{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 1 50 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 1 24 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Tim es" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 3" -1 5 1 {CSTYLE "" -1 -1 "Ti mes" 1 12 0 0 0 1 1 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Warning" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output " -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE " " -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Norma l" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 0 "" }{TEXT 256 18 "Su rfaces de B\351zier" }}{PARA 257 "" 0 "" {TEXT 257 49 "Application \+ \340 la conception d'une coque de bateau" }}{PARA 0 "" 0 "" {TEXT -1 484 " L'objectif du projet est de concevoir une coque compl \350te de bateau (inspir\351e du Luna Rossa italien) \340 l'aide de su rfaces de B\351zier. On pourra alors d\351terminer son poids en foncti on du niveau de sa ligne de flottaison. Attention! La fen\352tre Maple doit occuper la largeur de l'\351cran sinon, la mise en page est mauv aise et des lignes de texte vont s'intercaler entre les lignes de comm ande des proc\351dures. (ces lignes de texte sont pour cela supprim \351es sur le document imprim\351)" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 32 "Surfaces de B\351zier: g\351n\351ralit\351s." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 32 "D\351finition, proc\351dures, exemples" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 699 " \+ La m\351thode des courbes de B\351zier permet de lisser une ligne br is\351e constitu\351e des points de contr\364le. De la m\352me fa\347o n, l'utilisation des surfaces de B\351zier permet de lisser un treilli s de points. On entre les points dans un tableau M:=(Pij) de dimension m x n o\371 les Pij sont les points de contr\364le qui constituent le treillis. On va alors associer \340 chaque couple (t,s) du pav\351 [0 ;1]x[0;1] le point B(t,s) d\351fini comme suit: B(t,s) est le point B' (s) de la courbe de B\351zier param\351tr\351e par s dont les points d e contr\364le sont les points Bk(t). Pour k variant de 1 \340 m, Bk(t) est le point de la courbe de B\351zier param\351tr\351e par t dont le s points de contr\364le sont les Pkj, j variant de 1 \340 n." }}{PARA 0 "" 0 "" {TEXT -1 106 " On peut alors visualiser sur un exem ple simple cette m\351thode. On utilise pour cela la proc\351dure " } {TEXT 258 7 "grille " }{TEXT -1 48 "qui trace le treillis de points et la proc\351dure " }{TEXT 259 7 "surface" }{TEXT -1 56 " qui renvoie l e graphe de la surface de B\351zier associ\351e." }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "### WARNING: persistent store makes one-argument readlib obso lete\nwith(plots):readlib(bernstein):with(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been redefined\n" }} {PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names norm and tra ce have been redefined and unprotected\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "segment:=proc(M,N,x,X,y,Y,z,Z)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "plot3d([M[1]+t*(N[1]-M[1]),M[2]+t*(N[2]-M[2]),M[3]+t *(N[3]-M[3])],t=0..1,s=0..1,view=[x..X,y..Y,z..Z]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(segmentG R6*%\"MG%\"NG%\"xG%\"XG%\"yG%\"YG%\"zG%\"ZG6\"F/F/-%'plot3dG6&7%,&&9$6 #\"\"\"F8*&%\"tGF8,&&9%F7F8F5!\"\"F8F8,&&F66#\"\"#F8*&F:F8,&&F=FAF8F@F >F8F8,&&F66#\"\"$F8*&F:F8,&&F=FHF8FGF>F8F8/F:;\"\"!F8/%\"sGFN/%%viewG7 %;9&9';9(9);9*9+F/F/F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "g rille:=proc(A,x,X,y,Y,z,Z)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "local i,j,p,n_line,n_col;\nn_line:=rowdim(A);\nn_col:=coldim(A);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "for i to n_line do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 " for j to n_col do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 " if i=n_line and j=n_col then" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 " p[i,j]:=segment(A[i,j],A[i,j],x,X,y,Y,z, Z)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 " elif i<>n_line and \+ j=n_col then" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 " p[i,j] :=segment(A[i,j],A[i+1,j],x,X,y,Y,z,Z)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 " elif i=n_line and j<>n_col then" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 57 " p[i,j]:=segment(A[i,j],A[i,j+1],x,X,y, Y,z,Z)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 108 " else \+ p[i,j]:=segment(A[i,j],A[i+1,j],x,X,y,Y,z,Z),segment(A[i,j],A[i,j+1 ],x,X,y,Y,z,Z)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 " fi;" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 " od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "seq(seq(p[i, j],i=1..n_line),j=1..n_col);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end; " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'grilleGR6)%\"AG%\"xG%\"XG%\"yG% \"YG%\"zG%\"ZG6'%\"iG%\"jG%\"pG%'n_lineG%&n_colG6\"F4C&>8'-%'rowdimG6# 9$>8(-%'coldimGF:?(8$\"\"\"FBF7%%trueG?(8%FBFBF=FC@)3/FAF7/FEF=>&8&6$F AFE-%(segmentG6*&F;FMFQ9%9&9'9(9)9*30FAF7FI>FK-FO6*FQ&F;6$,&FAFBFBFBFE FRFSFTFUFVFW3FH0FEF=>FK-FO6*FQ&F;6$FA,&FEFBFBFBFRFSFTFUFVFW>FK6$FenF]o -%$seqG6$-Feo6$FK/FA;FBF7/FE;FBF=F4F4F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "surface:=proc(M,xm,xM,ym,yM,zm,zM,c)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "local n_line,n_col,i,ii,jj,j,kk,k,xx,yy,zz,p1,X, Y,Z,xn_line,xn_col;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "n_line:=rowd im(M);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "n_col:=coldim(M);" } {TEXT -1 130 " les 3 coordonn\351es des point s de contr\364le vont \352tre trait\351es s\351par\351ment dans des li stes index\351es X,Y et Z" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 " for ii to n_line do X[ii]:=[];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 60 " f or i to n_col do X[ii]:=[op(X[ii]),op(1,M[ii,i])] od od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 " for jj to n_line do Y[jj]:=[];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 60 " for j to n_col do Y[jj]:=[op(Y[jj]),op(2 ,M[jj,j])] od od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 " for kk to n _line do Z[kk]:=[];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 60 " for k to \+ n_col do Z[kk]:=[op(Z[kk]),op(3,M[kk,k])] od od;" }{TEXT -1 72 " \+ on d\351finit ensuit les fonctions qui permettent de d\351termine r " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "xn_col:=unapply(bernstein(n_c ol-1,r->A[(n_col-1)*r+1],t),t,A);" }{TEXT -1 39 " un point d'une \+ courbe de B\351zier.." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "xn_line:=u napply(bernstein(n_line-1,r->AA[(n_line-1)*r+1],t),t,AA); " }{TEXT -1 39 "..et par suite d'une surface de B\351zier." }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 68 " xx:=unapply(xn_line(s,[seq(xn_col(t,X[xi]),xi=1 ..n_line)]),t,s);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 68 " yy:=unappl y(xn_line(s,[seq(xn_col(t,Y[yi]),yi=1..n_line)]),t,s);" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 68 " zz:=unapply(xn_line(s,[seq(xn_col(t,Z[zi]) ,zi=1..n_line)]),t,s);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "plot3d([x x(t,s),yy(t,s),zz(t,s)],t=0..1,s=0..1,color=c,view=[xm..xM,ym..yM,zm.. zM],axes=normal):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(surfaceGR6*%\"MG%#xmG%#xMG%#ymG%#yMG%#zmG %#zMG%\"cG63%'n_lineG%&n_colG%\"iG%#iiG%#jjG%\"jG%#kkG%\"kG%#xxG%#yyG% #zzG%#p1G%\"XG%\"YG%\"ZG%(xn_lineG%'xn_colG6\"FAC->8$-%'rowdimG6#9$>8% -%'coldimGFG?(8'\"\"\"FOFD%%trueGC$>&806#FN7\"?(8&FOFOFJFP>FS7$-%#opG6 #FS-Ffn6$FO&FH6$FNFX?(8(FOFOFDFPC$>&816#F]oFV?(8)FOFOFJFP>F`o7$-Ffn6#F `o-Ffn6$\"\"#&FH6$F]oFdo?(8*FOFOFDFPC$>&826#F_pFV?(8+FOFOFJFP>Fbp7$-Ff n6#Fbp-Ffn6$\"\"$&FH6$F_pFfp>84-%(unapplyG6%-%*bernsteinG6%,&FJFOFO!\" \"R6#%\"rGFA6$%)operatorG%&arrowGFA&%\"AG6#,&*&,&T#FOFOFiqFOFHFOFOFOFO FAFA6$F1FJ%\"tGFhrFar>83-Fcq6%-Ffq6%,&FDFOFOFiqRF[rFAF]rFA&%#AAGFbrFAF A6$F0FDFhrFhrFbs>8,-Fcq6%-Fjr6$%\"sG7#-%$seqG6$-Faq6$Fhr&FT6#%#xiG/Fct ;FOFDFhrFjs>8--Fcq6%-Fjr6$Fjs7#-F]t6$-Faq6$Fhr&Fao6#%#yiG/FcuFetFhrFjs >8.-Fcq6%-Fjr6$Fjs7#-F]t6$-Faq6$Fhr&Fcp6#%#ziG/FbvFetFhrFjs-%'plot3dG6 (7%-Fes6$FhrFjs-FgtFiv-FfuFiv/Fhr;\"\"!FO/FjsF]w/%&colorG9+/%%viewG7%; 9%9&;9'9(;9)9*/%%axesG%'normalGFAFAFA" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 " Les proc\351dures \351tant entr\351es, d\351ter minons un treillis de points:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "M:=array(1..3,1..3,[[[0,1,0],[0,2,0.4],[0,3,1]],[[2,1,1],[2,2,0 .5],[2,3,1]],[[5,2,1],[5,3,1],[5,4,0.5]]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG-%'matrixG6#7%7%7%\"\"!\"\"\"F+7%F+\"\"#$\"\"%!\" \"7%F+\"\"$F,7%7%F.F,F,7%F.F.$\"\"&F17%F.F3F,7%7%F8F.F,7%F8F3F,7%F8F0F 7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "dessin11:=grille(M,0,5 ,0,4,0,2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "dessin12:=sur face(M,0,5,0,4,0,2,red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "display(dessin11,dessin12);" }}{PARA 13 "" 1 "" {GLPLOT3D 400 300 300 {PLOTDATA 3 "62-%%MESHG6#X.%)anythingG6\"6\"^[[[[]\\q^`r\":\":\"$0 0000000000000003FF0000000000000000000000000000000000000000000003FF0000 000000000000000000000000000000000000000003FF00000000000000000000000000 00000000000000000003FF0000000000000000000000000000000000000000000003FF 0000000000000000000000000000000000000000000003FF0000000000000000000000 000000000000000000000003FF00000000000000000000000000000000000000000000 03FF0000000000000000000000000000000000000000000003FF000000000000000000 0000000000000000000000000003FF0000000000000000000000000000000000000000 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