Simulating Life of Virtual Plants, Fishes and Butterflies

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1 Simulating Life of Vitual Plants, Fishes and Butteflies 1 3 Simulating Life of Vitual Plants, Fishes and Butteflies Hansudi Nose, Daniel Thalmann Comute Gahics Lab, Swiss Fedeal Institute of Technology CH-1015 Lausanne, Switzeland 3.1. Intoduction In this chate, we esent a geneal methodology fo simulating the life and behavio of ceatues and thei eaction to thei envionment. In a lage sense, we can attibute a behavio to each object. Dead objects, like balls, stones,... move (behave) in thei envionment accoding to hysical laws. Plants, as all othe living ceatues, obey also Newton's laws (gavitation, wind,..), but besides, they gow, eoduce themselves and eact to thei envionment (light, temeatue,... ). The behavio of animals is in addition influenced by emotional and instinctive comonents. Males and females ae attacted by each othe, o as an othe examle, a edato is attacted by his victim, which on the othe hand is eelled by the edato. Fo most of the humans and a lot of animals thei behavio is stongly detemined by thei vision, which is a vey imotant channel, by which the envionment is eceived. Finally, the highest level of behavio includes intelligence, which is only attibuted to humans. Ou aoach is based on L-systems, foce fields and synthetic vision. With this aoach, high level hysical and behavioal animation is ossible. Vision based actos find thei way without collisions in a L- system envionment to given destinations. Dynamically ceated objects, moving in comlex 3D vecto foce fields can inteact with each othe and banched stuctues, simulating behavio of heds, flocks, schools o some hysical effects. Poduction systems and L-gammas, as intoduced by Pusinkiewicz and Lindenmaye (1990) ae vey oweful tools fo ceating images. Fom a use-defined axiom and a set of oduction ules, the comute ceates images with a comlexity deendent only on the numbe of times the oductions ae alied. The theoy of L-systems has been mainly used fo the visualization of the develoment and gowth of living oganisms like lants, tees and cells. In a evious ae (Nose et al. 1992), we esented the softwae ackage LMobject which ealizes a timed and aametic L-system with conditional and seudo stochastic oductions fo animation uoses. With this softwae ackage a use may ceate ealistic o abstact shaes, lay with vaious tee stuctues and geneate a vaiety of concets of gowth and life develoment in the esulting animation. To extend the ossibilities fo moe ealism in the ictues, we added extenal foces, which inteact with the L-stuctues and allow a cetain hysical modeling. Extenal foces can also have an imotant imact in the evolution of objects. Pusinkiewicz and Lindenmaye (1990) have oosed two simle cases of extenal foces. In the fist method, the 3D tutle which inteets the symbolic gamma may be aligned hoizontal to a vecto eesenting the gavity. Thus, an object (a tee, fo examle) is able to "feel" the gavity and to eact. The second case is secific to lant and tee modeling and allows the simulation of toism, which is esonsible fo the bending of banches towads light souces.

2 Simulating Life of Vitual Plants, Fishes and Butteflies 2 In ou extended vesion, tee stuctues can be defomed in an elastic way and animated by time and lace deendent vecto foce fields. The elasticity of each aticulation can be set individually by oductions. So, the bending of banches can be made deendent on the banches' thickness, making animation moe ealistic. The foce fields too, can be set and modified with oductions. This kind of inteaction is based on the incile of toism as descibed by Pusinkiewicz and Lindenmaye (1990). Futhe, we intoduced a thid tye of foce inteaction, that affects L-stuctues. This simulates the dislacement of objects in any vecto foce field deendent on time and osition. An object's movement is detemined by a class of diffeential equations Eq.(3.1), which can be set and modified by oductions. The mass of the tutle, which eesents the object, can be set as well by using a secial symbol of the gamma. x = 1 m f x (t, x, y,z, x, y, z, X,Y, Z) y = 1 m f y(t, x, y, z, x, y, z, X,Y, Z) (3.1) z = 1 m f z (t, x, y, z, x, y, z,x,y, Z) To solve the diffeential equation system, we evolve an initial value oblem fo Eq.(3.1) using the 4thode Runge Kutta Method. This vecto foce field aoach is aticulaly convenient fo simulating the motion of objects in fluids (ai, wate) as descibed by Wejchet and Haumann (1991). Behavioal animation was studied in detail by Reynolds (1987). He gives a good oveview of diffeent methods and its oblematic. Susan Amkaut and Michael Giad showed in the film "Euhythmy" (Giad and Amkaut 1990) a flock of bids flying aound and avoiding collisions between themselves and obstacles in thei envionment using a foce field animation system to ealize the simulation. Reulsion foces aound each bid and aound static objects ae esonsible fo collision avoiding. At the beginning of the animation, the sace field and the initial ositions, oientations, and velocities of objects ae defined and the est of the simulation is evolved fom these initial conditions. The foce field aoach in behavioal animation is well suited fo modeling an instinct diven, animal behavio fo a lage numbe of actos foming schools, heds o flocks. The behavio of intelligent actos, howeve, needs moe sohisticated techniques. To simulate intelligent o human behavio in ath seaching and obstacle avoidance in a synthetic envionment, we have develoed a synthetic vision based global navigation module (Renault et al. 1990; Nose et al. 1994). The task of a navigation system is to lan a ath to a secific goal and to execute this lan, modifying it as necessay to avoid unexected obstacles (Cowley 1985). With the use of synthetic vision we simulate the way a eal human eceives the envionment. Thus, the fist ste in the simulation of a natual and intelligent behavio is done. Moeove, in a L-system defined envionment, whee thee is no 3D geometic database of the envionment because the wold exists only afte the execution of oduction ules, synthetic vision gives an elegant and fast way to ovide infomation about the envionment to the acto. In the L-system based animation system, descibed in (Nose et al. 1992), only a global foce field detemining the system of diffeential equations of all objects could be defined in the axiom o in the oduction ules. Thus, only one movement tye could be defined at a given time. Collision detection and object inteaction wee not ossible. In ou new aoach, each geneated object has its own foce field acting on othe objects o banching stuctues. This extension allows collision detection and behavioal animation. In addition, each object has its own diffeential equation, which detemines his movement in the global foce field given by the contibutions of all othe objects. So, at the same time seveal tyes of movements ae ossible (falling ale, juming ball, school of fish,...). In this chate, we also esent the imlementation of synthetic vision into the L-system based animation softwae, whee the behavio of an acto is at a highe level detemined by an automata, which allows an acto to find a ath fom his cuent osition to a given destination in known o unknown envionments by avoiding collisions. Seveal actos can be geneated and esonalized in the axiom o in a oduction ule. Moe details about ou synthetic vision aoach may be found in anothe ae (Nose et al. 1994), including the visual memoy eesentation by an octee and the ath seaching algoithms (3D), woking with heuistics, used to esonalize an acto's behavio. In Section 3.2 we esent the fomal

3 Simulating Life of Vitual Plants, Fishes and Butteflies 3 definition of ou L-system, exlained in (Nose et al. 1992). Section 3.3 and 3.4 descibe some theoetical asects of the new extensions. In Section 3.5 some imlementation details ae given, as well as the semantic of the symbols 'C' and 'M' of the fomal gamma which eesent the behavioal featues of foce field (symbol 'C') and vision based (symbol 'M') animation. As at of tutle inteeted L-systems (o LOGO), these symbols can be consideed as high level tutle oeation symbols, in contast to lowe level, standad symbols like 'f' (fowad) o '+' (otate), which only advance the tutle o otate it aound an axis by given values. Ou new aoach allows, fo examle, an easy animation of flocks of butteflies, of schools of fishes o of ath seaching actos in a L-system modeled envionment. Simle hysical simulations, like exlosions o bouncing balls ae also feasible The L-system In this section, we intoduce the fomal descition of ou L-system; it defines how an object at a cetain age is deived by the deivation function D fom an initial wod (= axiom = sequence of aametic symbols) and the oduction ules. The symbols of the alhabet ae inteeted by means of a 3D tutle, whose cuent oientation is given by a local othonomal coodinate system with the axis H(heading), L(left) and U(u). Some symbols contol the osition and the oientation of the tutle's coodinate system, othes eesent geometical imitives, dawn in the tutle's coodinate system, and othes efom secial oeations. Fo eades not familia with L-systems we highly ecommend the lectue of "The Algoithmic Beauty of Plants" (Pusinkiewicz and Lindenmaye 1990) whee diffeent tyes of L-systems ae well intoduced. Moe details about ou imlementation and extension may be found in (Nose et al. 1992). We stat the fomal descition of the L-system by giving some elementay definitions. V: an alhabet R + : the set eal ositive numbes N : the set intege ositive numbes Σ : a set of fomal aametes E( Σ): the aithmetic exessions in Polish notation C( Σ): the locical exessions in Polish notation ω : an axiom P : P = { a,i a V,i N } the set of oductions Π: a,i [ 0,1] a seudo -statistical distibution with π( a,i ) =1 (a, x o,..., x 6 ) V R 7 a aametic symbol The oductions of an L-system descibe how symbols of a timed and aametic sting ae elaced if they ass thei maximal age duing evolution of time. P V R 7 C(Σ) Π (V E( Σ) 7 ) * : the oduction sace a,i P : a single oduction ( ) a,i : a,α,x 1,..., x 6 Cond(α,x 1x 2,x 3,T) and Π ( a,i ) (a 1, f 10,..., f 16 )...(a n, f n0,..., f n6 ) α: maximal age of symbol a f j 0 initial age of symbol a j x 1,x 2,x 3 3 aametes x 4,x 5,x 6 values of the evaluated gowth functions f j 0 + t local age of the symbol T global time f j 1,..., f j3 aamete exessions f j 4,..., f j6 gowth functions f jk f jk( f j 0, x 1, x 2,x 3,t,T ) a,i a conditional and seudo -statistical oduction i

4 Simulating Life of Vitual Plants, Fishes and Butteflies 4 In the above definition the symbol a is elaced with a cetain obability unde a given condition by a aametic sting given by the ight at of the oduction. The way, how this elacement is done, is contolled by the deivation function D and its mathematical axioms. ( ) * R ( V R 7 ) * D: V R 7 Axiom 1: The develoment of each symbol is indeendent of each othe in time D( ( a 1,x 0,...,x 6 )...( a n,x 0,...,x 6 ),t) = D( ( a 1,x 0,...,x 6 ),t)...d ( a n,x 0,,...,x 6 ),t Axiom 2: The Gowth of a symbol befoe teminal age. if x 0 +t <α then D ( a,x 0,...,x 6 ),t ( ) = a,x 0 +t,x 1,x 2,x 3,f 4,f 5,f 6 The gowth functions ae evaluated. ( ) ( ) Axiom 3: Alication of stochastic oduction at teminal age ( ( ) = TRUE) then with a obability π( a,i ) ( ) = D( a 1, f 10,...,f 13,x 4,x 5,x 6 ( )) if (x 0 + t α) Cond α,x 1,x 2,x 3,T D ( a,x 0,...,x 6 ),t ( )... ( a n,f n0,..., f n3,x 4,x 5,x 6 ),t α x 0 The initial age and aamete exessions f 10,...,f 13 ae evaluated. Axiom 4: Selection of andom numbes fo the oductions [( ) mod z] andom_numbe= and_table x + y[ x] and_table: table of size z with unifom andom values between 0 and 1 x : ecusion deth of function D y [ x] : osition of the oduction in the deivation tee of D at the x deth. To guaantee unde cetain conditions a continuous gowth of the lants the axiom 4 has to be added (Nose et al. 1992) if stochastic oductions ae used in the lant definition by oduction ules Behavioal Modeling using Foce Fields In a "foce field animation system" the 3D wold has to be modeled by foce fields. Some objects have to cay eulsion foces, if thee should not be any collision with them. Othe objects can be attactive to othes. Many objects ae both attactive at long distances and eulsive at shot distances. The shaes and sizes of these objects can vay, too. Sace fields like gavity o wind foce fields can geatly influence animation sequences o shaes of tees. The system of diffeential equations, given in Eq.(3.2), descibes the movement of a oint object i with mass m i in a foce field. The global foce field is given by the sum of all objects' contibutions f kj. The individual at g ki of each object detemines it's behavio in the global field, x i = 1 m i y i = 1 m i g xi (x i, y i, z i, x i, y i, z i, t) + g yi (x i, y i, z i, x i, y i, z i,t) + j, j i j, j i f xj (x i, y i, z i, x j, y j, z j, x j, y j, z j, ij, t) f yj ( x i, y i, z i, x j, y j, z j, x j, y j, z j, ij,t)

5 Simulating Life of Vitual Plants, Fishes and Butteflies 5 z i = 1 m i g zi ( x i, y i, z i, x i, y i, z i,t) + j, j i whee : i = index of an object m i = the mass of object i x i, y i, z i = osition comonents of object i x i, y i, z i = velocity comonents of object i x i, y i, z i = acceleation comonents of object i t = time = distance between object i and object j ij f zj ( x i, y i, z i, x j, y j, z j, x j, y j, z j, ij,t) (3.2) The behavio of an object in the global foce field is detemined by a edefined cuve (e.g. sline, fixed) o it's individual at g of Eq.(3.2). The tems of g can deend on the object's actual osition and it's seed. Seed deendent tems can be used to model fiction oeties. The osition vaiables allow it to make the object's behavio osition deendent. Toism foces act on the aticulations of banching stuctues (Pusinkiewicz and Lindenmaye 1990). The bending of banches is simulated by a otation of the tutle in diection of the toism foces. If ou dynamically ceated objects have to inteact with banching stuctues, then thei foce fields have to be added to the toism foce. The following equations descibe this inteaction in detail. M = H F m = M A = H F β = sin 1 A ( ) / e toque vecto H toque ( e =elasticity of aticulation) ( ) / ( F ) otation vecto ( ) angle between tutle heading α = β m F = f toisme + f i i esulting otation angle global foce field H et F (3.3) The effective otation angle is ootional to the toque m, oduced by the global foce by acting on the tutle's heading vecto, but it neve exceeds the angle between the tutle heading and the foce vecto at the tutle's osition Vision Based Animation In ou aoach to synthetic vision (Nose et al. 1994), a dynamic occuancy octee gid seves as a global 3D visual memoy and allows an acto to memoize the envionment that he sees and to adat it to a changing and dynamic envionment. His easoning ocess allows him to find 3D aths based on his visual memoy by avoiding imasses and cicuits. The global behavio of the acto is based on the navigation automata, shown in Figue 3.1, eesenting the automata of an acto, who has to go fom his cuent osition to diffeent laces, memoized in a list of destinations. He can dislace himself in known o unknown envionments. Afte the initialization of his memoy and his vision system (off -> mem_set -> vision_set) and the definition of his destinations (end_oint_set) the ath seaching is initialized. Befoe stating a seach, he looks aound and memoizes what he sees (look_aound). Then, he ties to find a ath by easoning (seach_new_ath). If he finds one, he follows it (move). If he encountes an obstacle while moving, he stos, tuns aound, looks fo a new ath and estats moving. This ocess is eeated until he

6 Simulating Life of Vitual Plants, Fishes and Butteflies 6 aives at his destination. Thee, he takes the next destination fom the list and estats the whole ocedue descibed above. M(0) M(1) off mem_set vision_set M(5) end_oint_set M(6) exloe exloe and set ath_end ath_a_set M(11) move go and set ath_end look_aound_set if no ath found if no goal if ath found seach_new_ath if (ath_end) get new goal if (no goal) state = off else seach ath look aound look_aound state action, tansition Figue 3.1. Global navigation automata of the acto. In unknown envionments, if he uses a conditional heuistic, he cannot find a ath to an invisible destination ( behind an obstacle, fo examle). In this case, he stats to exloe his envionment accoding to his heuistic, which will lead him to his destination, if thee is any ath Imlementation of the Behavioal Featues We integated the hysical modeling and behavioal animation featues in the LMobject softwae ackage (Nose et al. 1992). This L-system is a timed aametic context-fee gamma with conditional and seudostochastic oductions. All gamma symbols have thee aametes and thee gowth functions. It should be noted, that neithe the fomal object, no the gahical object need to be stoed comletely in memoy. Only the fomal descition of the cuent state of the deivation function needs to be maintained on a stack, esulting in memoy equiements ootional to the ecusion deth of the deivation function. Each time the deivation functions encountes a leaf of the deivation tee, it inteets the symbol and executes immediately the coesonding action. This action can be, fo examle, a tutle oeation, a foce field declaation, a camea maniulation o the dawing of a gahical imitive in the tutle's coodinate system. The actions ae detemined by the semantic of the gamma. The new extension allows us, to dynamically ceate new oint objects with a secial symbol C of the gamma. Each time the deivation function of the L-system which deives the object fom the axiom and the oduction ules encountes this symbol C, it acts on a global inteaction table containing the tace of all geneated objects. This inteaction table has the following stuctue: tyedef stuct { shot id_deth, id_lage; float osvel[6]; cha *foce[3], *eqdiff[3], *tye; float mass } Tinteaction; Tinteaction InteactionTable[MAX_OBJECTS] The ecusive deivation function identifies with its fouth axiom each object in the deivation tee and etuns the unique object identification id_deth and id_lage.

7 Simulating Life of Vitual Plants, Fishes and Butteflies 7 If the object is encounteed fo the fist time, it is aended to the global inteaction table. The initial osition (tutle) and the seed (fom the symbol C) ae coied into the table osvel and the foce (*foce), the diffeential equation (*eqdiff(3)) and the tye (*tye) ointes ae diected to the coesonding chaacte exessions in the aametic symbol C. This way, the mass of the object can be set as well. If the object aleady exists in the inteaction table, the deivation function stats an iteation ste of the numeical solution of the object's diffeential equation by egading all the contibutions of the othe foce fields of the same tye in the global inteaction table. Then, the tutle is laced at the esulting osition. The iteation ste is calculated by the method of Runge Kutta of degee fou and solves the system of diffeential equations given in Eq.(3.2). This system descibes the Newtonian movement of a oint with a mass m in the 3D sace unde the influence of a 3D vecto foce field. As the osition and seed of each ceated object ae stoed in the global inteaction table, the elative distances ij fom one object i to an othe object j, can be calculated and thus used in the exessions of the foce fields and diffeential equations. The semantic of the aametic symbol C C (t) (x) (y) (z) (f1) (f2) (f3) deends on the aamete x. If x = 1, the exessions f1, f2 and f3 define the thee comonents of the vecto foce field of the cuent object. If x = 2, the cuent object's diffeential equation is detemined in the same way. If x = 3, the value of y sets the mass, and the object's movement is stated with an initial seed given by the evaluated exessions f1, f2 and f3. Sometimes it is useful, that an object caying a foce field moves along a edefined ath. If x = 4, the diffeential equation of the cuent object is not evaluated, but the object is laced at the osition given by (f1, f2, f3). The fi's, fo examle, can be timed sline defining a 3D ath. The integation of the vision module into the L-system is ealized via the secial symbol M of the gamma. The semantic of M deends also on it's aamete x. In Figue 3.1, we can see its influence on the state of the acto. With x = 0, the symbol M (t) (0) (y) (z) (f1) (f2) (f3) initializes and scales the visual octee memoy of an acto. The values of f1 and f2 detemine the cube including the whole scene and f3 sets the esolution of the octee (maximal tee deth). M (t) (1) (y) (z) (f1) (f2) (f3) emits to esonalize the actos vision system. With f1 and f2 the nea and fa cliing is set, and f3 gives the ange of inteest within which the acto adats his memoy to changes in the envionment. With the symbol M (t) (6) (y) (z) (f1) (f2) (f3) the ath seaching and exloing ocedue can be esonalized. The value of f1 gives the distance, the acto eviews along his actual found ath, to detect collisions with his envionment. By f2 the size of voxels to be examined duing ath seaching and exloing is set. With f3, the use can detemine one of the edefined heuistics to use in the ath seaching o exloing algoithm. It is also ossible to set initial ositions and look at oints, destinations and moving seeds of the actos by using the coesonding aametic symbol M. So, a use can define and esonalize one o seveal actos in the axiom. He can give him seveal destinations, and then the animation will evolve accoding to the global navigation automata shown in Figue 3.1. We also imlemented the ability to lace and oient the camea accoding to an object's osition and velocity so that nice animation fom the esective of an acto with vision o a foce field influenced object can be ealized.

8 Simulating Life of Vitual Plants, Fishes and Butteflies Alications Gowing Tee Stuctue In this section we illustate the woking incile of the deivation function D by deiving the fomal object of a continuously gowing simle tee stuctue at seveal ages. We conside the alhabet V = {, q, +, -, [, ] } with all chaactes inted in bold. and q ae timed and aametic symbols with gowth functions. To establish the link with the fomal descition of Section 3.2 we use the following notation. (, a, x, f(x, t) ) the aametic symbol the symbol a the aamete x0 which is the actual o initial age of the symbol. x the aamete x1 f(x,t) the gowth function f 4 t the time The symbols and q eesent line segments with a length of the actual value of the gowth function. The line segment is dawn fom the actual osition of the tutle in diection of its heading vecto H. Afte having dawn the line segment the tutle is laced at its end without changing its diection. The symbols + and - of the alhabet V eesent otations of the tutle aound its U vecto by an angle of +-90 degees. The backets [ and ] coesond to ush and o oeations of the tutle state (osition and oientation) which ae intoduced to delimit banches. Thus, the backets enable the constuction of tee stuctues. The tee stuctue is defined by an axiom and one oduction fo the symbol. Axiom: (, a=0, x=1, f=6xt) Poduction: (, α = 1) > (Condition = TRUE, obability = 1) (q, a=0, x=x, f=6x+t) [ + (, a=0, x = x/2, f=6xt) ] [ - (, a=0, x=x/2, f=6xt) ] The axiom coesonds to a line segment (symbol ) with initial age a=0, a aamete x=1 and a linea gowth function f = 6xt = 6t (as x =1). So, the segment will linealy gow until it eaches its maximal age α = 1 given at the left side of the oduction. The oduction is alied with a obability of 1.0 without any econdition. Figue 3.2 illustates some alications of the oduction q q q q age = 0.99 age = 1.99 age = 2.99 Figue 3.2. Continuous gowth of a simle tee. The line segment at maximal age of 1 has to be elaced by a segment q of the same length at its initial age of a=0. So, the aamete x of the symbol (which is elaced) is assed without any change. The gowth function fq=6x+t satisfies this continuity condition as f(t=1, x=1) = fq(t=0, x=1) = 6. The tem t is esonsible fo a futhe linea but smalle gowth ate of the segment q. So each banch of the segment continues to gow. The two new banches should be a facto 2 shote then the eceding one. Theefoe, the aamete x is assed to the new symbols by dividing it by 2 ( x = x/2). As they stat thei gowth fom zeo the same gowth function has to be used as fo the symbol in the axiom. Thus, a continuous and consistent futhe elacement is guaanteed duing the following iteations. In the next aagah, we ovide some details about deivation stes of the fomal object. We show how the deivation function D woks and when the coesonding axioms of the deivation function D ae

9 Simulating Life of Vitual Plants, Fishes and Butteflies 9 alied. To avoid some confusion the eade is asked to distinguish between the enumeated mathematical axioms of the deivation function D and the "sting" axiom of the tee definition. Object age t = 0.5: D(Axiom, 0.5)= D((, 0, 1, 6xt), 0.5) = (Axiom 2, as a+t = 0.5 < α=1, the gowth function is evaluated) (, a+t, x, 6xt) = (, 0+0.5, 1, 6*1*0.5) = (, 0.5, 1, 3) Object age t = 0.9 D(Axiom, 0.9) =... = (, 0.9, 1, 5.4) Object age t = 1.5 D(Axiom, 1.5) = D((, 0, 1, 6xt), 1.5) = (Axiom 3, as a+1.5 = = 1.5 > α = 1, the aamete x fom the eceding symbol is assed to the symbols of the oduction by evaluating the coesonding exessions) D((q, 0, x, 6x+t) [ + (, 0, x/2, 6xt) ] [ - (, 0, x/2, 6xt) ],1.5-(α-a)) = D((q, 0, 1, 6x+t) [ + (, 0, 0.5, 6xt) ] [ - (, 0, 0.5, 6xt) ], 0.5) = (Axiom 1) D((q, 0, 1, tx+t), 0.5) [+D((, 0, 0.5, 6xt), 0.5) ][- D((, 0, 0.5, 6xt), 0.5)] = (Axiom 2, as a+t = < α=1, the gowth functions ae evaluated by using the evaluated aamete x) (q, 0.5, 1, 6.5) [ + (, 0.5, 0.5, 1.5) ] [ - (, 0.5, 0.5, 1.5) ] School of Fishes In the following animation fishes with mass m=1 ae geneated at the same lace at a egula ate. Thei eulsive foce field is given by Eq.(3.4). All these fishes ae attacted by a moving object, the bait, with an attactive foce field at long distances and a eulsive one at shot distances (Eq.(3.5)). The movement of each fish is damed by Eq.(3.6). Figue 3.3 illustates the bait's foce field. Figue 3.3. The bait's foce field.

10 Simulating Life of Vitual Plants, Fishes and Butteflies 10 The fish's foce field has about the same fom at shot distances. At long distances howeve, it emains zeo. ( ) = n f fish with = ( X object x ), = X object x X object : Position of the object (fish) x : osition in the the foce field and n = : distance fom the foce field cente of the object f bait (3.4) ( ) = ( ) n (3.5) g fish ( x ) = 3 x (3.6) The bait moves along a given 3D ath sline. Each new geneated fish joins immediately the school, following the bait. The stongly eulsive comonents of the foce fields at shot distances of all moving objects, event collisions. In Figue 3.4 (see Colo Section), we can see some ictues fom the whole animation sequence. Figue 3.5 shows ats of the axiom and the oduction ules used fo the above animation sequence. Axiom... /* some symbols, descibing the envionment It follows the definition of the bait */ f () () () () (0) (0) (0) /* the symbol f laces the tutle at (0,0,0) and detemines the initial bait osition*/ C () (1) () () ((4-2500*3^(-0.005^2) * (X-x)) ((4-2500*3^(-0.005^2) * (Y-y)) ((4-2500*3^(-0.005^2) * (Z-z)) /* Symbol C with aamete x=1 defines the bait's foce field. (X,Y,Z) ae the coodinates of the cuent object (in this case the bait). (x,y,z) is the osition of an object, feeling the foce field. The vaiable is given by =sqt((x-x)^2 + (Y-y)^2 + (Z-z)^2), the distance of a fish to the bait. */ C () (4) () () (sline_1(t)) (sline_2(t)) (sline_3(t)) /* Symbol C with aamete x=4 laces the tutle (= the bait) at the osition (sline_1(t), sline_2(t), (sline_3(t)) detemined by a edefined timed 3D sline */ z ()() () () () () () /* the symbol z is a gem fo the bait. It is an abitay geometical figue, defined by the coesonding oduction ule, not given hee. It is dawn in the tutle coodinate system */ x () () () () () () () /* The symbol x is a gem fo a fish foce field declaation */... /* some moe symbols, descibing something abitay */ EndAxiom Poduction1 x (maximal_age = 1) >/*symbol x to be elaced by the following symbols*/ /* it follow the aametic symbols, defining the behavio of a fish */ f () () () () (50) (50) (50) /* the symbol f laces the tutle (=fish) at the osition (50, 50, 50) and thus detemines the initial osition of a fish */ C () (1) () () ((-2500*3^(-0.005^2) * (X-x)) ((-2500*3^(-0.005^2) * (Y-y)) ((-2500*3^(-0.005^2) * (Z-z))

11 Simulating Life of Vitual Plants, Fishes and Butteflies 11 /* Symbol C with aamete x=1 defines the fish's foce field, felt by othe objects. */ C () (2) () () (-3u) (-3v) (-3w) /* Symbol C with aamete x=1 defines the individual at of each geneated fish of the Eq.(2). The vecto (u,v,w) eesents the velocity of the cuent fish. */ C () (3) (1) ()(5) (3) (2) /* Symbol C with aamete x=3 stats the evolution of the geneated fish with the initial velocity (5, 3, 2) and the mass 1 */ y (1) () () () () () () /* Symbol y eesents a gem of a fish with an initial age of 1 */ x () () () () () () () /* The symbol x is a gem fo a fish foce field declaation */ EndPoduction1 Figue 3.5. Pats of axioms and oduction ules fo the fish animation. This textual descition of the axiom and the oduction ae inteeted and contolled by the deivation function D of the L-system. In this case, at each time inteval of 1 (1 = maximal symbol age of a fish gem x) a new fish is geneated and immediately stats it's tyical behavio. It follows the bait and avoids collisions Tee-Ball Inteaction In this examle, we show the inteaction of a ball and a tee. The ball moves towads a tee. As it caies a eulsive "velocity" foce field, the tee and the banches bend unde it's influence and ty to avoid collision with the ball. The ball's foce field is shown in Eq.(3.7) f ball, x = ( ) x (3.7) Hee, the foce field is always in the diection of the ball's velocity, even behind the ball, so wind, caused by the movement of the ball is simulated. Figue 3.6 (see Colo Section) shows some ictues of the animation sequence A Buttefly in a Flowe Field In the animation sequence, illustated in Figue 3.7 (see Colo Section), a buttefly seaches his way though a flowe field, guided by his vision. It is an examle of the use of 3D heuistic seach (Nose et al. 1994). The flowes ae animated by a wind foce field. The buttefly is modeled by some symbols of the gamma just as the flying moto. The fist destination of the buttefly is the eflecting shee. When it aives thee, it looks aound and seaches fo a ath to the second destination at the othe end of the flowe field. Guided by its vision, it is able to avoid collisions and to find a ath, as well as to memoize in its visual memoy eveything it sees Conclusion With ou cuent L-system based animation system, simulations in all of the above mentioned domains can be ealized and combined in one sequence. As Eq. (3.2) is based on Newton's laws, hysical simulations ae immediate. Gowing and eoduction featues ae inheent in the L-system. We can ealize, fo examle, animation sequences of a gowing tee, oducing ales, which contain gems fo new tees. These gems develo to new tees, afte the ales have fallen to the gound. Thus, a whole foest can develo fom a single ale (Nose et al. 1992) Beside the hysical modeling, the foce field aoach allows as well to simulate instinctive o emotional behavio (attactive, eellent foces) of individuals o gous and to

12 Simulating Life of Vitual Plants, Fishes and Butteflies 12 inteact hysically with lants and dead objects. Animation accoding to the following descition ae ossible: "A small lake is oulated with lants, edatos and a fish. The fish sawns. The sawn floats away in a wate cuent. Afte some time the sawn develos to fish which join to a school following a guide. They avoid the edatos. " The vision based featues, allow the actos to avoid collisions and to simulate intelligent behavio. The global navigation automata eesents 'intelligent' behavio. It allows an acto to find, fo examle, the exit of a maze, even if thee ae imasses and cicuits, and to memoize the toology of the seen envionment. This leaned infomation he can use in futue ath seaching. In a futue develoment, we lan to imove the foce field animation module. Since most objects' foce fields have only shot distance anges, it would save much calculation time if only the foce field contibutions of neaby objects had to be consideed. To solve this oblem, we lan to intoduce a dynamic sace subdivision with octees, in which the foce fields can be laced. Thus, neighboing objects could easily be identified. With such an aoach much moe comlex animation would be feasible at still easonable calculation times. The synthetic vision module eesents a vey univesal and oweful tool fo futue behavioal models. By designing and combining new automata and including moe sohisticated actos with walking, seaking and gasing featues, high level scit based animation can be ealized in a extemely dynamic envionment with atially autonomous actos. Acknowledgements This eseach was sonsoed by the Swiss National Foundation fo Scientific Reseach. Refeences Cowley JL, Navigation fo an Intelligent Mobile Robot (1985) IEEE Jounal of Robotics and Automation, Vol. RA 1, No 1, Giad M, Amkaut S (1990) Euhythmy: Concet and Pocess, Jounal of Visualization and Comute Animation, Vol. 1, Nose H, Renault O, Thalmann D, Magnenat Thalmann D, Vision-Based Navigation fo Synthetic Actos, (Submitted fo ublication, 1994) Nose H, Thalmann D, Tune R (1992) Animation based on the Inteaction of L-systems with Vecto Foce Fields, Poc. CGI '92, Singe, Tokyo Pusinkiewicz P, Lindenmaye A (1990), The Algoithmic Beauty of Plants, Singe-Velag Renault O, Magnenat Thalmann N, Thalmann D (1990), A Vision-based Aoach to Behavioal Animation, Jounal of Visualization and Comute Animation, Vol 1, No 1, Reynolds C (1987), Flocks, Heds, and Schools: A Distibuted Behavioal Model,Poc. SIGGRAPH 1987, Comute Gahics, Vol.21, No4, Wejchet J, Haumann D (1991), Animation Aeodynamics, Poc. SIGGRAPH 1991, Comute Gahics, Vol.25, No4,