scale) mus be edied individually, and iming curves can only be modified a heir conrol poins by using heir angens. The range of operaions is also limied o a single spline segmen. However, he sysem was developed in cooperaion wih radiional animaors who find i a convenien and precise way o specify and visualize moion. The Pixar model is implemened in many commercial animaion sysems oday [12]. The Inkwell sysem developed a he Apple Advanced Technology Group includes a se of ools for digially filering densely spaced conrol poins in iming curves [7]. The filers, applied over an arbirary range of moion, provide high level conrol of he overall characerof an animaion. Animaors can modify he gain, decayor oscillaion of heir digially sampled moion curve by filering wih an infinie impulse response filer. To he animaor, hese parameers are undersood inuiively as magniude, wiggle and lag. Their sysem also has a cosine blending funcion o blend changes from a single frame wih an arbirary range surrounding he modified poin. The drawbacks of heir sysem are he lack of a direc manipulaion inerface o he filering and blending, he separaion of moion ino one-dimensional channels and he inabiliy o saisfy precise goals using he high level filers. A large number of researchers are pursuing echniques for he direc manipulaion of spline curves and surfaces over arbirary ranges. Leas-squares echniques [4], consrain-based echniques [1][3][17] and oriened paricle sysems [15] are he major areas currenly being explored. These echniques are all eiher highly compue-inensive, sensiive o he number and spacing of conrol poins, or do no allow precise conrol of he effecive range of consrains. Oher specific drawbacks of applying consrains o saisfy displacemen funcions are discussed in secion 7. 3 Separaing Spaial and Temporal Conrols In order o saisfy our independen emporal and spaial goals, we represen an objec s changing posiion over ime by wo curves. The firs curve, Q,ishemoion pah describing a pah hrough space along which an objec ravels. Q is represened as a parameric funcion of u: Q(u) =hx(u);y(u);z(u)i

4 The second curve, S, ishemoion graph, a funcion of disance vs. ime which maps from a ime value o a disance raveled along he moion graph, s: S() =s We wan o use he graph S o deermine he parameric posiion along Q a a given ime. Our direc manipulaion ools can hen be phrased as geomeric consrains on eiher S or Q. Since he curve Q is parameerized by is naural parameer, u, and he moion graph gives us disance values in erms of arc-lengh (s), we mus creae a mapping from s o u for he curve Q. This problem has no analyic soluion, bu can be solved numerically. A complee discussion of his problem and several approximae soluions can be found in [5] and [16]. In his paper we will assume for simpliciy ha we have a funcion A which maps from u o s: A(u) =s, and ha he u value for a paricular arc-disance s can be deermined by A 1 (s). Using his equaion, we can now express he posiion of an objec as a funcion of ime (figure 1): P () =Q(A 1 (S())) In pracice, a velociy curve V () is more commonly used as he moion graph. This enables animaors o more easily visualize suble changes in velociy. The curve V () can be inegraed wihou difficuly o deermine S(). For ease of manipulaion, he graph S is ofen represened as a wo-dimensional parameric curve, raher han a implici funcion of. Animaors find his represenaion easier o manipulae and capable of finer conrol of moion wih fewer conrol poins [9]. The deails of inerpreing a wo-dimensional spline curve as an implici funcion, and assuring ha i remains one-o-one are discussed in [11]. For simpliciy we will refer o S as a one-dimensional implici funcion hroughou he res of his paper. Q arc-lengh S s Q( u) u = A 1 ( s) ime Figure 1: Moion conrol wih separae space and ime curves. 4 Tools for Moion Conrol Our direc manipulaion ools require ha consrains such as reach his poin a his ime or increase velociy a his insan are saisfied. We can express hese consrains as mahemaically precise goals for he shape and derivaive of he moion graph S and he moion pah Q. In addiion o he specific consrain a a single poin in ime, we also wish o blend he change smoohly ino an arbirary range surrounding he curren poin in ime. Previous approaches o his problem proceed by firs saisfying he hard consrains, hen relaxing (in he case of consrains) or filering (in he case of digial signal processing) o blend he change ino surrounding areas of he curve. We choose a simpler approach ha preserves he fine deails ofhe curve andallows preciseconrolover he range. We inroduceadisplacemen funcion F which represens a pre-filered goal which, when composed wih a curve, resuls in he consrains being saisfied and smoohly blended over he specified inerval. Composing he curves F and can be a non-rivial problem when he funcion F is expressed in erms oher han he naural parameerizaion of he curve. Secion 7 discusses several echniques for he saisfacion of hese goals in such cases. Given a displacemen funcion F () which represens he desired change in he curve () over he ime inerval [ ; ], we can express he resuling curve () as he addiion of he wo funcions: () =()+F ;8: (1) The above funcion assumesha F is defined over he domain [0; 1] so ha F can be easily applied o differen ime inervals. No maer which mehod we use o saisfy his goal, he resuling curve will preserve he coninuiy of he original curve if he displacemen funcion F is also coninuous o he desired degree. The following sub-secions show how we consruc displacemen funcions o saisfy a spaial or emporal goal. In he illusraions of hese echniques, we show he graph S along wih he 3D scene. However, keep in mind ha he animaor does no need o see or edi he graph in order o visualize and modify he animaion. In he nex secion, we show how hese ools are applied using he principles of direc-manipulaion in a 3D view.

5 Q β α s S α β ime Figure 2: Temporal ranslaion. Wihin he specified inerval [ ; ], he animaor drags he cone objec along he lengh of he moion pah Q. The iming around he objec changeso mainain he desired posiion a he curren ime (middle, righ). The moion graph S, below each image, shows he applicaion of he posiional displacemen funcion F a he curren ime (indicaed by he black riangle). 4.1 Temporal Translaion The emporal ranslaion echnique for moion conrol allows he animaor o change he posiion of an objec a ime o a new poin along he moion pah Q. This posiional goal is saisfied by ranslaing S() up or down a he poin corresponding o he curren ime, while mainaining he coninuiy of S over he range being modified (figure 2). To implemen his operaion, we consruc a displacemenfuncion F which will mainain coninuiy over he specified range and give a maximum displacemen value of 1 a ime. These goals can be expressed mahemaically as: F = 1 F (0) = 0 F (1) = 0 F 0 (0) = 0 F 0 (1) = 0 If higher degrees of coninuiy are desired, hen he higher order derivaives a he endpoins mus also equal zero. The funcion is applied over he specified range of he moion graph, scaled by he desired displacemen, s: S() =S()+ sf ; 8 : We can represen he displacemen funcion F as a wo segmen Bézier curve. By adjusing he angens of he curve F, he displacemen funcion s shape can be changed o achieve differen qualiies of ineracion. The parameers k, b 1,and b 2 conrol he widh and amoun of blending a he wo endpoins of he funcion. Each parameer can vary from 0 o 1, represening he minimum and maximum values he angens can have for a given value of (figure 3). 4.2 Spaial Translaion The spaial ranslaion mehod achieves he effec of direcly manipulaing he posiion of an objec by modifying he underlying moion pah. Given a change in posiion for he objec, q, a displacemen funcion is consruced which will modify Q so ha is posiion a ime passes hrough Q(u) + q,whereu is he parameer value along Q a ime. The displacemen funcion used is he same as ha used for emporal ranslaion, bu applied o he moion pah raher han he moion graph. We wish o have he displacemen funcion fall off wih respec o ime, raher han wih he naural parameerizaion of he curve. In his case, he applicaion of he funcion is slighly more complicaed. The displacemen

6 1 F() k b 1 b 2 ( 00, ) α β α 1 Figure 3: Consrucion of posiional displacemen funcion Q β α s S α β ime Figure 4: Spaial ranslaion. By applying he poin displacemen funcion along he lengh of he curve Q, we can direcly manipulae he animaed objec. A lef is he original spaial pah and graph. In he nex wo images, we have dragged he objec o a new posiion by applying he poin displacemen funcion o he mouse dela. Applying a scaling filer allows us o eiher mainain he duraion of he segmen (middle) or mainain he velociy of he segmen (righ). funcion F () is consruced as a funcion of disance vs. ime. In order o apply his funcion o he moion pah Q over he ime inerval [ ; ], F mus be convered o a funcion of disance vs. u (he naural parameer o he curve Q) by ransforming via he arc-lengh funcion A,andS(): S 1 (A(u)) F (u) =F This is a ransformaion from ime o disance o parameric space. In pracice, his funcion canno be efficienly implemened, since here are no analyic expressions for eiher S or A. A fas approximaion of F can be deermined by sampling F a even inervals of, 2 [ 0 ::: n] where n is seleced based on he number of frames in he inerval. For each value of we calculae he parameer value u and consruc a one-dimensional spline F from he se of poins F (u0) :::F(u n). Addiional conrol poins may be required a he ends of he spline so ha he curve s derivaives equal 0 a he endpoins. The displacemen funcion can hen be applied in a manner similar o he emporal ranslaion mehod: u u Q(u) =Q(u)+ qk F ;8u: u uu u u Where u and u are he parameer values along Q corresponding o imes and. The consan k is a scaling consan which ensures ha F (ui) =1, so ha he posiion of he curve Q a u i will exacly equal Q(u i)+ q: k= 1 F(u i)

7 Since he addiion of F and Q changeshe oal arc-lengh of he curve Q, he graph S mus be scaled o mainain he characer of he animaion. If we wish o mainain he duraion of he edied segmen along Q, hen he graph mus be adjused so ha he disance raveled wihin he inerval [ ; ] of S is modified o equal he new disance from o along Q. This involves a scaling along he verical axis of S. If insead, we wan o mainain velociy, hen he graph S mus be scaled horizonally in, changing he duraion of he segmen wihin he inerval [ ; ]. These scaling operaions are described in he following secion. Figure 4 shows he resuls of hese wo operaions. 4.3 Scale Operaions We presen hree scale operaions in his secion. The firs wo mehods (ime/arc-lengh scale and arc-lenghscale) are only used in conjuncion wih he spaial ranslaion mehod o mainain velociy or duraion. The hird mehod (ime scale) is applied direcly by he animaor o change he lengh of an animaion segmen Time/Arc-lengh Scale The ime/arc-lengh scale funcion is applied in combinaion wih he spaial ranslaion mehod. Is purpose is o mainain he velociy along a segmen of he moion pah given a change in arc-lengh s by varying he duraion of he segmen. In simpler erms, an animaor drags he animaed objec hrough space and wans he velociy a he given ime o remain he same, and he surrounding moion o mainain he same characer, while allowing he duraion of he enire segmen o vary. We assume ha he original segmen lies wihin he range [ ; ]and ha we wish o mainain coninuiy a he boundaries of his region. We need o change he duraion by an amoun proporional o he change in arc-lengh. Since he raios of s o oal arc-lengh (s end) and o oal duraion ( end) are equal, we can easily compue and hen calculae he new version of S, S by uniformly scaling he inerior segmen: S() = 8 < S() : S( )+ S + 0 S( ) b S( )+ b b end Where = 1 + b = + b end = end Arc-lengh Scale An animaor may wish o scale he arc-disance raveled over a specified segmen while mainaining he shape of he moion pah and he oal lengh of he animaion. This siuaion occurs while direcly-manipulaing he objec s spaial posiion. Since he arc-disance ravelled changes as he animaor drags he objec, and he animaor wishes o mainain he duraion of he segmen, he velociy of moion along he moion pah mus be changed. In his case, mainaining coninuiy becomes a more difficul problem. We mus scale he seleced segmen [ ; ] in s and simulaneously compress he surrounding segmens in s, hen displace using a funcion F which resores coninuiy over he seleced region. The new curve can be deermined by he following equaion: S() = 8 >< >: S() S( )+(S() S( )+ S() S( ) S( )) + F 0 end The displacemen funcion F mus be coninuous o he desired degree and have he following properies: F (0) = 0 F (1) = 0 F 0 (0) = S0 () S 0 ( ) F 0 (1) = S0 ( b ) S 0 ( ) One choice for he represenaion of his funcion, consruced from four Bézier segmens, is shown in figure 5. We provide he high-level parameers k and m which correspond inuiively o he range of influence and he magniude of change for he displacemen funcion.

8 F() m k k m 0 1 Figure 5: Consrucion of ime scale displacemen funcion Time Scale The mos common scale operaion is simply o change he duraion of a segmen wihou modifying he moion pah, so ha he oal arc-lengh remains fixed. The chosen segmen is scaled in ime by and displaced in s by a funcion F which resores coninuiy over he region [ ; ]: S()= 8 >< >: S() S + +F b 0 b S( ) b b end Where he funcion F is idenical o he ime scale displacemen funcion, applied over he range [ ; ]. 4.4 Velociy Conrol Q β α s S α β ime Figure 6: Velociy Conrol. An objec moving a a consan velociy is shown a lef. By applying he velociy displacemen funcion o he ime graph (middle, righ), we can change he velociy a ime wihou changing he duraion of he segmen or he posiion of he objec a ime. The velociy conrol mehod modifies he moion graph o achieve a specific velociy goal a a given ime. A displacemen funcion is applied o he curve S a he poin in ime, mainaining he shape of he underlying spaial curve and he duraion of he emporal segmen. An illusraion of he velociy funcion in use can be seen in figure 6. This funcion is applied only

9 o he graph S, mainaining he shape of Q. The displacemen funcion F mus have he following properies: F = v F (0) = 0 F (1) = 0 F 0 (0) = 0 F 0 (1) = 0 Where v is he desired change in velociy of he graph S a poin. We show our own consrucion of he velociy conrol displacemen funcion from Bézier curves in figure 7. F() k b 2 m b 1 0 α Figure 7: Consrucion of velociy displacemen funcion β α k 5 Applying Moion Tools Using Direc Manipulaion To achieve direc manipulaion we mus mainain a one-o-one correspondence beween a user s mouse moions and he changes made o objecs in he visible scene [8]. Suprisingly, many curren modeling and animaion sysems sill do no mainain such correspondence for heir ediing ools. In pracice, he direc-manipulaion process amouns o a bi of rigonomery which maps from camera-space mouse-delas o he 3D space of he objecs in he scene. In our case, we mus deermine a specific change in posiion, ime or velociy from a user s mouse moion, which we hen apply using he previous algorihms. 5.1 Direc Manipulaion for Spaial Translaion Direc manipulaion for he spaial ranslaion ool is sraighforward. We use a mehod which allows manipulaion of he objec in a plane parallel o he film plane of he viewing camera. Suppose he iniial posiion of an objec is a q (figure 8). We projec subsequen mouse delas ono a plane parallel o he film-plane which passes hrough he poin q, by calculaing he inersecion of he ray hrough he new screen-space posiion wih he film-plane. This gives us he new poin in space q 0. By subracing q from q 0, we can deermine he vecor, q, which we use in he applicaion of he displacemen funcion. Oher mehods for direc manipulaion are possible, which are only briefly described here. The firs projecs he mouse delas ono he hree axes corresponding o he objec-space basis of he objec being manipulaed. This allows manipulaion in hree spaial dimensions, in conras o he parallel-plane mehod. However, i is very sensiive o he camera view and orienaion of he objec, someimes making i impossible o move along a paricular axis, or vacillaing beween choices of axis. A hird mehod works similarly, bu projecs ono he world-space basis. 5.2 Direc Manipulaion for Temporal Translaion For he emporal ranslaion ool, we mus deermine s, he relaive change in arc-disance for he objec a ime. We assume ha a user has seleced he objec o manipulae, and wishes o drag he objec along he moion pah by applying he emporal ranslaion funcion o he moion graph. We know he parameric posiion, u a which he objec iniially lies along Q. As he user drags in camera-space, we compue he closes poin on he pah Q o he ray formed by exending he

10 screen/film plane q' plane parallel o screen inersecing objec posiion q mouse dela q Figure 8: Mapping mouse posiion o objec posiion eyepoin hrough he poin on he film plane (his is he same ray we used o deermine q 0 in he previous echnique). Using he parameer value u new, corresponding o he new poin on he pah, we can now calculae he change in arc-disance wih he arc-lengh funcion A: s = A(u new) A(u) 5.3 Direc Manipulaion for Velociy Conrol For our direc manipulaion inerface o velociy conrol, we would like he ick-marks surrounding he objec posiion o rack he mouse posiion in he same way ha he objec racks he mouse posiion during emporal ranslaion. We can accomplish his by using he value s, as compued in he mehod for emporal ranslaion. We assume ha he user clicks on he objec, viewed a ime i and drags in eiher direcion along he curve o increase or decrease velociy. The ick a he nex ime sep i+1 should be moved by s. We can deermine v, he discree change in velociy from his informaion (figure 9): s v = i+1 i s Q s s = S( i+ 1 ) s = S( i ) i i+ 1 Figure 9: Mouse racking for direc manipulaion of velociy 5.4 Specificaion of Range To specify he range [ ; ] over which he displacemen funcions are applied, we allow he animaor o place wo bars along he lengh of he curve Q which indicae he sar and end of he modified region. To manipulae hese bars, we use a mehod similar o he direc manipulaion echnique for emporal ranslaion. As he user drags on he bar, we deermine he closes poin on he curve Q o he mouse posiion. This new poin is hen chosen as he new posiion for he bar, and he orienaion of he bar is deermined by he Frene frame of Q a ha poin. We can hen calculae he ime-value a he endpoins o specify he range of he moion graph S o be modified. To indicae he sar and end of he range, we color he wo bars green and red, respecively. This inerface can become edious when making muliple modificaions o differen objecs along differen curves. A simpler, bu less flexible inerface for range specificaion migh allow he animaor o choose a fixed range, which is auomaically calculaed whenever he animaor changes he curren ime or objec. For example, he animaor migh decide ha she always wans a range of five frames abou he manipulaed emporal poin o be modified.

11 6 Visualizing Temporal Change We employ several echniques for visualizing emporal change in our sysem. The simples echnique involves drawing a poin or line a equal inervals of ime along he moion pah and is found in many commercial sysems. Poins are someimes difficul o disinguish in a complicaed scene. We chose o use shor lines or ick-marks, which provide sronger visual cues, bu presen a problem in hree dimensions. If he lines are drawn a a fixed orienaion relaive o he curve, hey are difficul o see from cerain viewing angles. A simple soluion is o always draw he lines perpendicular o he film plane of he camera. Oher soluions involve drawing hree-dimensional objecs a each poin, such as a vecor, a disk or a 3D axis. If we wan o visualize changing orienaion along he curve, an axis is paricularly useful. As a more accurae visualizaion, we can draw copies of he animaed objec a equal inervals of ime along he curve. As ime recedes in eiher direcion, he ransparency of he copies increases. This allows he mos complee visualizaion by he animaor, as all he informaion is visually presen (figure 10). The animaor can easily edi he objec a differen poins in ime, by simply clicking on he version of he objec o modify. This represenaion can become visually cluered and slow for complicaed objecs. One soluion o his problem is o draw he animaed objec a wider inervals of ime, for example every 1/4 second. We can also use a simpler version of he animaed objec for he copies, even resoring o bounding boxes. Figure 10: Ghosing used o visualize change over ime 7 Implemenaion of Displacemen Funcions A popular and naural approach o saisfying geomeric goals for spline curves is consrained opimizaion. However, we encounered several major problems when we aemped o use his mehod o implemen our displacemen funcions. The firs drawback is ha we canno always represen he consrain as an analyic funcion, since he consrains are phrased in erms of arc-lengh or ime raher han he naural parameerizaion of he splines Q and S. We insead have o approximae our consrains wih a series of poin goals. Our second problem is he range over which he consrains are applied. The boundaries of his region are limied o he conrol poins along he curve. Finally, he ime o arrive a a soluion using consrained opimizaion is oo grea for ineracive manipulaion and he final soluion is only approximae. For a more deailed discussion of hese problems see [11]. We insead use wo more predicable and mahemaically less complex mehods. The firs mehod works when he spline o be displaced is represened as a digially sampled curve wih N samples wihin he inerval [ ; +N] such ha +N =. We can sample F a he same frequency and rewrie equaion 1 as S() = NX i=0 S( +i)+f( i) This is he mehod we currenly use for modifying he moion graph S. The drawback of his represenaion is ha he high-level conrols afforded by splines are los when we discreize he curve. This drawback can be remedied by promoing he discree represenaion back o a spline represenaion. This can be achieved using wo differen mehods. The firs simply involves fiing a spline hrough every discree poin in he graph, bu his is likely o inroduce redundan conrol poins along many secions of he curve. The second mehod involves using a curve fiing algorihm which finds he bes spline curve inerpolaing he se of poins wihin a given olerance [10]. Finally, we can apply his echnique direcly o

12 he conrol poins of an inerpolaing spline o achieve an approximae soluion o our goal, which is he soluion we choose when we modify Q. The second mehod of applying he displacemen funcion does no involve any discreizaion of he original curve. This mehod is only applicable when boh he original curve S and he displacemen funcion F are represened by Bézier curves. Two properies of hese curves allow us o direcly add he conrol poins of he splines o produce he sum of he wo funcions. These properies are: 1. The sum of wo Bézier curves, Q 1 and Q 2, wih equal number of conrol poins is exacly equal o he curve obained by adding heir conrol poins. 2. Subdivision of Bézier curves can be accomplished wihou affecing he shape or coninuiy of he original curve. Using hese properies, and assuming ha S and F boh begin and end on inerpolaed poins, we can simply subdivide F so ha i has he same number and spacing of conrol poins as he region of S being displaced, hen add he conrol poins of he wo splines. 8 Resuls Our research presens a complee se of ools for visualizing and ediing moion in a hree-dimensional scene. Every userinerface acion has a disinc and reversible visual reacion direcly proporional o he user s movemens. The animaor has precise conrol of he range over which an operaion can be blended ino he surrounding moion. This allows animaors o refine moion a a high level, raher han consanly resoring o individual frame-by-frame adjusmen. By re-parameerizing he moion curve by arc-lengh, animaors can hink and see in he naural erms of disance vs. ime. We believe our se of direc manipulaion ools is simple o learn and easy use, alhough we require furhur user sudies o deermine his wih more confidence. This research has applicaions ouside of deskop animaion sysems. In he emerging world of virual realiy, he user does no have access o a keyboard or windowing sysem which are currenly essenial componens of oday s muliple-view animaion sysems. Using our echniques, a single sereo view is sufficien for visualizing and ediing moion a he same ime. Direc manipulaion is he mos naural means of ineracing in VR and our echniques are easily exended o accommodae a freely moving hree-dimensional poin of manipulaion, raher han a wo-dimensional cursor. Moion capure and performance animaion are seadily growing areas of compuer animaion. Our ediing echniques are especially well suied o modifying sampled moion of his ype, since he moion daa is naurally sampled a he frame rae of he animaion and is difficul o edi. We have experimened briefly wih moion capure using our sysem. A segmen of animaion is capured, hen convered o our space/ime curve represenaion. Obaining a spaial curve is accomplished by fiing a spline hrough he sampled spaial poins o creae he spaial pah. The emporal curve is creaed by forward differencing he sampled poins o deermine disance raveled vs. ime. The moion can hen be edied using our ools. Several problems are currenly inheren o our sysem. The arc-lengh evaluaion process is a major compuaional boleneck. If we approximae arc-lengh oo coarsely in order o speed ineracion, hen he resuls may no precisely mach he final animaion. Choosing he correc displacemen funcions is currenly more of an ar han a science. We would like o provide simpler parameerizaion o he user or auomaically deermine he opimal displacemen funcion, if here is such a hing. Our bigges problem is he spaial direc manipulaion of he animaed objec. The applicaion of a displacemen funcion o he moion curve only has he desired effec when here are sufficien conrol poins wihin he edied region. If he poin of direc manipulaion is far from any conrol poin, he curve can behave uninuiively. If a conrol poin is added a he poin of direc manipulaion, his problem disappears; however we hen face wo oher problems adding conrol poins changes he shape of some spline ypes, and he number of conrol poins can quickly become unmanageable. Large changes may also disor he original moion curve in a manner uninended by he animaor, desroying he small deails of moion. 9 Fuure Work We would like firs and foremos o improve he represenaion of he moion pah and moion graph. For he moion pah, we would like a spline parameerized direcly by arc-lengh, or a faser and more robus mehod for approximaing arc-lengh. We would like o find a spline ype ha allows he applicaion of displacemen funcions o be expressed analyically so ha he resuls of displacemen are precise and coninuous. As possible soluions o hese problems we are considering curves represened as NURBS and oher schemes using Bézier curves which auomaically add and remove conrol poins. A more radical soluion migh involve represening he curves as a one-dimensionaloriened paricle sysem wih coninuiy properies auomaically mainained a each paricle along he curve. Geomeric modeling using hese echniques has already been explored by Richard Szeliski [15]. A wavele represenaion for he curves is also possible, alhough preliminary resuls of wavele spline manipulaion show many of he same problems of oher consrain-based echniques [3]. The mos serious drawback is he difficuly of precisely specifying he range over which a modificaion has effec.

13 Alhough our echniques are easily applicable o oher one-dimensional parameers such as color or single channels of scale and roaion, here are currenly no mehods o visualize and edi roaion or scale over ime hrough direc manipulaion. For roaion we are hopeful ha a quaernion represenaion migh be a fruiful visualizaion scheme. The roaion of he objec could be represened as a pah along he surface of a sphere surrounding he objec, and ick-marks adjused along his pah. Scale migh be visualized using addiional pahs showing he exen of he objec, or by using handles exending from he poins along he moion curve. Insead of drawing enire ghos copies of he animaed objec o visualize emporal change, we are considering mehods o visualize he leadingor railing edge of he objec as i fades emporally. This echnique is inspired by radiional animaors echniques for moion blur. We do no ye have a mehod for his ype of visualizaion, bu are looking a boh image processing and polygonal approximaions for possible soluions. Acknowledgmens I would like o graefully acknowledge he suppor of Professor Andy van Dam, who allowed me he opporuniy and ime o pursue his research. Professor John Hughes provided many simulaing discussions and helped me develop he underlying mahemaical echniques. Cindy Grimm, Brook Conner and many oher members of he Brown Graphics Group provided excellen feedback on several drafs of his paper. This research was sponsored in par by NASA, he NSF/ARPA Science and Technology Cener for Compuer Graphics and Scienific Visualizaion, Sun, Auodesk, Taco Inc., NCR, HP, IBM, DEC, Apple, Microsof and Adobe Sysems. REFERENCES [1] Richard H. Barels and John C. Beay. A echnique for he direc manipulaion of spline curves. In Proceedings of Graphics Inerface 89, pages 33 39, June [2] Elecric Image version 2.0. Pasadena, CA, [3] Adam Finkelsein and David Salesin. Muliresoluion curves. In Compuer Graphics (SIGGRAPH 94 Proceedings), pages , July [4] Barry Fowler and Richard Barels. Consrain-based curve manipulaion. IEEE Compuer Graphics and Applicaions, 13(5):43 49, Sepember [5] Brian Guener and Richard Paren. Compuing he arc lengh of parameric curves. IEEE Compuer Graphics and Applicaions, 10(3):72 78, May [6] D. H. Kochanekand R. H. Barels. Inerpolaing splines for keyframe animaion. In Graphics Inerface 84 Proceedings, pages 41 42, [7] Peer C. Liwinowicz. Inkwell: A D animaion sysem. In Compuer Graphics (SIGGRAPH 91 Proceedings), volume 25, pages , July [8] Gregory M. Nielson and Dan R. Olsen, Jr. Direc manipulaion echniques for 3D objecs using 2D locaor devices. In Proceedings of 1986 Workshop on Ineracive 3D Graphics, pages , [9] Samuel J. Leffler William T. Reeves Eben F. Osby. The Menv modelling and animaion environmen. Journal of Visualizaion and Compuer Animaion, 1(1):33 40, Augus [10] Philip J. Schneider. An algorihm for auomaically fiing digiized curves. In Andrew Glassner, edior, Graphics Gems, pages Academic Press, San Diego, CA, [11] Sco S. Snibbe. Gesural conrols for compuer animaion. Maser s hesis, Brown Universiy, Deparmen of Compuer Science, [12] Sofimage version Monreal, Canada, [13] Sco N. Sekeee and Norman I. Badler. Parameric keyframe inerpolaion incorporaing kineic adjusmen and phasing conrol. In Compuer Graphics (SIGGRAPH 85 Proceedings), volume 19, pages , July [14] TDI Explorer version Culver Ciy, CA, [15] R. Szeliski D. Tonnesen and D. Terzopoulos. Curvaure and coninuiy conrol in paricle-based surface models. In SPIE Geomeric Mehods in Compuer Vision II, [16] Alan Wa and Mark Wa. Advanced Animaion and Rendering Techniques: Theory and Pracice. Addison-Wesley Publishing Company, [17] William Welch and Andrew Wikin. Variaional surface modeling. In Compuer Graphics (SIGGRAPH 92 Proceedings), volume 26, pages , July 1992.

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