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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 256 10 "L-syst¸mes" }}{PARA 
257 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 12 "Felice Rong
a" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 257 12 "Introduction" }}{PARA 0 "
" 0 "" {TEXT -1 345 "Le formalisme des L-ystemes est du a Aristid Lind
enmayer (1968), dont le but premier etait de modeliser la croissance d
es plantes. Il s'est revele par la suite que les L-systemes permettait
 d'engendrer une large classe d'objets fractals, c'est-a-dire de figur
es (dans le plan ou dans l'espace) dont une partie a une structure sem
blable au tout. " }}{PARA 0 "" 0 "" {TEXT -1 248 "Nous proposons ici u
n d'algorithme issu des L-systemes, qui convient pour engendrer des fi
gures fractales dans le plan, puis une variante qui convient pour mode
liser la croissance des plantes (qui seront aussi representees par des
 figures planes). " }}{PARA 0 "" 0 "" {TEXT -1 239 "Le mechanisme de b
ase consiste a se donner des regles pour engendrer une chaine de carac
teres (ou mot) comportant un certain nombre de symboles de base (alpha
bet), puis une interpretation des mots de facons a engendrer une image
 graphique." }}{PARA 0 "" 0 "" {TEXT -1 30 "Plus precisement on se don
ne :" }}{PARA 0 "" 0 "" {TEXT -1 35 "   1) un ensemble fini de symbole
s " }{TEXT 258 1 "V" }{TEXT -1 25 " (alphabet); par exemple " }{TEXT 
259 9 "V=\{f,p,m\}" }}{PARA 0 "" 0 "" {TEXT -1 70 "   2) un mot de dep
art, appele axiome ou seed en anglais; par exemple " }{TEXT 260 5 "fpf
pf" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 44 "   3) des regles de
 reecriture; par exemple " }{TEXT 261 19 "f -> fpfmfmffpfpfmf" }}
{PARA 0 "" 0 "" {TEXT -1 142 "   4) le nombre d'etapes, chaque etape c
onsistant a appliquer la ou les regles de reecriture, en commencant pa
r le mot de depart donne sous 2)" }}{PARA 0 "" 0 "" {TEXT -1 170 "   5
) l'interpretation graphique d'un mot. Generalement, cela se fait sous
 forme d'ordres a donner a une tortue qui se deplace sur une feuille d
e dessin ou sur un ecran. " }}{PARA 0 "" 0 "" {TEXT -1 14 "Par exemple
 : " }}{PARA 0 "" 0 "" {TEXT 262 44 "f => avancer d'une distance fixee
 en tracant" }{TEXT -1 136 " (cette distance dependra du nombre d'etap
es afin de ne pas sortir de la feuille de dessin - cela se fait automa
tiquement avec Maple).  " }}{PARA 0 "" 0 "" {TEXT 263 45 "p => rotatio
n d'un angle alpha donne d'avance" }{TEXT -1 73 " (p pour plus, le sym
bole + pouvant etre problematique dans un programme)" }}{PARA 0 "" 0 "
" {TEXT 264 31 "m => rotation d'un angle -alpha" }{TEXT -1 16 " (m pou
r moins)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
171 "L'image est engendree en transformant un certain nombre de fois l
e mot de depart selon les regles e reecriture, puis a faire interprete
r par la tortue le mot ainsi obtenu." }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 132 "References: \"Lindenmayer Systems\", p
ar P. Prusinkiewicz et J. Hanan, springer Verlag, Lecture Notes in Bio
mathematics no. 79 (1989)." }}{PARA 0 "" 0 "" {TEXT -1 153 "Le code Ma
ple a ete tire de : \"Maple, a comprehensive introduction\", par Roy N
icolaides et Noel Walkington, Cambridge University Press, Cambridge (1
996)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 266 10 
"Exercice 1" }{TEXT -1 63 ". Reprendre l'exemple donner ci-dessus en c
ours d'explication :" }}{PARA 0 "" 0 "" {TEXT -1 8 "axiome: " }{TEXT 
265 5 "fpfpf" }{TEXT -1 13 "     regle : " }{TEXT 267 44 "f -> fpfmfmf
fpfpfmf   alpha=Pi/2 (90 degres)" }}{PARA 0 "" 0 "" {TEXT -1 167 "Ecri
re le mot obtenu en appliquant la regle 1,2 fois. Tracer sur une feuil
le de papier les dessins correspondants (faire un dessin separe pour l
a regle de reecriture)." }}{PARA 0 "" 0 "" {TEXT -1 46 "Quelle est la \+
longueur du mot apres n etapes ?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{SECT 1 {PARA 4 "" 0 "" {TEXT 268 8 "Solution" }{TEXT 269 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "Apres 2 e
tapes :" }}{EXCHG {PARA 12 "" 1 "" {XPPMATH 20 "6#7gn%\"fG%\"pGF$%\"mG
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F$F%F$F&F$F%F$F%F$F&F$F&F$F$F%F$F%F$F&F$" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 20 "Dessin de la regle :" }}}{EXCHG {PARA 0 "" 0 "" {INLPLOT 
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{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "Apres 3 etapes :" }}}{EXCHG {PARA 
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*F.7$F(F.7$F2F.7$F2F*7$F(F*F'-%'COLOURG6&%$RGBG$FioF3F(F(-%(SCALINGG6#
%,CONSTRAINEDG-%*AXESSTYLEG6#%%NONEG" 2 246 246 246 2 0 1 0 2 9 0 1 1 
1.000000 45.000000 45.000000 10030 0 10056 10074 0 0 0 20040 0 12010 
0 255 0 0 255 0 0 1 0 0 0 0 104 95 0 0 0 0 0 0 }{MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 270 20 "I. Figures fractales
" }}{PARA 0 "" 0 "" {TEXT -1 43 "On prend'alphabet de l'exemple ci-des
sus : " }{TEXT 284 9 "V=\{f,p,m\}" }{TEXT -1 179 ". Un peu plus loin d
ans ce paragraphe on utilisera encore 2 lettres, x et y, qui ne seront
 pas interpretees par la tortue, mais serviront uniquement pour les re
gles de reecriture." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 241 "Voici une procedure qui prend comme parametres le mot \+
de depart, la regle de substitution, le nombre d'etapes et l'angle alp
ha et qui retourne le dessin correspondant. Notez que si le mot compor
te des lettres qui ne sont pas dans l'alphabet " }{TEXT 285 1 "V" }
{TEXT -1 38 ", elles sont tout simplement ignorees." }}{PARA 0 "" 0 "
" {TEXT -1 12 "La variable " }{TEXT 271 3 "mot" }{TEXT -1 126 " est un
e liste; elle represente le mot, qui est l'axiome,  dans lequel on ope
re les substitutions prescrites par la procedure " }{TEXT 272 5 "regle
" }{TEXT -1 27 " un nombre d'etapes egal a " }{TEXT 273 6 "etapes" }
{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 44 "Efforcez-vous de compre
ndre cette procedure." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "des
sine:=proc(axiome::list,regle::procedure,etapes::integer,alpha::realco
ns)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "local i,mot,pts,tt,xx,yy;" }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "mot:=axiome;" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "for i to etapes do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
19 "mot:=map(regle,mot)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od;" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "tt:=0;" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 6 "xx:=0;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "yy:=0;" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "pts:=[xx,yy];" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 15 "for i in mot do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 
"if i=p then tt:=tt+alpha;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 " elif
 i=m then tt:=tt-alpha;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "  elif i
=f then " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "   xx:=xx+cos(tt);" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "   yy:=yy+sin(tt);" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 19 "   pts:=pts,[xx,yy]" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 3 "fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od;" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 41 "plot([pts],axes=none,scaling=CONSTRAINED)
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end;" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>%(dessineG:6&'%'axiomeG%%listG'%&regleG%*procedureG'%'
etapesG%(integerG'%&alphaG%)realconsG6(%\"iG%$motG%$ptsG%#ttG%#xxG%#yy
G6\"F:C*>8%9$?(8$\"\"\"FA9&%%trueG>F=-%$mapG6$9%F=>8'\"\"!>8(FK>8)FK>8
&7$FMFO?&F@F=FC@(/F@%\"pG>FJ,&FJFA9'FA/F@%\"mG>FJ,&FJFAFY!\"\"/F@%\"fG
C%>FM,&FMFA-%$cosG6#FJFA>FO,&FOFA-%$sinGF`oFA>FQ6$FQFR-%%plotG6%7#FQ/%
%axesG%%noneG/%(scalingG%,CONSTRAINEDGF:F:" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 47 "regle1:=a->subs(f=(f,p,f,m,f,f,p,f,p,f,m,f),a);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'regle1G:6#%\"aG6\"6$%)operatorG%&ar
rowGF(-%%subsG6$/%\"fG6.F0%\"pGF0%\"mGF0F0F2F0F2F0F3F09$F(F(" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "Par exemple :" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 39 "dessine([f,p,f,p,f,p,f],regle1,3,Pi/2);" }}
{PARA 13 "" 1 "" {INLPLOT "6%-%'CURVESG6$7i`p7$\"\"!F(7$$\"\"\"F(F(7$F
*F*7$$\"\"#F(F*7$$\"\"$F(F*7$F1F.7$F.F.7$F.F17$F*F17$F*F.7$F(F.7$$!\"
\"F(F.7$F:F*7$F(F*F'7$F:F(7$F:F:7$$!\"#F(F:7$$!\"$F(F:7$FDFA7$FAFA7$FA
FD7$FA$!\"%F(7$F:FJ7$F:$!\"&F(7$F:$!\"'F(7$F(FQ7$F(FN7$F*FN7$F*FJ7$F(F
J7$F(FD7$F(FA7$F:FA7$F:FDFHFIFLFMFPFSFTFU7$F*FQ7$F.FQ7$F.$!\"(F(7$F.$!
\")F(7$F1F\\o7$F1Fin7$$\"\"%F(Fin7$FaoFQ7$F1FQ7$F1FN7$F1FJ7$F.FJ7$F.FN
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Exercice 2" }{TEXT -1 62 ". Dessinez l'exemple de l'exercice 1 a l'aid
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" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
124 "La courbe de Von Koch est obtenue par la regle suivante. On part \+
d'un segment et on le remplace par 4 segments, comme suit :" }}}
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$$!#6F(Fir7$Fj_lF]s7$$!#7F(F]s7$F^`lFir7$$!#8F(Fir7$Fb`lFdr7$$!#9F(Fdr
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{TEXT -1 130 "L'exemple suivant est la vatiante de Hilbert de la courb
e de Peano. Il utilise l'alphabet \{f,p,m,x,y\} et 2 regles de reecrit
ures :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "regle3:=a->subs(
\{x=(m,y,f,p,x,f,x,p,f,y,m),y=(p,x,f,m,y,f,y,m,f,x,p)\},a);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%'regle3G:6#%\"aG6\"6$%)operatorG%&arrowGF(
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0 "" 0 "" {TEXT -1 223 "Efforcez-vous de comprendre cet exemple. Cela \+
aide de partir des mots [x] et [y] et de faire une ou deux etapes seul
ement, en appliquant pour commencer seulement la regle de substitution
 pour x, respectivement celle pour y." }}}{SECT 0 {PARA 258 "" 0 "" 
{TEXT 277 11 "Exercice 4." }}{PARA 0 "" 0 "" {TEXT -1 89 "Trouver le c
odage de la courbe suivante, qui est une variante du triangle de Sierp
inski :" }}{EXCHG {PARA 0 "" 0 "" {INLPLOT "6%-%'CURVESG6$7fhm7$\"\"!F
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`lF(F(-%*AXESSTYLEG6#%%NONEG-%(SCALINGG6#%,CONSTRAINEDG" 2 219 219 
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}}}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 278 25 "II. Croissance de plantes" 
}}{PARA 0 "" 0 "" {TEXT -1 696 "Dans le paragraphe precedent, tous les
 dessins obtenus sont des courbes d'un seul trait, puisque chaque depl
acement de la tortue est tracant. Cela ne convient par pour simuler la
 croissance de plantes, qui presentent des branchements. Pour cela il \+
est necessaire d'introduire deux nouveaux symboles, le crochet gauche \+
et droit, notes respectivement l et r, et une nouvelle regle pour leur
 interpretation : lorsqu'on trouve un crochet gauche, on met sur une p
ile la position et l'orientation de la tortue; lorsqu'on rencontre un \+
crochet droit, la position et orientation du crochet gauche correspond
ant sont enleves de la pile, et la tortue se met, sans tracer, dans ce
tte position et orientation." }}{PARA 0 "" 0 "" {TEXT -1 35 "Voici une
 variante de la procedure " }{TEXT 281 7 "dessine" }{TEXT -1 70 " qui \+
permet de traiter les crochets; efforcez-vous de la comprendre : " }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "machin:=proc(axiome::list,ru
le::procedure,etapes::integer,theta::realcons)" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 43 "local i,lst, plante,pts,tt,xx,yy,xyt,stack;" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "pla
nte:=axiome;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "for i to etapes do
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "plante:=map(rule,plante)" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 9 "tt:=Pi/2;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "xx:=0;" }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 6 "yy:=0;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 
"pts:=[xx,yy];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "lst:=NULL;" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "stack:=[];" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "for i in plante do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
25 "if i=p then tt:=tt+theta;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 " e
lif i=m then tt:=tt-theta;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 " elif
 i=l then stack:=[[xx,yy,tt],op(stack)];" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 54 "elif i=r then if nops(stack)<1 then ERROR('Stackato')
;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 " fi;" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 15 "               " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 
"   xyt:=stack[1];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "   stack:=sub
sop(1=NULL,stack);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "   xx:=xyt[1]
;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "   yy:=xyt[2];" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 14 "   tt:=xyt[3];" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 18 "   lst:=lst,[pts];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "   pt
s:=[xx,yy];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 " elif i=f then " }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "   xx:=xx+cos(tt);" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 18 "   yy:=yy+sin(tt);" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "   pts:=pts,[xx,yy]" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 3 "fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od;" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 47 "plot(\{lst,[pts]\},axes=NONE,scaling=CONSTRAINED)" 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end;" }}{PARA 12 "" 1 "" {XPPMATH 
20 "6#>%'machinG:6&'%'axiomeG%%listG'%%ruleG%*procedureG'%'etapesG%(in
tegerG'%&thetaG%)realconsG6+%\"iG%$lstG%'planteG%$ptsG%#ttG%#xxG%#yyG%
$xytG%&stackG6\"F=C,>8&9$?(8$\"\"\"FD9&%%trueG>F@-%$mapG6$9%F@>8(,$%#P
iG#FD\"\"#>8)\"\"!>8*FT>8'7$FSFV>8%%%NULLG>8,7\"?&FCF@FF@,/FC%\"pG>FM,
&FMFD9'FD/FC%\"mG>FM,&FMFDF`o!\"\"/FC%\"lG>Fhn7$7%FSFVFM-%#opG6#Fhn/FC
%\"rGC*@$2-%%nopsGF]pFD-%&ERRORG6#.%)StackatoG>8+&Fhn6#FD>Fhn-%'subsop
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FenF^r/%%axesG%%NONEG/%(scalingG%,CONSTRAINEDGF=F=" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 44 "rule1:=a->subs(f=(f,l,p,f,r,f,l,m,f,r,f),a)
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279 10 "Exercice 5" }{TEXT -1 62 ". Trouver le codage du dessin suivan
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{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Modifiez le code de la procedure \+
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}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "Pouvz-vous en trouver le code ?
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