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1. THE L C PLANT MODELLING LANGUAGE 41 L C is well suited to the specification of models that incorporate a single aspect of plant function within a developing plant structure An open problem is the construction of comprehensive models that incorporate several aspects such as genetics partitioning of different resources hormonal control biomechanics and development The challenge is to devise a methodology supported by appropriate language constructs that would make it possible to build such models in a well structured manner with different model components specified implemented and tested independently and easily combined into final synthetic models ACKNOWLEDGEMENTS We thank Lynn Mercer Jim Hanan and Alla Seleznyova for comments on the manuscript The support of this research by the Human Frontier Science Program and the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged REFERENCES Allen M T Prusinkiewicz P and DeJong T M 2005 Using L systems for modeling source sink interactions architecture and physiology of growing trees the L PEACH model New Phytologist 166 3 869 880 Fournier M 1989 M canique de l arbre maturation poids propre contraintes climatiques Institut National Polytechnique de Lorraine Nancy Ph D dissertation Institut National Polytechnique de Lorraine Frijters D 1976 An automata theoretical model of the vegetative and flowering development2. scan the branch in the distal direction Given the adjusted orientation H of internode i first find the vector H that forms angle 7 with respect to H Figure 1b The vector E H 41 H is the difference between the orientation of internode i 1 calculated in the previous iteration step and the orientation that would be required to achieve equilibrium The adjusted orientation of internode i 1 is then defined as H we Normalize H 41 KE Parameter k controls the amount of adjustment and thus affects the speed of convergence to the solution and function Normalize restores the result to the unit length Furthermore the magnitudes of difference vectors are accumulated to form an error measure N S i 0 36 P PRUSINKIEWICZ ET AL error S maxerror n gt MAX NR PropagateLeft PropagateRight gt Grow gt Stop Figure 2 The sequence of phases in the simulation of a growing and bending branch Similar graphs are useful when defining the sequence of phases in other L C models as well The iteration ends when the error e falls below an assumed tolerance threshold Within this tolerance the orientations of the internodes and positions of the nodes then satisfy Equations 5 In the complete model of a growing branch this is an appropriate time to simulate a developmental step The cycle of computation is illustrated in Figure 2 The resulting L C progra
3. e a graphical browser for organizing and accessing models on both local and remote machines Prusinkiewicz 2004 The simulation programs cpfg and lpfg are at the heart of both L studio and Vlab Their design has been guided by two key objectives a the programs should be suitable for modelling and simulating a wide range of structures and developmental processes in plants and b the programs should support diverse visualization techniques from schematic to realistic These objectives are addressed by allowing users to specify models in specialized programming languages which are based on the formalism of L systems Lindenmayer 1968a 1968b 1971 The modelling language for the cpfg simulator was developed first Prusinkiewicz and Hanan 1990 Prusinkiewicz and Lindenmayer 1990 Hanan 1992 and makes it possible to specify simple models quickly and concisely The evolution of the cpfg 27 J Vos L F M Marcelis P H B de Visser P C Struik and J B Evers eds Functional Structural Plant Modelling in Crop Productione 27 42 2007 Springer Printed in the Netherlands 28 P PRUSINKIEWICZ ET AL language has been surveyed by Prusinkiewicz 1999 with subsequent additions in particular decomposition and interpretation rules described by Prusinkiewicz et al 2000 The currently available constructs have been listed by M ch et al 2005 The language of the Ipfg simulator was designed more recently Karwowski 2002 Karwowski and
4. the derivation direction Productions with a new context field inconsistent with the current derivation direction are ignored FORMAT OF L C PRODUCTIONS The productions considered in the previous section were specified using a mathematical L system notation In L C the arrow is replaced by the keyword produce which leads to production specification of the form B m lt lt A n produce B m 1 In general an L C production has the syntax predecessor production body where the predecessor is specified as discussed in the previous section and the production body is a block of C code including one or more statements produce parametic string op Although the mathematically inspired arrow notation may appear more elegant than the L C notation the latter is more flexible as a programming construct For example an L C production may include several alternative successors as in the construct if m 3 0 produce C m B m else produce B m In addition to assigning consecutive numbers to consecutive modules B m this production inserts a module C m in front of every B m each time m is divisible 30 P PRUSINKIEWICZ ET AL by 3 The traditional notation would require two separate productions to express the same idea which might be inefficient if the computations preceding production application were much longer than the simple incrementation m L C also supports a variant of the produce statement denoted by ke
5. this production is difficult to read The corresponding L C code is struct data float x1 x2 x3 x4 x5 module A data module L data module R data j L dl lt A d gt R dr d x5 1 0 produce A d In this example as in most simple programs the L C code is longer than the equivalent cpfg code Nevertheless the use of structured module parameters offers several advantages THE L C PLANT MODELLING LANGUAGE 31 e Long lists of parameter modules can be avoided This results in a more legible code e The L C code is less error prone In cpfg it is easy to introduce an error by inadvertedly skipping a parameter in a long parameter list e The L C code is easier to modify For example if an additional parameter x6 is needed to characterize modules L R and A in L C it suffices to extend the definition of the structure data In contrast in cpfg it is necessary to include x6 explicitly in the parameter list associated with each occurrence of these modules CONTROL OF L SYSTEM PROGRAM EXECUTION In principle the notion of L systems leads to a declarative programming style Each production is a statement of the form If a module and its neighbours match the production predecessor then subsequent actions will be performed as specified in the production body These actions thus depend on the state and context of the module to which they apply rather than a control flow mechanism as found in imperative languages Nev
6. vgroup statements concerns the execution of the affected productions In addition to group 0 only one THE L C PLANT MODELLING LANGUAGE 33 production group specified by the latest call to the UseGroup function applies to any particular simulation step In contrast interpretation rules in several visual groups can be executed in each simulation step provided that windows associated with these groups are open An L C programmer may control which windows are open using the function call UseView view Additional constructs are provided in L studio and Ipfg to determine the default size and position of the windows and to open and close them using menus INTERACTION WITH THE MODELS Computational models are often used in simulated experiments in which model attributes are modified to address what if type questions Models expressed in L C are conducive to such experiments since the user can modify any aspect of the model by changing the L system code In addition L studio and vlab provide interactive tools that provide an interface for manipulating the numerical parameters and functions incorporated in the model These tools include user configurable virtual control panels and graphical function editors Prusinkiewicz 2004 The user may also explore models by directly manipulating their visual representations on the screen Such manipulations may mimic physical operations such as pruning girdling or pulling branches The fundame
7. with the new context that are incompatible with the current direction of derivation However in many models it is convenient to control the applicable production set explicitly Frijters and Lindenmayer 1974 Frijters 1976 To this end L C supports statements of the form group group_id which divide the set of productions into subsets called groups or tables in L system theory cf Rozenberg 1973 Ginsburg and Rozenberg 1975 A group statement can be inserted before any production and assigns a numerical label group_id to the 32 P PRUSINKIEWICZ ET AL subset of productions that follow This label remains in effect until the next group statement or the end of the production list The group statements are used in conjunction with the function UseGroup group_id which is typically called within the StartEach block and defines the group of productions to be used in the forthcoming step By definition productions in group 0 are used in all steps The notion of groups lends itself to the division of the simulation into a sequence of phases each characterized by the use of a specific production group The sequence of phases that constitute a simulation may be fixed but it may also be determined dynamically with the next phase depending on the outcome of the previous phase For example such a situation may occur if L system productions are applied iteratively until some criterion of convergence is met It may then be desirable to inte
8. CHAPTER 3 THE L C PLANT MODELLING LANGUAGE P PRUSINKIEWICZ R KARWOWSKI AND B LANE Department of Computer Science University of Calgary Calgary Canada Abstract L C is a modelling language that combines features of L systems and C It extends the L system formalism with the notion of fast transfer of information and supports a number of standard programming constructs absent from its predecessor the cpfg language These include modules with structured parameters productions with multiple successors and user definable functions Visualizations of L system models can be enhanced using multiple views and the selective display of frames These features extend the overall range of simulation models that can be conveniently expressed using L systems and are particularly advantageous when creating and visualizing complex plant models A biomechanical model of a growing pendulous branch is given as the key example INTRODUCTION L studio Windows and the Virtual Laboratory Linux and Mac are related plant modelling packages distributed by the University of Calgary Canada Each of the L studio and Vlab systems consists of a two L system based simulation programs cpfg and lpfg b a modelling environment that provides auxiliary modelling tools and a graphical interface for creating and manipulating models c a library of programs for simulating environmental processes that affect plant development d a set of sample models and
9. Prusinkiewicz 2003 Karwowski and Lane 2006 to facilitate the specification of complex plant models e g Miindermann et al 2005 Allen et al 2005 We call this language L C because it combines features of L systems and the C programming language Here we overview key features of L C in a manner that complements and updates its earlier presentation by Karwowski and Prusinkiewicz 2003 and we illustrate the discussed features in the context of a complete L C program a model of a growing branch that bends due to gravity FAST INFORMATION TRANSFER An essential component of functional structural plant models is transport of information through the organism Perttunen et al 1996 Allen et al 2005 A simple L system example would be assigning consecutive numbers to a sequence of modules in a string This task can be accomplished using the context sensitive production B m lt A n gt B m 1 as illustrated below for a string of N 4 modules B 0 A Q A 0 A Q B 0O B l A 0 A O B 0 B 1 B 2 A Q B 0 B 1 B 2 B 3 We observe that it takes N 1 derivation steps for the information to propagate to the end of an N element string as indicated by the arrows This is a consequence of determining the next state of each module as a function of the current state of its left neighbour Although each module in the string is rewritten in each derivation step the state of only one module is changed The remaining modules are not affecte
10. angle alpha V3 VecRotate V3f a float alpha V3 c c xX a x cos alpha a y sin alpha c y a x sin alpha a y cos alpha Og 05 return c Analogous definitions of cross product vector length and vector normalization should go here Declarations of structures modules and variables struct InternodeData float s internode length float mass node mass float kappa rotational spring constant float sigma_mass total mass to the right float torque torque from masses to the right V3 E P proximal node position V3 H internode orientation module Internode InternodeData module Apex int int number of internodes module Axes for visualizing coordinate axes initial internode data computation phase distance from equilibrium current color index InternodeData iid int Phase float error int color Enumeration of computation phases define PropagateLeft 1 accumulate masses torques define PropagateRight 2 update angles positions define Grow 3 append internode Organization of computation Start At the beginning of simulation initialize non zero variables iid s S internode length iid mass MASS node mass iid kappa KAPPA vot spring constant iid H x 1 internode orientation 38 P PRUSINKIEWICZ ET AL Phase PropagateLeft initial phase
11. color 1 initial color index UseView Branch Display view Branch UseView Torques Display view Torques StartEach At the beginning of simulation step set group and derivation direction UseGroup Phase set current group if Phase PropagateLeft depending on the phase Backward derive right to left else ox Forward left to right error 0 Also clear cumulative error EndEach At the end of simulation step determine next phase switch Phase consider prev phase case PropagateLeft after propagating left Phase PropagateRight propagate right break case PropagateRight after propagating right if error gt maxerror if error above limit Phase PropagateLeft propagate left else otherwise DisplayFrame display branch Phase Grow and grow break case Grow after growing Phase PropagateLeft propagate left color increment color index break derivation length 1000000 Axiom Axes Internode iid Apex 0 Accumulate masses and torques using fast information transfer to the left group PropagateLeft Internode id gt gt Internode idr id sigma_mass id mass idr sigma_mass id torque idr torque VecCrossProd idr H id sigma_mass Gravity z produce Internode id Update node angles and internode positions using fast information transfer to the right grou
12. d L C includes a construct that makes it possible to accelerate this information transfer if the string is rewritten sequentially from left to right during a single derivation step Under this assumption the next state of a module can be calculated as a function of the just calculated next state of its left neighbour rather than the current state of that neighbour The flow of information is then represented by the arrows in the following derivation step B O A O A 0 A 0 B O B 1 B 2 B 3 SANA LS THE L C PLANT MODELLING LANGUAGE 29 In order to refer to this new context in L system productions we use the symbol lt lt A production that defines the above derivation can thus be written as B m lt lt A n gt B m 1 In general the predecessor of a production using new context in a derivation proceeding from left to right has the following format new left context lt lt strict predecessor gt right context In an analogous manner if a derivation step is performed from right to left a production predecessor with new context has the general format left context lt strict predecessor gt gt new right context The Jleft context and right context fields are optional Motivated by the above example we refer to L systems that use the new context construct as L systems with fast information transfer Note that a production may only have one new context field i e either lt lt or gt gt depending on
13. ertheless L C also includes elements of the imperative programming style The body of each production is specified as a sequence of statements based on C Furthermore there are four blocks of statements that are performed at specific points of the derivation at the beginning of the simulation Start at the beginning of each derivation step StartEach at the end of each step EndEach and at the end of the simulation End These blocks were already defined for cpfg Hanan 1992 but play a more significant role in L C because they may include statements that affect the flow of the simulation One such pair of statements are calls to the predefined functions Forward and Backward These functions are typically used within the StartEach block and determine whether the derivation will proceed left to right or right to left in the forthcoming step As we have seen in the section Fast information transfer the direction of the derivation is of critical importance in the case of fast information transfer because left new context can only be used when the derivation proceeds left to right and right new context can only be used when the derivation proceeds right to left As the applicability of productions with new context depends on the direction of the derivation only a subset of the production set may apply in a given derivation step In the case of fast information transfer this subset may be established implicitly by ignoring productions
14. ki R and Lane B 2006 LPFG user s manual http algorithmicbotany org Istudio LPFGman pdf Karwowski R and Prusinkiewicz P 2003 Design and implementation of the L C modeling language Electronic Notes in Theoretical Computer Science 86 2 1 19 Lindenmayer A 1968a Mathematical models for cellular interaction in development Part 1 Filaments with one sided inputs Journal of Theoretical Biology 18 3 280 299 Lindenmayer A 1968b Mathematical models for cellular interactions in development Part 2 Simple and branching filaments with two sided inputs Journal of Theoretical Biology 18 3 300 315 42 P PRUSINKIEWICZ ET AL Lindenmayer A 1971 Developmental systems without cellular interaction their languages and grammars Journal of Theoretical Biology 30 455 484 Lindenmayer A 1974 Adding continuous components to L systems Jn Rozenberg G and Salomaa A eds L Systems Springer Verlag Berlin 53 68 Lecture Notes in Computer Science no 15 M ch R 2005 CPFG version 4 0 user s manual Available http algorithmicbotany org Istudio CPFGman pdf 30 May 2006 Miindermann L Erasmus Y Lane B et al 2005 Quantitative modeling of Arabidopsis development Plant Physiology 139 2 960 968 Perttunen J Siev nen R Nikinmaa E et al 1996 LIGNUM a tree model based on simple structural units Annals of Botany 77 1 87 98 Press W H Teukolsky S A Vetterling W T et a
15. l 1992 Numerical recipes in C the art of scientific computing 2nd edn Cambridge University Press Cambridge http www nrbook com a bookcpdf php Prusinkiewicz P 1999 A look at the visual modeling of plants using L systems Agronomie 19 3 4 211 224 Prusinkiewicz P 2004 Art and science for life designing and growing virtual plants with L systems Acta Horticulturae 630 15 28 Prusinkiewicz P and Hanan J 1990 Visualization of botanical structures and processes using parametric L systems In Thalmann D ed Scientific visualization and graphics simulation Wiley Chichester 183 201 Prusinkiewicz P Hanan J and M ch R 2000 An L system based plant modelling language In Nagl M Schiirr A and Miinch M eds Applications of graph transformation with industrial relevance international workshop AGTIVE 99 Kerkrade The Netherlands September 1 3 1999 Springer Berlin 395 410 Lecture Notes in Computer Science no 1779 Prusinkiewicz P and Lindenmayer A 1990 The algorithmic beauty of plants Springer Verlag New York Roberts J C 2000 Multiple view and multiform visualization Jn Erbacher R Pang A Wittenbrink C et al eds Visual data exploration and analysis VII San Jose CA 24 26 January 2000 IS amp T Bellingham 176 185 Proceedings of SPIE no 3960 Rozenberg G 1973 TOL systems and languages Information and Control 23 4 357 381 Taylor Hell J 2005 Biomechanics in b
16. lor color LineTo3f id P 165 166 167 ea 5 draw coordinate axes 168 169 nproduce SetColor 255 SB F 20 EB 170 produce SB Right 90 F 25 EB 171 The Grow group consists of two productions The first production lines 133 140 describes the addition of a new internode to the branch and stops the simulation once the maximum branch length has been reached The second production lines 142 146 handles interaction with the model If the user clicks on an internode the symbol MouseIns is automatically inserted before the corresponding Internode module in the L system string The production handles this situation by incrementing the mass of the internode threefold and removing the MouselIns symbol from the string This operation can be used to investigate the impact of a fruit load on the branch shape for example 40 P PRUSINKIEWICZ ET AL The remainder of the code lines 148 171 is devoted to the visualization of simulation results Two windows are used for this purpose The first window is associated with the vgroup Branch and shows the shape of the growing branch The second window is associated with the vgroup Torques and shows the distribution of torques along consecutive nodes of the branch These views are obtained using different interpretations of the same module Internode The last tule with predecessor Axes defined in the axiom plays an auxiliary role of displaying coordinate axes in the plo
17. m is listed on the following pages The program begins with the definitions of constants variables and functions and declarations of data structures and modules lines 1 to 53 The three dimensional vector V3 which appears for the first time in the definition of the gravity vector line 10 is one of the types declared in the L C header file lpfgall h An example of a user defined function follows lines 12 22 Lines 55 97 present a typical example of the organization of computation The statements in the Start block lines 57 67 initialize variables set the initial phase of the computation and activate two different views of the model at the beginning of the simulation The statements in the StartEach block lines 69 77 specify the production group and derivation direction to be used in the forthcoming simulation step Finally the statements in the EndEach block lines 79 97 determine the sequence of simulation phases according to Figure 2 As the number of iterations needed to reach the equilibrium is not known in advance the results are displayed on demand using the DisplayFrame function line 89 once the error drops below a threshold value maxerror line 86 For similar reasons the number of simulation steps needed for the branch to grow to a desired length is not known in advance The simulation is thus terminated by a call to the Stop function once the branch has reached the maximum prescribed length lines 136 139 The deri
18. ntal operation underlying these manipulations is the selection of a module within the graphical representation of the model When the user selects a module with the mouse a reserved module MouselIns in L C is automatically inserted before the symbolic representation of the selected module in the L system string The modeller specifies the response to this event using a production that includes Mouse Ins in the predecessor A BIOMECHANICAL EXAMPLE To illustrate L C constructs in the context of a complete program let us consider the biomechanical model of a growing pendulous branch proposed by Jirasek et al 2000 according to the ideas of Fournier 1989 Jirasek et al observed that the forces and torques involved in the bending as well as the resulting reorientations and displacements of internodes can be considered signals that propagate between plant modules and have local effects This observation led to L system implementations of the biomechanical model first in cpfg Jirasek 2000 and later in L C Taylor Hell 2005 The L C implementation makes use of almost all the programming constructs specific to this language and therefore provides a good example of its features In the model below we ignore for simplicity the effects of tropisms and secondary growth We also assume that the model operates in 2D 34 P PRUSINKIEWICZ ET AL Figure 1 Geometry of branch bending Left representation of a branch with N 5 internodes and
19. of Hieracium murorum L Biological Cybernetics 24 1 1 13 Frijters D and Lindenmayer A 1974 A model for the growth and flowering of Aster novae angliae on the basis of table lt 1 0 gt L systems In Rozenberg G and Salomaa A eds L Systems Springer Verlag Berlin 24 52 Lecture Notes in Computer Science no 15 Ginsburg S and Rozenberg G 1975 TOL schemes and control sets Information and Control 27 2 109 125 Hanan J S 1992 Parametric L systems and their application to the modeling and visualization of plants University of Regina Regina Ph D dissertation University of Regina http algorithmicbotany org papers hanan dis 1992 pdf Jirasek C A 2000 A biomechanical model of branch shape in plants expressed using L systems University of Calgary Calgary M Sc thesis University of Calgary Jirasek C A Prusinkiewicz P and Moulia B 2000 Integrating biomechanics into developmental plant models expressed using L systems In Spatz H C and Speck T eds Plant biomechanics 2000 proceedings of the 3rd plant biomechanics conference Freiburg Badenweiler August 27th to September 2nd 2000 Thieme Stuttgart 615 624 http algorithmicbotany org papers biomechanics00 pdf Karwowski R 2002 Improving the process of plant modeling the L C modeling language University of Calgary Calgary Ph D dissertation University of Calgary http algorithmicbotany org papers radekk dis2002 pdf Karwows
20. otanical trees University of Calgary Calgary M Sc thesis University of Calgary http algorithmicbotany org papers juliath th2005 pdf
21. p PropagateRight Internode idl lt lt Internode id V3 NewEquilVector VecRotate idl H id torque id kappa V3 DifferenceVector NewEquilVector id H error VecLength DifferenceVector id H VecNormalize id H relax DifferenceVector THE L C PLANT MODELLING LANGUAGE 39 125 id P idl P id s id H 126 produce Internode id 127 128 129 Append new internode unless maximum number reached 130 Handle interaction with the model 131 132 group Grow 133 Internode id lt Apex n 134 135 id P id s id H Calculate new internode position 136 if n lt MAX_N if internode limit not exceeded 137 produce Internode id Apex n 1 append 138 else otherwise 139 Stop terminate simulation 140 141 142 MouseIns Internode id If node selected by mouse 143 144 id mass 3 increase its mass 3 times 145 produce Internode id 146 147 148 Visualize simulation results 149 150 interpretation 151 group 0 display irrespective of phase 152 153 vgroup Branch specification of view Branch 154 Internode id draw an internode as a line 155 and a circle 156 nproduce ea a 157 produce LineTo3f id P Circle id mass 158 159 160 vgroup Torques specification of view Torques 161 Internode id extend the plot of torques 162 163 id P y 0 045 id torque 164 produce SetCo
22. rpret and visualize the result of a simulation graphically only after convergence has been achieved To this end L C supports the function DisplayFrame which is typically called within the EndEach block to display the result of the latest simulation step Furthermore as the number of iterations may be difficult to define a priori L C supports the function Stop which terminates the execution of the simulation at the end of the derivation step in which it has been called Examples of the constructs discussed in this and the following sections will be given in the context of a complete L C model Section A Biomechanical example MULTIVIEW VISUALIZATION In some applications it is useful to display different aspects views of a simulation concurrently Roberts 2000 For example one view may realistically represent a developing plant while another shows corresponding statistical information in the form of a dynamically updated table or a histogram In L C different views can be displayed in separate windows the contents of which are specified using subsets of interpretation rules Prusinkiewicz et al 2000 Each subset is called a visual group and is identified by the statement vgroup view A vgroup statement assigns the label view to the subset of interpretation rules that follow it A visual group is terminated by the next vgroup statement or the end of the production list An important difference between the group and
23. t of torques With a few additional lines of code we could also place numerical scales and labels on the axes A snapshot of simulation results is shown in Figure 3 Figure 3 Visualization of the model of a growing and bending branch The simulation was run in a stroboscopic mode in which consecutive simulation stages are superimposed Left view changes of the branch shape over time The mass of the node represented by the enlarged circle has been increased interactively Right view distribution of torques along the branch Grey levels represent developmental stages lengths of the branch and visually associate branch shapes with the corresponding torque plots CONCLUSIONS The most significant conceptual advancement in L C compared to previous L system based languages is fast information transfer It significantly speeds up many simulations especially those of functional structural and biomechanical models which rely on the propagation of hormones resources or mechanical forces through the plant In addition the specification of complex models is facilitated through the use of modules with structured parameters and the division of productions into groups L C also provides the modeller with the wealth of programming constructs available in C The biomechanical model of a growing pendulous branch presented in the section A Biomechanical example illustrates the use of these features in the context of a complete L C program
24. the symbols used in the derivation of the formula for static equilibrium Right the adjustment of internode orientation during the relaxation process The point of departure in the model construction is the derivation of equations that characterize branch shape in static equilibrium We represent the branch as a sequence of internodes Figure la beginning with a fixed internode and terminated by an apex The internodes are numbered from 1 fixed proximal internode to N distal internode and connect nodes numbered 0 to N Each internode is represented as a vector r s H where the number s denotes internode length and the unit i vector H denotes internode orientation The position of node j with respect to node i 0 Si lt j lt N is thus described by the vector J R j r 1 k i l We assume that the branch would be straight in the absence of gravity but bends downward in the presence of gravity To model this bending we assume that each node i is an elastic joint subject to a torquet This torque is caused by gravity acting on the overhanging part of the branch positioned distally with respect to node i To simplify calculations we assume that the mass of the branch is concentrated in its nodes Let F m g denote the force acting on node i with mass m under gravitational acceleration g The torque 1 is the sum of the torques exerted by the individual masses positioned distally with respect to node i N t DR
25. vation length statement line 99 serves as a safeguard specifying an upper limit on the number of steps The Axiom statement line 100 specifies the initial structure First is the Axes module which is used to draw coordinate axes in the Torques view Following it is the branch which initially consists of a single internode followed by an apex The L system productions are divided into three groups The PropagateLeft and PropagateRight groups lines 102 127 with a single production each iteratively compute the shape of the branch according to the mathematical analysis presented earlier in this section The conciseness of these productions illustrates the expressive power of key constructs of L C context sensitive productions fast information transfer modules with structured parameters and vector operations THE L C PLANT MODELLING LANGUAGE 37 WODIHUAPWNH OAAAAAR ASA CODINHUARWNE include lt math h gt C C math header include lt lpfgall h gt declarations L C variables structures functions etc const int MAX N 30 max number of internodes const float S 1 0 internode length const float MASS 0 1 mass of a node const float KAPPA 4500 0 spring constant const float maxerror 0 01 error limit const float relax 0 5 relaxation coefficient const V3f Gravity 0 9 81 0 gravity vector Sample definition of functions on vectors Rotation of vector a in xy plane by
26. xF 2 j itl The above equation expresses torques in global terms in the sense that it incorporates influences of masses m that may be positioned far away from node i Nevertheless torques acting on consecutive joints can be related to each other in local terms Specifically the torque acting on node i 1 is equal to THE L C PLANT MODELLING LANGUAGE 35 N N N t UR xF Ra xF RG xF r xF X hr R xF ji j i l j i l 3 N N 41 x gt F R xF 41 xT 1 j i j i l where N N T XF F 9 F F_ T 4 j i 1 j i In static equilibrium a torque of magnitude 7 z acting on the node i rotates internode r by angle 7t with respect to internode r The parameter x is the rotational spring constant associated with node i The geometry of the branch is thus described by the equations H Rotate H Pa P s H l 5 where the function Rotate H changes the orientation of vector H by angle a in 2D and P is the position of mass mj To find the equilibrium we use a relaxation method Press et al 1992 pp 754 755 which in this case consists of an iterative application of two steps Step 1 Given a sequence of internodes r s H calculate torque 7 acting on each node To this end scan the branch in the proximal direction and apply Equations 3 and 4 to consecutive nodes Step 2 Given the torques 7 adjust the orientation of each internode To this end
27. yword nproduce In contrast to produce the execution of nproduce does not terminate the application of a production This makes it possible to construct the successor of a production piece by piece For example using nproduce the previous production can be simplified to the form B m lt lt A n M if m 3 0 nproduce C m produce B m STRUCTURED PARAMETERS AND MODULE DECLARATIONS In the above examples we have only considered modules with a single numerical parameter L C extends the previous definition of parametric L systems Lindenmayer 1974 Prusinkiewicz and Hanan 1990 Prusinkiewicz and Lindenmayer 1990 Hanan 1992 by allowing for the use of parameters of different types including user definable data structures To make this possible L C modules are declared before use Declaration specifies the number and types of parameters that are associated with the given module type with the following syntax module identifier parameter listyy To illustrate the usefulness of structured parameters let us consider the following context sensitive production in cpfg notation L x11 x12 x13 x14 x15 lt A x1 x2 x3 x4 x5 gt R xrl1 xr2 xr3 xr4 xr5 gt A x1 x2 x3 x4 x5 1 This production operates on a module A with five real valued parameters and increments the value of the last parameter by if module A appears in the context of modules L and R Note that due to the relatively high number of module parameters
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