`
`COMPUTER GRAPHICS Proceedings, Annual Conlarnnco Series. 1999
`
`Animation of Plant Development
`
`Przernyslaw Prusinkiewicz
`
`Mark S. Hammel
`
`Eric Mjolsness
`
`Department of Computer Science
`Deparu'nent of Computer Science
`University of Calgary
`University of Calgary
`Calgary. Alberta. Canada T2N 1N4 Calgary, Alberta. Canada T2N 1N4
`
`Department of Computer Science
`Yale University
`New Haven, CT 06520-2153
`
`ABSTRACT
`
`intervals. This creates several problems if a smooth animation of
`development is sought [27. Chapter 6]:
`
`This paper introduces a combined discreteicontinuous model of
`plant development that integrates L-system-style productions and
`differential equations. The model is suitable for animating simu—
`lated developmental processes in a manner resembling time-lapse
`photography. The proposed technique is illustrated using several
`developmental models. including the flowering plants Compunnia
`rapunculoides, Lwirnis coma-aria. and Hieracr'ton ronbelintum.
`CR categories: E42 [Mathematical Logic and Formal Lan-
`guages]: Grammars andOLher Rewriting Systems: Parallel rewrit-
`trig system, 1.3.? [Computer Graphics]: Thee-Dimensional
`Graphics and Realism: Animation, 1.6.3 [Simulation and Mod-
`eling]: Applications. 1.3 [Life and Medical Sciences]: Biology
`Keywords: animation through simulation. realistic image synthe-
`sis. modeling of plants. combined discretet'continuous simulation,
`Lrsystem, piecewisecontinuous differential equation.
`
`1
`
`INTRODUCTION
`
`Time-lapse photography reveals the enormous visual appeal of de-
`veloping plants. related to the extensive changes in topology and
`geometry during growth. Consequently.
`the animation of plant
`development represents an attractive and challenging problem for
`computer graphics. Its solution may enable us to retrace the growth
`of organs hidden from view by protective cell
`layers or tissues.
`illustrate prooesses that do not produce direct visual effects, and
`expose aspects of development obscured in nature by concurrent
`phenomena. such as the extensive daily motions of leaves and flow-
`ers. Depending on die application. different degrees of realism may
`be sought. ranging from diagrammatic representations of develop-
`mental mechanisms to photorealistic recreations of nature's beauty.
`Known techniques for simulating plant development, such as L—
`systems [[6, 27, 28, 31]. their variants proposed by Aono and Ku—
`nii [l ], and the AMAP software {4. 10]. operate in discrete time,
`which means that the state of the model is known only at fixed time
`
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`
`0 Although. in principle. the time interval can be arbitrarily
`small. once it has been chosen it becomes a part of the model
`and cannot be easily changed. From the viewpoint of com—
`puter animation, it is preferable to specify this interval as
`an easy to control parameter, decoupled from the underlying
`model.
`
`o The continuity criteria responsible for the smooth progression
`of shapes during animation can be specified more easily in
`the continuous time domain
`
`I It is conceptually elegant to separate the model of develop-
`ment. defined in continuous time. from its observation. taking
`place in discrete intervals.
`
`Smooth animations of plant development have been created by
`Miller (a growing coniferous tree [19]). Sims (artificially evolved
`plant-like structures [30]). and Prusinkiewicz er. al.
`(a growing
`herbaceous plant Lyehnir coronaria [24]). but the underlying tech-
`niques have not been documented in the literature. Greene proposed
`a model of branching structures [12] suitable for animating accre-
`tive growth [11]. but this model does not capture the non-accretive
`developmental processes observed in real plants.
`This paper introduces a mathematical framework for modeling
`plants and simulating their development in a manner suitable for
`animation. The key concept is the integration of discrete and con~
`tinuous aspects of model behavior into a single formalism, called
`diferentiol L-systems (dL-systems), where L-system-style produc-
`tions express qualitative changes to the model [for example. the
`initiation of a new branch). and differential equations capture con-
`tinuous processes. such as the mdual elongation of internodes.
`The proposed integration of continuous and discrete aspects of de—
`velopment into a single model has several predecessors.
`Barrel [2] introduced piecewise-continuous ordinary dtferentiol
`equations (P0085) as a framework for modeling processes de-
`scribed by differential equations with occasionally occurring discon-
`tinuities. PODEs lasts a formal generative mechanism for specifying
`changes to system configuration resulting from discrete events. and
`dierefore cannot be directly applied to simulate the development of
`organisms consisting of hundreds or thousands of modules.
`Fleischer and Barr [7'] addressed this limitation in a model of mor-
`phogenesis consisting of cells developing in a continuous medium.
`
`0 1993
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`The configuration of the system is determined implicitly by its ge-
`ometry. For example. in a simulated neural network. a synapse is
`formed when a growing dendrite of one cell reaches another cell.
`Mjolsness at. al. [21] pursued an alternative approach in a connec-
`tionist model of development. Differential equations describe the
`continuous aspects of cell behavior during interphase (time between
`cell divisions). while productions inspired by L-systems specify
`changes to the system configuration resulting from cell division
`and death. The connectionist model makes it possible to consider
`networks with arbitrary topology (not limited to branching struc—
`tures). but requires productions that operate globally on the entire
`set of cells constituting the model. This puts a practical limit on the
`number of compoaents in the system.
`Fracchia er. cl. [9] {see also [27, Chapter 7]) animated the devel-
`opment of cellular layers using a physically-based model in which
`differential equations simulate cell growth during the interpbase,
`and productions of a map L-sysrem capture cell divisions. The pro—
`ductions operate locally on individual cells. making it possible to
`simulate the devel0pment of arbitrarily large layers using a finite
`number of mles. Unfortunately. this technique does not seem to
`extend beyond the modeling of cellular layers.
`Toned brystems [27, Chapter 6] were introduced specifically as a
`formal framework for constructing models of branching structures
`developing in continuous time. They operate under the assumption
`that no information exchange between coexisting modules takes
`place. This is a severe limitation. as interactions between the mod-
`ules are known to play an important role in the development of
`many plant species [14. 27, 28]. A practical application of timed
`L—systems to animation is described by Noser at at. [22].
`The model of development proposed in this paper combines ele»
`ments of PODEs. the connectiortist model. arid L-systems. The
`necessary backgrotmd in L-systems is presented in Section 2. Sec-
`tions 3 and 4 introduce the definition ofdifferentia] L—systems and
`illustrate it using two simple examples. Section 5 applies combined
`discretei'mntinuous simulation techniques to evaluate dL-systents
`over time. Section 6 focuses on growth functions. which char-
`acteriae continuous aspects of model development. Application of
`differential L—systems to the animation of the development of higher
`plants is presented in Section 1', using the models of a compound
`leaf and three herbaceous plants as examples. A summary of the
`results and a list of open problems conclude the paper.
`
`2 LaSYSTEMS
`
`An extensive exposition of L—systcms applied to the modeling of
`plants is given in [27]. Below we summarize the main features of
`L-systerns pertinent to the present paper.
`We view a plant as a linear
`or branching structure com-
`posed of repeated units called
`modules. An Lr-systcm de-
`scribes the development of
`this
`struchire in terms of
`rewriting rules or produc-
`tions. each of which replaces
`the predecessor module by
`zero, one. or more succes-
`
`F1gure 1: Example of a typical
`bsystem production
`
`sor modules. For example.
`the production in Figure i re-
`
`3
`
`_._D L
`
`A
`
`B
`
`L
`
`I
`
`places apex A by a structure consisting of a new apex A. an internode
`I . and two lateral apices B supported by leaves L.
`In general. productions can be context fine and depend only on the
`replaced module. or context-sensitive and depend also on the neigh-
`borhood of this module. A developmental sequence is generated
`by repeatedly applying productions to the consecutively obtained
`structures.
`In each step. productions are applied in parallel to all
`parts of the structure obtained so far.
`The original formalism of L-systerns [[6] has a threefold discrete
`character [17]:
`the modeled structure is a finite collection of mod-
`ules. each of these modules is in one ofa finite number ofstates. and
`the development is simulated in discrete derivation smps. An exten-
`sion called permit-ii: L—systems [25. 27] increases the expressive
`power of L-systems by introducing a continuous characterization of
`the module states. Each module is represented by an identifier de-
`noting the module type (one or more symbols starting with a letter)
`and a state vector of zero. one. or more numerical parameters. For
`instance, M = A(5,9.5} denotes a module M of type A with two
`parameters on = 5 and w: = 9.5. fonning thevectorw = (5. 9.5).
`The interpretation of parameters depends on the semantics of the
`module definition. and may vary from one module type to another
`For example. parameters may quantify the shape of the module. its
`age. and the concentration of substances contained within it.
`In the formalism of lrsystems. modeled
`structures are represented as strings of
`modules. Branching suuctures are cap-
`tured using bracketed strings, with the
`matching pairs of brackets [ and ] delim-
`iting branches. We visualize these struc-
`tures using a turtle interpretation of strings
`[23. 28]. extended to strings of modules
`with parameters in [13. 25. 2'7]. A prede-
`fined interpretation is assigned to a set of
`reserved modules. Some of them represent
`physical parts of the modeled plant. for ex-
`ample a leaf or an internode. while others
`represent local properties. such 3 the magnitude of a branching
`angle. Reserved modules frequently used in this paper are listed
`below:
`
`Figure 2: Thrtle
`interpretation of
`a sample string
`
`F(=)
`+(o), —[a)
`
`line segment of length 2.
`orientation change of the following line by :l:o
`degrees with respect to the preceding line,
`a predefined surface X scaled by the factor a.
`@Xla}
`The interpretation of a string of modules proceeds by scanning it
`from tell to rightand considering the reserved modules as commands
`that maneuvcr a LOGO-style turtle. For example, Figure 2 shows
`the turtle interpretation of a sample string:
`
`F(1j[+{45)oL[o.r51]F(o.a)[—(so)@L(o.s)]F(o.s)@K[l}.
`
`where symbols @L and @K denote predefined surfaces depicting a
`leaf and a flower.
`
`3 DEFINITION OF dL—SYSTEMS
`
`Differential Lrsystems extend parametric L—systerns by introducing
`continuous time flow in place of a sequence of discrete derivation
`steps. As long as the parameters is of a module Alw) remain in the
`domain of legal values ’94. the module develops in a continuous
`
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`EOMEUTEEEBAPHICS Proceedings. Annual Conference Series. 1993
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`5
`
`way. Once the parameter values reach the boundary CA of the
`domain DA. a production replaces module AW} by its descendants
`in a discrete event. The form of this production may depend on
`which segment (3,4 l of the boundary of 1),; has been crossed.
`M
`For example. module
`MginFigureSis cre-
`r._‘__...
`9L¢
`ated at time to as one
`of
`two descendants
`of the initial module
`M:. It develops in the
`interval [tmtaL and
`ceases to exist at time
`in. giving rise to two
`new modules Ma and
`M5. The instant is is
`the time at which parameters of M2 reach die boundary of its do-
`main of legal states 13. A hypothetical trajectory of module M2 in
`its parameter space is depicted in Figure 4.
`In order to formalize the above description. let us assume that the
`modeled structure consists of a sequence of modules (an extension
`to branching struttum is straightfonvard if a proper definition of
`context is used [27. 28]]. The state of the structure at time t is
`represented as a string:
`
`+__._‘“3 =
`
`
`
`time
`{a
`‘F
`t]
`Figure 3: Fragment of the lineage tree
`of a hypothetical modular structure
`
`'1 = A|(W1)A3(Wa) I
`
`I
`
`I A5(Wn).
`
`The module Ai_1[w.-_1) immediately preceding a given module
`Adwr} in the string it is called die lefi neighbor or left context
`of Adm). and the module Audra“) immediawa following
`Adi-we) is called its right neighbor or right context. When it is
`inconvenient to list the indioes. we use the symbols <. >, andfor
`subscripts i. r to specify the context of Mn), as in the expression:
`
`.4“er < AU!) ) Ar(Wu).
`
`The continuous behavior of A(w) is described by an ordinary dif—
`ferential equation that determines the rate of change dwidt of pa-
`rameters w as a function of the current value of these parameters
`and those of the module's neighbors:
`
`initial state of
`
`3‘ time I :1“
`
`Domain ofiegai values
`of the parameter vector at
`.
`'4'
`.' .
`
` module M2 = am
`‘ 1N)" Turkish—9‘. eye)?
`
`Trajmtory wt!)
`"w a. w—n-r
`..
`..
`
`
`
`
`
`C3
`
`:-
`.
`
`
`
`'
`Pmsibieci discontinuity of wit)
`Trajectory reaches boundary t?
`resulting from the application
`"
`
`segment C: - a production
`of a production to the contest
`-any;
`
`
`deteirntnes' the descendants
`
`andtheirstmeattimer=
` gagggm.)
`
`1
`
`Figure 4: A hypothetical trajectory of module M2 in its parameter
`space
`
`boundary CA: The initial value of parameters assigned to a module
`Br“,- (WM) upon its creation is determined by a flirtation ham.
`which takes as its arguments the values of the parameters W: . W. and
`w. at the time immediately preceding production application:
`
`t
`t-
`We; = [in] ha”-{Wr(t),W(t),Wr(t)).
`"c
`
`(A stronger
`The vector WM must belong to the domain 1331”"
`condition is needed to insure that the number of productions applied
`in any finite interval [ht + At] will be finite.)
`in summary. a differential L~system is defined by the initial string
`of modules on and the specification of each module type under
`consideration. The specification of a module type A consists of
`four components:
`
`dw
`I -— IA (Wt. Wt “1")-
`
`(Uaflmfmpa >.
`
`where:
`
`The above: equation applies as long as the parameters or are in the
`domain DA characteristic to the module type A. We assume that DA
`is an open set. and specify its boundary Ca as the union of a finite
`number m 2 1 ofnoru‘ntersectiug segments CM. is = 1, 2, .
`.
`.
`. m.
`The time is at which the trajectory of module MW) reaches a
`segment C,“ of the boundary of Dar satisfies the expression:
`
`[in WU) €- CA..-
`t—rtfl
`
`The replacement of module AU?) by its descendants at time tg is
`described by a production:
`
`p,” : 1410*” < Ah?) } Ar(W.-) _§
`Bb,1(wk.l)3k‘2(wk.2) *
`- -Br=.m,, {Warns}-
`
`The module A{w) is called the strict predecessor and the sequence
`oft-nodules Bk‘1(Wfi_]}Bk.1(Wk_2) - - - Bitmktwtmk) is called the
`successor of this production. The index it emphasizes that difl‘ta'ent
`productions can be associated with individual segments Cab of the
`
`I the open set “DA is the domain of legal parameter values of
`modules of type A,
`
`l the set C4 = CA1 U U C4,“ is the boundary of‘Da.
`consisting of nonintersecting segments Ca“ .
`.
`. ,CA." .
`
`I the function fat specifies a system of difi'erential equations
`that describe the continuous behavior of modules of type A
`in their domain of legal parameter values “DA .
`
`. ,pam} captures the
`a the set of productions P4 = {pa1.. .
`discrete behavior of modules of type A.
`
`A production that E P5 is applied when the parameters of a module
`M of type A reach segment (3,4" of the boundary CA. At this time
`module M disappears. and zero, one. or more descendant modules
`are created. The functions ham. embedded in productions 13A,:
`determine the initial values of parameters in the successor modules.
`
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`SIGGRAPH 93. lltnahoim1 gslitornia, 1_-6 August 199g
`
`1
`Fri
`Fro),
`\
`__b
`____
`_U_, —.>
`ctM‘i) WI \IF'G’
`
`For
`
`Figure 5: Initial steps in the construction of a dragon curve
`
`4 EXAMPLES OF (IL-SYSTEMS
`
`We will illustrate thenotion of adL-system using two sample models
`suitable for animating the development of the dragon curve and the
`filamentous alga Anabaeno carer-ado.
`
`4.1 A dL~systern model of the dragon curve
`
`In the discrete case. consewtive iterations of the dragon Curve
`(described. for example. in [27. Chapter 1]) can be obtained by the
`following parametric L-systern:
`
`Lu”: ——F‘.(1)
`p1:
`F..(s} -+ 443%) ++n{a:§%)-
`pa: Fits} —9 Hits?) — -Fi{5°§)+
`
`Assuming that symbols + and ~— reprcsent turns of 5:45“, this L-
`system encodes a Koch commotion [18, Chapter 6] that repeatedly
`substitutes sides of an isosceles right—angled triangle for its by-
`potenuse (Figure 5). Subscripts l and r- indicate that the triangle is
`formed respectively on the left or right side ofthe oriented produces»
`sor segment. A corresponding dL-systern that generates the dragon
`curve through the continuous progression of shapes indicated in
`Figure 6 is given below:
`
`inltlalstrlng: —-F.(l, 1}
`Elms):
`ifm<ssolve:—f=-%‘§§=G
`ifs = s produce -F.{O, 312;) + Fido. s) + FrflL 3y?)—
`Fl(c,s):
`tfz<ssolvefi—j=%,j—:=u
`its: = s produce +F,(D,s%} — F343, 5] — H(fl,s’;—E)+
`Fh(:e,s):
`at
`c
`i‘zbflleEd—t=—%I§£:U
`if: = 0 produces
`
`The operation of this model starts with the replacement of the initial
`module F. (l. 1) with the string:
`
`'FPUJI g) + Film! 1) + Flu]! J72.)__)
`which has the same turtle interpretation: 3 line segment of unit
`length ". Next. the horizontal line segment represented by module
`Fr. decreases in length with thespeed 2—: = —%. while the diagonal
`segments represented by modules F. and F} elongate with the speed
`is = 39%. The constantTdetermines [helifetime OthE. modules:
`after time T. the module Ft reaches zero length and is removed from
`
`the string (replaced by the empty string 5). while both modules F.
`and Fr reach their maximum length of 335 and are replaced by their
`respective successors. These successors subsequently follow the
`same developmental pattem.
`It is not accidental that the predecessor and the successor of the
`productions for F, (a:I s) and Fifi, s) have identical geometric in-
`terpretations. Since productions are assumed to be applied instan-
`taneously. any change of the model‘s geometry introduced by a
`production would appear as a discontinuity in the animation.
`In
`general, correctly specified productions satisfy continuity criteria
`[27, Chapter 6], which means that they conserve physics] entities
`such as shape, mass, and velocity of modules.
`
`4.2 A dL—system model of Anoboona catenula
`
`The. continuously developing dragon curve has been captured by
`a contest—free dL-systern. in which all productions and equations
`depend only on the strict predecessor module. A simple example
`of a context-sensitive model inspired by the development of the
`blue-green alga Anobacno entertain [3. 20. 27] is given below.
`Anabaeno forms a nonhranching filament consisting of two clue“
`of cells: vegetative cells and heterocyrtr. A vegetative cell usu-
`ally divides into two descendant vegetative cells. However,
`in
`some cases a vegetative cell differentiates into a heterocyst. The
`spacing between beterocysts is relatively constant. in spite of the
`continuing growth of the filament. Mathematical models explain
`this phenomenon using a biologically motivated hypothesis that
`the distribution of heterocysts is regulated by nitrogen compounds
`produced by the heterocysts, diffusing from cell to cell along the
`filament. and decaying in the vegetative cells.
`If the compound
`concentration in a vegetative cell falls below a specific level. this
`cell differentiates into a heterocyst (additional factors are captured
`by more sophisticated models}. A model opaating in continuous
`time according to this description can be captured by the following
`dL-system:
`
`Inn-[Bl string: Fhlxmz. Cmnn}Fa (Zr-Luz, Cmo=JFh (Emma: cut-ea}
`F(z;,cg) < Fu(:,c] > F(:r,c‘.):
`twice“: 5: C}Cmin
`
`solvefif =T2‘gri" = l‘D-(ct+r:F -2c}—,uc
`“a: = emu 85 c ) cw“-n
`produce Fu(k$mzy c}F.,((l — mm, c}
`iIC 2 ‘Hindu
`produce Fn(£,c)
`Fh(:l:,c)i
`solve % = ”tam“ -— 2], if = rake“, - c)
`
`Vegetative cells E, and beterocysts Fr. are characterized by dieir
`length :5 and concentration of nitrogen compounds c. The differ-
`ential equations for the vegetative cell F.i indicate that while the
`cell length .1: is below the maximum value x..." and the compound
`concentration c is above the threshold cm... the cell elongates ex-
`ponentially according to the equation 11—: = rm, and the compound
`concentration changes according to the equation:
`dc
`E—D-(cr+c.-—2c}—uc.
`
`111:: mole interprets the first paterneter us the apparent length. and ignores the
`sccondparameter.
`
`The first term in this equation describes diffusion of the compounds
`through the cell walls Following Fir-kit law [5. page 404], the
`
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`COMPglEFt GRAPHICS Proceedings. Annual Conference Series. logo
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`//// l\t\\
`
`Figure 7: Diagrammatic representation of the development of An-
`abnena entertain. simulated using a dL-System with the constants
`set to the following values: 5cm” : 1. cm“ = 255. em... = 5.
`D : it : 0.03. 1- : 1.01. i: : 0.37. r: = 0.1.1": = 0.15. The
`development was recorded from L...“ = 200 to in,“ = 5?5 at
`the intervals or : 1. Developmental stages are shown as horizon-
`tal lines with the colors indicating the concentration c of nitrogen
`compounds. Dark brown represents cmtn; white represents emu.
`
`According to this paradigm, the evaluation can be viewed as a dyv
`rirtmt'c process governed by a scheduler: a part of the simulation
`program that monitors the state of the model. advances time. and
`dispatches the activities to he performed.
`In the absence of dis-
`crete events (productions). the scheduler repeatedly advances time
`by the time .rtr'tre All During each slice. the differential equations
`associated with the modules are integrated numerically (using an
`integration technique appropriate for the equations in hand). thus
`advancing the state of the structure from p(t) to ntt + Art). Ifthe
`scheduler detects that a discrete event should occur (Le. a produc-
`tion should be applied) at time t" within the interval [l, t + At). this
`interval is divided into two subintervais [t, t’) and [t’. t + At). The
`differential equations are integrated in the interval [t. t') and yield
`parameter values for the production application at time t'. The pro—
`duction determines the initial values for the differential equations
`associated with the newly created modules; these equations are in,
`tegrated in the remaining interval [t’, t + At). Each ofthe intervals
`[L t”) and [t’, t. -+ At) is subdivided further if more discrete events
`occur during ll, t + At).
`Plant structures generated using dL-systems may consist of large
`numbers (thousands) of modules. If many modules are replaced at
`different times t." during the interval [L i‘. + At). the global advance—
`ment of time may require an excessive subdivision of this interval.
`leading to a slow evaluation of the model. This problem can be
`solved by detecting and processing events the interval [t. t + At)
`individually for each module. The increase of simulation speed is
`obtained at the expense of accuracy. since the state of the context
`of a module replaced at time if E (Li 4— Al) must be approxi-
`mated. l‘or example, by its state at time t. No accuracy is lost in the
`contextsfree case.
`
`[n the above description we assumed that the scheduler is capa-
`ble of detecting each instant t' at which a discrete event occurs.
`If the differential equations are sufficiently simple, we can solve
`them analytically and determine time t' explicitly.
`in general. we
`need numerical techniques for special event location in piecewise-
`continuous ordinary differential equations. as described by Shams
`pine et. at. [29]. and Barzel [2. Appendix Cl.
`
`355
`
`
`
`Figure 6: Development of the dragon curve simulated using it til.-
`system, recorded in time intervals At 2 iii". Top left: Superim-
`posed stages 0 — 8. top right: stages 3 ~— 16. bottom row: stages
`16 — 24 and 24 — 32.
`
`rate of diffusion is proportional to the differences of compound
`concentrations. c, — c and c; — ct. between the neighbor cells and
`the cell under consideration. The term in: describes exponential
`decay of the compounds in the cell.
`In addition to the differential equations, two productions describe
`the behavior of a vegetative cell.
`If the cell reaches maximum
`length arm” while the concentration C is still above the threshold
`cm...“ the cell divides into two vegetative cells of length harm” and
`(l — elem“. with the compound concentration c inherited from
`their parent cell. Otherwise. if the concentration c drops down to
`the threshold Cmin . the cell differentiates into a heterocyst. Both
`productions satisfy the continuity criteria by conserving total cell
`length and concentration of nitrogen compounds.
`The last line of the model specifies the behavior of the heterocvsts.
`Their length and compound concentration converge exponentially
`to the limit values of arm“ and cm”. The heterocysts do not
`undergo any further transformations.
`
`Simulation results obtained using the above model are shown in
`Figure 7. The cells in the filament are represented as horizontal
`line segments with the colors indicating the concentration of nitro-
`gen compounds. Consecutive developmental stages are drawn one
`under another. An approximately equal spacing between the hete-
`rocysts (shown in white} is maintained for arty horizontal section.
`as postulated during model formulation.
`Note that for incorrectly chosen constants in the model, the spacing
`between heterocysts may be distorted: for example. groups of ad‘
`jaccnt vegetative cells may almost simultaneously differentiate into
`heterocysts.
`
`5 EVALUATION OF clL-SYSTEMS
`
`Although Figures ti and 7 were obtained using dLvsystems, we have
`not yet discussed the techniques needed to evaluate them. This
`term denotes the calculation of the sequence of strings ,uftJ) 2 up.
`pffin‘.) 2 in.
`,ufnétt] : ,rr” representing the states ofthc
`modeled structure at the desired intervals At. We address the prob—
`lem of dL-system evaluation in the framework of the combined
`discrete/continuous paradigm for system simulation introduced by
`Fahrland [6] and presented in a tutorial manner by Kreuti'ter [l5].
`
`355
`
`355
`
`

`

`
`SIGGRAFiH__93. Anaheim. California. 1—6_August 1993
`
`2mm:
`=03
`
`:01
`
`Z:1.0
`
`2.0
`
`2..., at
`
`3 .
`mm
`
`L1”lli$
`
`Figure 3: Examples of sigmoidal growth functions. a) A family of
`logistic functions plotted using r = 3.0 for different initial values
`on. b] A. cubic function G333".
`
`5 GROWTH FUNCTIONS
`
`Growth fimctt'rmr describe continuous processes such as the ex—
`pansion of individual cells. elongation of internodes. and gradual
`increase of branching angles over time. For example. the differ—
`ential equations included in the dL-system for the dragon curve
`(Section 4.1] describe linear elongation of segments F.- and Fr. and
`linear decrease in length of segments Fit. The dlxsystern model of
`Anaboenc {Section 4.2) assumes exponential elongation of cells.
`In higher plants, the growth functions are often of sigmoidcl (S-
`shaped) type. which means that they initially increase in value
`slowly, then accelerate, and eventually level off at or near the maxi-
`mum value. A popular example of a sigmoidal function is Velhurst’s
`logistic funCfion (of. [5. page 212]}. defined by the equation:
`1:
`d£=r(1— )2
`manna:
`dt
`with a properly chosen initial value no {Figure 8a). Specifically, so
`must be greater than zero. which means that neither the initial length
`nor the initini grown} rate of a module described by the logistic
`function will be equal to zero.
`In order to obtain a continuous
`progression of forms. it is often convenient to use a growth function
`that has zero growth rates at both ends of an interval 5" within
`which its value increases from amen (possibly zero) to am... These
`requirements can be satisfied. for example. by a cubic function of
`time. Using the Hermite form of curve specification [8. page 434].
`we obtain:
`
`“be
`I“)= —2T—:ha+ air—2:2 +$minr
`where Ar: = a...“ — 3min audt 6 [0,7']. The equivalent differ-
`ential equation is:
`
`at:
`Air;
`As:
`A:
`t
`E—-—5?§t2 +5——t- 5—7(1 Fig-)3
`with the initial condition to = am... In order to extend this curve
`to infinity (Figure 8b), we define:
`A:
`s—T-r (1
`[I
`
`__ 1
`5.):
`
`fortE[||J.T]
`fort E (T, +00).
`
`d1: —Gs=rill“
`dt
`
`Although the explicit dependence of the function G on time is
`questionable from the bioloy'ca] point of view {a plant module does
`not have a means for measuring time directly). parametric cubic
`functions constitute a well understood computer graphics tool {3.
`Chapter 112] and can be conveniently used to approximate the
`observed changes of parameter vatues over time.
`356
`
`356
`
`Figure 9: Development of a compound leaf simulated using a db
`system. Parameter values are: no : 4. on = 1.0. on. = 2.0, k =
`0.5. r, = 2.0. zamu = 3.0. r.- = LB, 2....“ = 3.0, an = 0.05.
`r'. = 2.0. s...” = 6.0. oo = 2.0. re. = 1.0. or...” = 60.0. and
`at = till}. The stages shown represent frames 50. 215. 300, 400.
`500,600,1111d9000fansniruated sequence.
`
`7 MODELING OF HIGHER PLANTS
`
`In this section we present sample applications of dis-systems to the
`mfintation of the development of higher plants.
`
`7.1 Pinnate Leaf
`
`A pinnate leaf provides a simple example of a atonement! branch-
`ing strucnn-e. Mompadr'al branching occurs when the apex of the
`main axis produces a succession of nodes bearing orgmr -— leaves
`or flowers — which are separated by imemodes.
`1n the case of
`plnnate leaves with the leaflets occurring in pairs (termed opposite
`arrangement). the essence of this process can be unpaired by the
`L—system production [27. page 71]:
`
`E. —> Fi[+@L][—@L]F..,
`
`where F. denotes the apex, Fr — an intemode. and @L — a
`leaflet. The dL-system model given below extends this Irsystern
`with growth functions that control the expansion of all components
`and gradually increase branching angles over time.
`
`initial sit-tug: F..{zn, no)
`F.{c.n) :
`ifs: < z”.
`
`z
`in...“ J 2"“d:
`
`=
`
`a
`
`solve—T: =r.. [I -
`if: = nu. 8!.
`11. > U
`Fromm Ft UM}[+(00}9L{8031[*(0=0)@L(30ll
`F.{(1- Home — I}
`if: = 3“; EL ‘5 = U
`
`produce F'rixifiusa)
`Fife): solveIiT: 2r. (1-
`‘
`zit-san- J I
`
`Lfs) :
`"if" ) 5
`sulve—‘7: = r. (l —
`Gran:
`in)”
`solved“3 =r° (l —
`
`:l:(ttr] :
`
`The apex F, has two parameters a and n which indicate its current
`length and the remaining number of inlemodes to be produced. The
`apex elongates according to the logistic function with parameters
`r (controlling growth rate) and on...“ (controlling the asymptotic
`apex length). Upon reaching the threshold length not. the 313“
`produces a pair of leaflets @L and subdivides into an internude
`F. of length hr: and a shorter apex of length {1 — He. Once the
`predefined number no of leaf pairs haVe been created.
`the apex
`
`356
`
`

`

`
`
`Figure 10: Development of the herbaceous plant Campanut'n m7
`pnncutoides. The snapshots show every '25:“ll frame of a computer
`animation. starting with frame 115.
`
`transforms itselfinto an lnternode and produces the terminal leaflet.
`The length of ititemodes, the size of leaflets. and the magnitude of
`the branching angles increase according to the logistic functions.
`Snapshots of the leai' development simulated by the above model
`are shown in Figure 9.
`
`7.2 Campanuia rapuncutofdes
`
`The inflorescence of Cantpnnnia ropuncutoia‘eo' (creeping bell—
`fiower) has a monopodial branching structure similar to that of a
`pinnate teal“. consequently, it is modeled by a similar dL-system:
`
`initial string: Fa (5:0, no)
`Ellen) :
`ifll.’ < am,
`
`produce Rtkr)[+(ag)@K]Fs((l — k)x,n — 1)
`ifzr.‘ =Jmt.
`3.! n: D
`produce Eta-hilt?
`solve fl—f : Small}
`solve ”Ii—E; : Gambit)
`
`Fl-(an) :
`+(a):
`
`The apex is assumed to grow at a constant speed. Cubic. growth
`functions describe the elongation of intemodcs and the gradual in-
`crease of branching angles. The combination of the linear growth
`of the apex with the cubic growth of the internodes results in first-
`ordcr continuity of the entire plant height (except when apex F“ is
`transformed into internode F. and terminal flower ©K).
`
`
`
`_§OI’yiPUTER GRAffi