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{SECT 0 {SECT 0 {PARA 3 "" 0 "" {TEXT -1 48 "CorrigŽ du TD 2 : Courbes
, plans, surfaces, etc." }}{PARA 256 "" 0 "" {TEXT -1 25 "Jean SŽbasti
en ROY, 1998." }}{PARA 0 "" 0 "" {TEXT -1 71 "Note : Les erreurs les p
lus frŽquentes proviennent de l'oubli de faire " }{TEXT 256 7 "restart
" }{TEXT -1 28 " et de recharger le package " }{TEXT 258 5 "plots" }
{TEXT -1 6 " avec " }{TEXT 257 10 "with(plots" }{TEXT -1 19 ") apr�s u
n restart." }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 37 "Ex 1 : Plans tange
nts ˆ un ellipso•de" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restar
t:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "Comme on va avoir besoin de
 quelques fonctions des packages '" }{TEXT 259 5 "plots" }{TEXT -1 6 "
' et '" }{TEXT 260 6 "linalg" }{TEXT -1 16 "' on les charge." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "with(plots):with(linalg):" }
}{PARA 7 "" 1 "" {TEXT -1 32 "Warning, new definition for norm" }}
{PARA 7 "" 1 "" {TEXT -1 33 "Warning, new definition for trace" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "Le vecteur directeur de la droite \+
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "vDroite:=[1,3,-2];" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(vDroiteG7%\"\"\"\"\"$!\"#" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "Et une Žquation paramŽtrique de ce
lle-ci (" }{TEXT 261 5 "evalm" }{TEXT -1 83 " (m pour 'matrices'), per
met entre autres de multiplier un vecteur par un scalaire)" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "eDroite:=evalm(t*vDroite);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(eDroiteG-%'vectorG6#7%%\"tG,$F)\"\"
$,$F)!\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "Les options graphiqu
es communes ˆ tous les graphes :" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 62 "opts:=view=[-2..2,-2..2,-2..2],scaling=constrained,ax
es=boxed;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%optsG6%/%%viewG7%;!\"#
\"\"#F)F)/%(scalingG%,constrainedG/%%axesG%&boxedG" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 27 "Le graphique de la droite :" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 54 "droite:=spacecurve(eDroite,t=-2..2,opts,shadi
ng=none):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 87 "Et celui de l'ellips
o•de, tracŽ en coordonnŽes cylindriques apr�s avoir ŽtŽ paramŽtrŽ :" }
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db]n$\"+--^vZFjgcl$!+oMn$=$Fjgcl7%$\"+(QpMn\"Fjgcl$\"+h\"3/-&Fjgcl$!+u
(QpM$Fjgcl7%$\"+S?5b<Fjgcl$\"+?hIl_Fjgcl$!+!3/-^$Fjgcl7%$\"+$pMn$=Fjgc
l$\"+zS?5bFjgcl$!+'QpMn$Fjgcl7%$\"+YtO=>Fjgcl$\"+Q?5bdFjgcl$!+#pMn$QFj
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%%NONEG-%(SCALINGG6#%,CONSTRAINEDG-%*AXESSTYLEG6#%$BOXG-%%VIEWG6%;Fid
\\n$\"\"#F)F_[^nF_[^n" 1 2 0 1 0 2 1 1 2 1 1.000000 45.000000 
45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "L'Žquation de l'el
lipso•de :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "eEllipsoide:=
x^2+y^2+2*z^2-1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,eEllipsoideG,**
$)%\"xG\"\"#\"\"\"\"\"\"*$)%\"yGF)F*F+*$)%\"zGF)F*F)!\"\"F+" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "Vecteur normal en tout point de l'
ellipso•de :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "grad(eEllip
soide,[x,y,z]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7%,$%\"
xG\"\"#,$%\"yGF),$%\"zG\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 164 "
On cherche ˆ trouver tous les (x,y,z) tels que ce vecteur soit colinŽa
ire au vecteur directeur de la droite, cad que le produit vectoriel de
s deux vecteur soit nul." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 
"crossprod(vDroite,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#
7%,&%\"zG\"#7%\"yG\"\"%,&%\"xG!\"%F(F.,&F*\"\"#F-!\"'" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 54 "Il faut rŽsoudre le syst�me prŽcŽdent (=[
0,0,0]) mais " }{TEXT 266 5 "solve" }{TEXT -1 20 " n'accepte que des '
" }{TEXT 262 3 "set" }{TEXT -1 14 "' et non des '" }{TEXT 263 6 "vecto
r" }{TEXT -1 22 "', donc on convertit :" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 15 "convert(%,set);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%
,&%\"xG!\"%%\"zGF&,&F'\"#7%\"yG\"\"%,&F*\"\"#F%!\"'" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 24 "Et on rŽsoud en x et y :" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 15 "solve(%,\{x,y\});" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#<$/%\"xG,$%\"zG!\"\"/%\"yG,$F'!\"$" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 59 "On enl�ve les accolades pour utiliser le resultat \+
dans un '" }{TEXT 264 4 "subs" }{TEXT -1 12 "' plus bas :" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "solution:=op(%);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%)solutionG6$/%\"xG,$%\"zG!\"\"/%\"yG,$F)!\"$" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "On substitue la solution dans l'Žq
uation de l'ellipso•de :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 
"subs(solution,eEllipsoide);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$)%
\"zG\"\"#\"\"\"\"#7!\"\"\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 
"Et on rŽsoud en " }{TEXT 267 1 "z" }{TEXT -1 119 " pour trouver tous \+
les points de l'ellipso•de en lesquels la normale est colinŽaire au ve
cteur directeur de la droite :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 20 "solutionZ:=solve(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*sol
utionZG6$,$*$-%%sqrtG6#\"\"$\"\"\"#\"\"\"\"\"',$F'#!\"\"F/" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 36 "Les valeurs des points en question :" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "subs(solution,z=solutionZ[1]
,[x,y,z]);subs(solution,z=solutionZ[2],[x,y,z]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7%,$*$-%%sqrtG6#\"\"$\"\"\"#!\"\"\"\"',$F%#F,\"\"#,$F%#
\"\"\"F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%,$*$-%%sqrtG6#\"\"$\"\"
\"#\"\"\"\"\"',$F%#F,\"\"#,$F%#!\"\"F-" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 165 "On cherche alors l'Žquation des plans perpendiculaires ˆ
 la droite, cad l'ensemble des points M(x,y,z) tq OM . vDroite = const
ante. On calcule plan0 = OM . vDroite :" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 32 "plan0:=dotprod([x,y,z],vDroite);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%&plan0G,(%\"xG\"\"\"%\"yG\"\"$%\"zG!\"#" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 122 "Les plans doivent passer par les points \+
prŽcŽdement trouvŽs, d'o� les valeurs des  constantes et les Žquations
 des plans :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "plan:= i ->
 plan0-subs(solution,z=solutionZ[i],plan0);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%planGR6#%\"iG6\"6$%)operatorG%&arrowGF(,&%&plan0G\"
\"\"-%%subsG6%%)solutionG/%\"zG&%*solutionZG6#9$F-!\"\"F(F(F(" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "Par exemple :" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 8 "plan(1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,
*%\"xG\"\"\"%\"yG\"\"$%\"zG!\"#*$-%%sqrtG6#F'\"\"\"\"\"#" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 40 "On trace les plans sans les parametrer :
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lan(1),x=-2..2,y=-2..2,z=-2..2,grid=[13,13,13],style=patchnogrid,opts)
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "plan2:=implicitplot3d(
plan(2),x=-2..2,y=-2..2,z=-2..2,grid=[13,13,13],style=patchnogrid,opts
):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "RŽsultat final :" }}}
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