{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 "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Input" 2 19 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 269 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE " " -1 270 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 280 "" 0 1 255 0 0 1 1 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 281 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 282 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 283 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 284 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 285 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Hea ding 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Title" 0 18 1 {CSTYLE "" -1 -1 " " 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 1 }3 0 0 -1 12 12 0 0 0 0 0 0 19 0 } {PSTYLE "Author" 0 19 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 8 8 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 0" -1 258 1 {CSTYLE "" -1 -1 "Monaco" 1 9 0 0 255 1 2 2 2 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Normal" -1 260 1 {CSTYLE "" -1 -1 "Ti mes" 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 "" 0 261 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" -1 262 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 "R3 Font 0" -1 264 1 {CSTYLE "" -1 -1 "Mona co" 1 9 0 0 255 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 }} {SECT 0 {PARA 18 "" 0 "" {TEXT -1 20 "Curvas e SUPERF\315CIES" }} {PARA 19 "" 0 "" {TEXT 279 0 "" }{TEXT -1 4 "Por " }{URLLINK 17 "Sylvi e Oliffson Kamphorst" 4 "http://www.mat.ufmg.br/~syok" "" }{TEXT -1 4 " da " }{TEXT 281 2 "UF" }{TEXT 280 1 "M" }{TEXT 282 1 "G" }}{PARA 260 "" 0 "" {TEXT -1 17 " Atualizado por " }{URLLINK 17 "Milton Proc \363pio de Borba" 4 "http://planeta.terra.com.br/educacao/miltonpb" " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 116 "Este texto pretende resumir para os alunos, as diversas possibilidades de \+ utiliza\347\343o do programa Maple no Estudo de:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 259 0 "" }{TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 " " {TEXT 262 19 "Gr\341ficos de Fun\347\365es" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "restart:with(plots):setoptions3d(scaling=constrain ed):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "f:=(x,y)->2*sin(x+y );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "plot3d(f(x,y),x=0..2* Pi,y=0..2*Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 267 26 "Superf\355cies Pa rametrizadas" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "restart:with(plots):setoptions3d(scaling=constrain ed):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "X:=(u,v)->[2*v*cos( u),2*v*sin(u),u]; # Helicoide" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "plot3d(X(u,v), u=0..2*Pi, v=0..1,style=patchnogrid,color=blue,li ghtmodel=light4);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Rampa:" }}{PARA 0 "" 0 "" {TEXT -1 95 "Neste exemp lo voce pode variar o raio interno do helicoide 0 < r < 2, o externo s endo sempre 2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "with(plots):setoptions3d(scaling=constrained);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "r:='r': r:=1;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "cilindro:= plot3d([r*cos(u), r*sin(u), v], \+ u=0..2*Pi, v=0..2,color=green):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 99 " rampa:= plot3d([((1-t)*r+ 2*t)*cos(u), ((1-t)*r +2*t)*sin(u), u/Pi],u= 0..2*Pi, t=0..1, color=blue):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "di splay(\{rampa, cilindro\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 260 22 "Superf\355cies Impl\355citas" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "implicitplot3d( x^3 + y^3 + z^3 + 1 = (x + y + z + 1)^3,x=-2..2,y=-2..2,\nz=-2..2,grid=[13, 13,13]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "F:=(x,y,z) -> x ^2 + y^2 - z;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "implicitpl ot3d( F(x,y,z)= 0 ,x=-2..2,y=-2..2,z=0..4);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 261 25 "Superf\355cies de Revolu\347\343o " } {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "restart:with(plots):setoptions3d(scaling=constrain ed):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "f:= x -> sin(x) + 2 ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "plot(f(x),x=-Pi..Pi,ax es=normal,labels=[x,y]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "plot3d([u, f(u)*cos(v), f(u)*sin(v)], u=-Pi..Pi, v=0..2*Pi, axes= normal, orientation=[-90,90], color=blue);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "f:= u-> exp(u); g:= u-> 5*cos(u);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "plot([f(u),g(u),u=0..1],labe ls=['x','z']);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "plot3d([f (u)*cos(v),f(u)*sin(v),g(u)],u=0..1,v=0..2*Pi);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 263 19 "Cones generalizados" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Seja " }{XPPEDIT 18 0 "alp ha;" "6#%&alphaG" }{TEXT -1 33 " = (x(t),y(t)) uma curva plana e " } {XPPEDIT 18 0 "p = (a, b, c);" "6#/%\"pG6%%\"aG%\"bG%\"cG" }{TEXT -1 14 " um ponto em " }{XPPEDIT 18 0 "R^3;" "6#*$%\"RG\"\"$" }{TEXT -1 25 ". Definimos a superf\355cie " }{XPPEDIT 18 0 "C(t,s);" "6#-%\"CG6$ %\"tG%\"sG" }{TEXT -1 51 " = (1 - s) p + s (x(t),y(t),0) ou em compone ntes " }{XPPEDIT 18 0 "C(t,s);" "6#-%\"CG6$%\"tG%\"sG" }{TEXT -1 4 " = (" }{XPPEDIT 18 0 "(1-s)*a;" "6#*&,&\"\"\"F%%\"sG!\"\"F%%\"aGF%" } {TEXT -1 3 " + " }{TEXT 265 3 "s x" }{TEXT -1 7 "(t), " }{XPPEDIT 18 0 "(1-s)*b;" "6#*&,&\"\"\"F%%\"sG!\"\"F%%\"bGF%" }{TEXT -1 23 " + s y(t), (1-s)c ). " }}{PARA 0 "" 0 "" {TEXT -1 24 "C \351 um cone no \+ caminho " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 17 " com v \351rtice em " }{TEXT 264 1 "p" }{TEXT -1 21 ". Abaixo, definimos " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 16 " (t) = (cos(t)- " } {XPPEDIT 18 0 "2*sin(t)^2;" "6#*&\"\"#\"\"\"*$-%$sinG6#%\"tGF$F%" } {TEXT -1 54 ", 2 + 2.3*sin(t)) , o ponto p e desenhamos o tra\347o de \+ " }{XPPEDIT 18 0 "alpha" "6#%&alphaG" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "a:=2; b:= 4; c:=1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "x:=cos(t)-2*sin (t)^2;y:=2+2.3*sin(t);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "plot([x,y ,t=-5..5]);" }}}{PARA 0 "" 0 "" {TEXT -1 20 "e desenhamos o cone:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "plot3d([(1-s)*a+s*x, (1-s)*b +s*y,(1-s)*c],s=-2..2,t=-Pi..Pi,axes=boxed);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 0 "" } {TEXT 268 20 "Superf\355cies Regradas" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "Uma s uperf\355cie regrada \351 definida a partir de uma curva " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 30 "(t) e um vetor unit\341rio \+ r(t)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "alpha := t -> [cos(t ), sin(t), t/5];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "r := t -> [0, c os(t), sin(t)];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "b1:= -1/2;b2:= 1 /2; a1:= 0; a2:=2*Pi; " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "Helicoide:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "#alpha := t -> [cos(t), sin(t), t/5]; r := t -> [-sin(t), cos(t),0 ]; a1:=0:a2:=3*Pi:b1:=0:b2:=1:" }}}{PARA 0 "" 0 "" {TEXT -1 84 "O tra \347o da superf\355cie \351 obtido com o deslocamento de um segmento r eto sobre a curva " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 149 " para t em (a1,a2) e na dire\347\343o do vetor r. A vari\341vel s em (b1,b2) localiza o ponto da superf\355cie sobre cada um dos segmen tos tra\347ados a partir de " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" } {TEXT -1 20 "(t) na dire\347\343o r(t)." }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "X:= (t,s) -> evalm(al pha(t) + s*r(t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "plot3d (X(t,s),t=a1..a2,s=b1..b2,style=patchnogrid);" }}}{PARA 0 "" 0 "" {TEXT -1 144 "Vamos agora mostrar graficamente o processo de constru \347\343o subdividinto grosseiramente o tra\347o da curva e para cada \+ divis\343o, tra\347ando o segmento." }{TEXT 269 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "div :=20:points := \{seq( a1 + i*(a2-a1)/di v, i=0..div)\}:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 163 "Rule := proc(t) \n polygonplot3d\n ([evalm(alpha(t) + b1*r(t)), e valm(alpha(t) + b2*r(t))], \n color=navy, thickness=2)\n \+ end:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "Rulings := map(Rule , points):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "Curva := spacecurve(a lpha(t), t=a1..a2, color=navy, thickness=2):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "display(\{Curva\} union Rulings);" }}}{PARA 0 "" 0 " " {TEXT -1 33 "Finalmente desenhamos tudo junto:" }{TEXT 270 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "Superficie:=plot3d(X(t,s), t =a1..a2, s=b1..b2,style=wireframe):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "display(Rulings, Curva,Superficie);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Sela:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "alpha := t -> [cos(t), sin(t), 0]; r := t -> \+ [cos(t), sin(t),2]; a1:=0:a2:=3*Pi:b1:=0:b2:=1:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "X:= (t,s) -> evalm(alpha(t) + s*(r(t+Pi/2)-alpha(t)));" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 83 "Superficie:=plot3d(X(t,s),t=a1..a2,s=b1..b2,style=p atchnogrid):display(Superficie);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "div :=16:points := \{seq( a1 + i*(a2-a1)/div, i=0..div)\}:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 144 " Rule := proc(t)\n polygonplot3d\n ([evalm(alpha(t )), evalm(r(t+Pi/2))], \n color=navy, thickness=2)\n \+ end:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "Rulings := map(Rule, po ints):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "Curva := spacecurve(\{alp ha(t),r(t)\}, t=a1..a2, color=navy, thickness=2):" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 31 "display(\{Curva\} union Rulings);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "display(Rulings,Curva, Superficie);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 261 "" 0 "" {TEXT -1 0 "" }{TEXT 283 47 "Tra \347ando superf\355cies e suas curvas coordenadas" }}{PARA 0 "" 0 "" {TEXT -1 14 "Inicializa\347\343o:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "restart:with(plots):setoptions(scaling=constrained): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "superficie:=[u,v,1-u^2- v^2];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "plot3d(superficie, u=-1..1,v=-1..1);" }}}{PARA 262 "" 0 "" {TEXT -1 34 "Podemos utilizar \+ a op\347\343o gr\341fica " }{HYPERLNK 17 "style" 2 "plot3d,style" "" }{TEXT -1 60 " que determina como o desenho \351 colorido. No modo pa dr\343o ( " }{TEXT 19 6 "patch " }{TEXT -1 70 ") acima, mostra uma tra ma corrspondente \340s curvas coordenadas. O modo " }{TEXT 19 10 "wire frame " }{TEXT -1 103 " colore o tra\347o da superf\355cie como se fos se uma trama de arame seguindo as curvas coordenadas. Observe:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "plot3d(superficie,u=-1..1,v= -1..1,style=wireframe);" }}}{PARA 0 "" 0 "" {TEXT -1 387 "As op\347 \365es gr\341ficas podem ser modificadas com o mouse, clickando com o \+ bot\343o esquerdo no gr\341fico para selecion\341-lo e depois com o bo t\343o direito para selecionar e modificar as op\347\365es style color axes e projection. Com o bot\343o esquerdo \351 poss\355vel rodar o d esenho. Ap\363s cada sele\347\343o \351 preciso clickar duas vezes com o mouse direito para que o gr\341fico seja novamente mostrado. EXPERI MENTE...." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "Observe agora as curvas coordenadas de outra parametriza\347\343o. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "superficie:=[v*cos(u),v* sin(u),1-v^2];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "plot3d(su perficie,u=0..2*Pi,v=0..Pi/2,color=blue);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{PARA 3 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 284 21 "Curvas em Superf\355cies" }{TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "restart:with(plots):setoptio ns3d(scaling=constrained);" }}}{PARA 0 "" 0 "" {TEXT -1 34 "Definindo \+ o dom\355nio da superf\355cie:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "u1:= 0;u2:= 2*Pi; v1:= 0;v2:= 4;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "dominio:= plot3d([u,v,0], u=u1..u2, v=v1..v2, color=blue,style =wireframe):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 85 "Definimos uma curva no dom\355nio. Para t em (t1,t2) a cu rva \351 dada por (f1(t), f2(t)). " }}{EXCHG {PARA 264 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "t1 := 0; t2 := 2*Pi; " }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "f1:= t -> t; f2:= t -> 0.5* t;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "DomCurva:= spacecurv e([f1(t), f2(t),0], t=t1..t2,color=yellow,thickness=2):\ndisplay(domin io,DomCurva);" }}}{PARA 0 "" 0 "" {TEXT -1 34 "Definimos a superf\355c ie e seu tra\347o" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "X:= (u, v) -> [cos(u), sin(u), v];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "Superficie := plot3d(X(u,v), u=u1..u2, v=v1..v2, color=blue):\ndis play(Superficie);" }}}{PARA 0 "" 0 "" {TEXT -1 18 "Definimos a curva \+ " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 16 " na superf\355ci e:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "alpha := t -> subs(u=f 1(t), v=f2(t), X(u,v));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 " CurvaSup:= spacecurve(alpha(t),t=t1..t2, color=yellow,thickness=2):" } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "display(Superficie,CurvaSup);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 3 "" 0 "" {TEXT 285 39 "Loxodr\364mica (exerc\355cio 5 da p\341gina 153)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "restart:with(plots):setoptions3d(sc aling=constrained);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "u1:= -Pi;u2:= Pi; v1:= 0;v2:= Pi;" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "dominio:= plot3d([u,v,0], u=u1..u2, v=v1..v2, color=blue,style=wireframe):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 264 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "t1 := -Pi/2; t2 := Pi/2; " }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 47 "f1:= t -> ln(cot(Pi/4-t/2)); f2:= t -> Pi/2 -t;" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "DomCurva:= spacecurve([f1( t), f2(t),0], t=t1..t2,color=yellow,thickness=2):\ndisplay(dominio,Dom Curva);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "X:= (u,v) -> [cos(u)*sin(v), sin(u)*sin(v), cos(v)]; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "Superficie := plot3d(X( u,v), u=u1..u2, v=v1..v2, color=blue):\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "alpha := t -> subs(u=f1(t), v=f2(t), X(u,v));" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "CurvaSup:= spacecurve(alpha( t),t=t1..t2, color=yellow,thickness=2):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "display(Superficie,CurvaSup);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{MARK "13" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }