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{TEXT -1 21 "Param\351trage du tore :" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 106 "restart:Digits:=4:with(plots):\nf:=(u,v)->sin (2*u)*cos(v)/a:\ng:=(u,v)->sin(2*u)*sin(v)/a:\nh:=(u,v)->sin(u):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "surf:=(u,v)->[f(u,v),g(u,v), h(u,v)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "a:=1:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "a:=4*sqrt(2):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 258 32 "On trace un arc sur la surface :" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "arcsurf:=t->surf(u(t),v(t)): " }{TEXT -1 0 "" }}}}{PARA 4 "" 0 "" {TEXT -1 30 "M\351thode particuli \350re au tore :" }}{EXCHG {PARA 0 "" 0 "" {TEXT 256 89 "On \351crit q ue le plan osculateur \340 l'arc trac\351 sur la surface est orthogona l \340 la surface." }{TEXT -1 291 "\nOn sait d'une part qu'en un point r\351gulier de la surface et de l'arc (ce qui est toujours le cas ici ), le vecteur tangent \340 l'arc est dans le plan tangent \340 la sur face et d'autre part que le plan osculateur \340 l'arc est dirig\351 p ar les deux premiers vecteurs d\351riv\351s du param\351trage de l'arc ." }}{PARA 0 "" 0 "" {TEXT 259 79 "Comme dans le cas pr\351sent, les \+ deux vecteurs dirigeant le plan tangent au tore" }{TEXT -1 86 " (vecte urs d\351riv\351es premi\350res par rapport aux deux param\350tres du \+ param\351trage du tore) " }{TEXT 260 16 "sont orthogonaux" }{TEXT -1 102 ", on se contente d'\351crire que le vecteur d\351riv\351e seconde de l'arc leur est orthogonal (condition (A))." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 29 "deriv2:=diff(arcsurf(t),t,t):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "plantang:=subs(u=u(t),v=v(t),[diff( surf(u,v),u),diff(surf(u,v),v)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %)plantangG7$7%,$*&#\"\"\"\"\"%F**(-%$cosG6#,$*&\"\"#F*-%\"uG6#%\"tGF* F*F*-F.6#-%\"vGF5F*F2#F*F2F*F*,$*&F)F**(F-F*-%$sinGF8F*F2F;F*F*-F.6#F3 7%,$*&#F*\"\")F**(-F@F/F*F?F*F2F;F*!\"\",$*&#F*FGF**(FIF*F7F*F2F;F*F* \"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "On v\351rifie donc l'ort hogonalit\351 des deux vecteurs directeurs du plan tangent :" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "On \351crit la condition (A) :" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "sys:=\{seq(add(p[i]*deriv2[ i],i=1..3),p=plantang)\}:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "sys:=factor(expand(combine(sys,trig)));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$sysG<$,0*&#\"\"\"\"\")F)-%%diffG6$-%\"uG6#%\"tG-%\"$ G6$F1\"\"#F)F)*()-F,6$F.F1F5F))-%$cosG6#F.\"\"$F)-%$sinGF=F)!\"\"*&#F) F5F)*(F;F)F?F)F7F)F)FA*&#F)F5F)*&F+F))F;\"\"%F)F)F)*&FFF)*&)F;F5F)F+F) F)F)*&#F)FIF)*()-F,6$-%\"vGF0F1F5F)F:F)F?F)F)FA*&F(F)*(FPF)F;F)F?F)F)F ),$*&F(F)*&F;F),**&-F,6$FSF2F)F:F)FA*,FIF)F8F)FQF)F?F)FLF)F)*&FfnF)F;F )F)**F5F)F8F)FQF)F?F)FAF)F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 257 34 "R\351solution (num\351rique) et graphe :" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 98 "On r\351sout num\351riquement (impossible formellement ic i) et pour les conditions initiales demand\351es :" }}}{EXCHG 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