{VERSION 5 0 "IBM INTEL NT" "5.0" }
{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 "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "Times" 0 1 
0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "Courier" 0 14 0 0 0 0 
0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "Times" 1 12 0 0 0 0 0 0 0 0 0 
0 0 0 0 1 }{CSTYLE "" -1 261 "Times" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 
}{CSTYLE "" -1 262 "Times" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "
" -1 263 "Courier" 0 1 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 "" -1 266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 
0 1 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" 
-1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 270 "" 0 1 0 
0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 272 "" 1 18 0 0 0 0 0 1 0 0 
0 0 0 0 0 1 }{CSTYLE "" -1 273 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 275 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
276 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 1 14 0 0 
0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 1 14 0 0 0 0 0 0 0 0 0 
0 0 0 0 0 }{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 "Heading 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 "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 
14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 8 2 0 0 0 0 0 0 -1 0 }
{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "Times" 1 12 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 "" 0 257 1 {CSTYLE "
" -1 -1 "Times" 1 12 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 "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 
0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 259 1 
{CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 
0 0 0 0 0 -1 0 }{PSTYLE "" 0 261 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 
0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 262 
1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 
0 0 0 0 0 0 -1 0 }}
{SECT 0 {PARA 258 "" 0 "" {TEXT 272 1 " " }}{PARA 262 "" 0 "" {TEXT 
275 21 "Geometr\355a Diferencial" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 261 "" 0 "" {TEXT 273 10 "CURVAS EN " }{XPPEDIT 18 0 "R^3;" "6#*
$%\"RG\"\"$" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" 
{TEXT 269 100 "Laboratorio confeccionado por  Mirta Iriondo, FaMAF, UN
C  y  Departamento de Matem\341tica, FCEyN, UBA." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 276 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 170 "En este labo
ratorio vamos a familiarizarnos con  los comandos para dibujar curvas \+
y familias de curvas en el espacio y calcular curvatura, torsi\363n y \+
el triedo de Frenet. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 
0 "" {TEXT -1 41 "Comience colocando el cursor en el signo " }{TEXT 
257 3 "(+)" }{TEXT -1 147 " en la secci\363n de abajo y cliquee con el
 mouse, luego coloque el cursor en la l\355nea de comandos (que aparec
er\341 en letras rojas) y apriete la tecla " }{TEXT 256 5 "Enter" }
{TEXT -1 220 " para ejecutarlos . Si el comando termina con punto y co
ma, Maple responder\341 con un eco de la tarea realizada en letras azu
les y si el comando termina con dos puntos Maple ejecuta el comando pe
ro no muestra el resultado." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 
1 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 258 14 "Los paquetes  " }{TEXT 
259 5 "plots" }{TEXT -1 3 " y " }{TEXT 263 6 "linalg" }}{PARA 0 "" 0 "
" {TEXT 260 32 "Primero vamos a leer el paquete " }{HYPERLNK 17 "plots
" 2 "plots" "" }{TEXT 261 111 " , que contiene distintas rutinas para \+
dibujar curvas, superficies y para realizar animaciones, y  el paquete
  " }{HYPERLNK 17 "linalg" 2 "linalg" "" }{TEXT 262 79 "  que contiene
 rutinas de \341lgebra lineal para trabajar con vectores y matrices." 
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 12 "with(plots):" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 
9 "Ejemplo I" }}{PARA 0 "" 0 "" {TEXT -1 31 "Considere  la h\351lice d
ada  por " }{XPPEDIT 18 0 "a = [cos(t), sin(t), t];" "6#/%\"aG7%-%$cos
G6#%\"tG-%$sinG6#F)F)" }{TEXT -1 72 ",  y calcule el triedo de Frenet,
 la curvatura y la torsi\363n de la misma." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "Comenzamos guardando en la vari
able " }{XPPEDIT 18 0 "a;" "6#%\"aG" }{TEXT -1 14 " esta funci\363n." 
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "a:=[cos(t),sin(t),t];" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 90 "Calcularemos ahora su velocidad y \+
aceleraci\363n para con ello calcular el triedo de Frenet  " }
{XPPEDIT 18 0 "[T, N, B];" "6#7%%\"TG%\"NG%\"BG" }{TEXT -1 4 " .  " }}
{PARA 0 "" 0 "" {TEXT -1 17 "En las variables " }{TEXT 264 2 "da" }
{TEXT -1 3 " y " }{TEXT 265 3 "dda" }{TEXT -1 44 ", guardamos la prime
ra y segunda derivada de" }{TEXT 266 2 " a" }{TEXT -1 17 ", respectiva
mente" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "da:=diff(a,t);" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "dda:=diff(da,t);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "La tangente la calculamos como" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "T:=simplify(evalm(da/sqrt(i
nnerprod(da,da))));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "Observe qu
e aqu\355 hemos usado el comando " }{HYPERLNK 17 "innerprod" 2 "innerp
rod" "" }{TEXT -1 205 "  para calcular el producto escalar entre vecto
res (en este caso la norma de la velocidad). Para calcular la binormal
 primero calculamos el producto cruz entre la velocidad y la aceleraci
\363n con el comando " }{HYPERLNK 17 "crossprod" 2 "crossprod" "" }
{TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "beta:=cros
sprod(da,dda);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Entonces la bin
ormal es " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "B:=simplify(ev
alm(beta/sqrt(innerprod(beta,beta))));" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 33 "y la correspondiente normal como " }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 28 "N:=simplify(crossprod(B,T));" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 70 "Ahora procedemos a calcular  la curvatura y la \+
torsi\363n respectivamente" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "kappa:=simplify(sqrt(innerprod(beta
,beta))/(innerprod(da,da))^(3/2));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 67 "tau:=simplify(innerprod(B,diff(dda,t))/sqrt(innerprod
(beta,beta)));" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 277 10 "Ejemplo II" 
}}{PARA 0 "" 0 "" {TEXT -1 193 "Dibuje una animaci\363n de la curva co
nsiderada en el Ejemplo I que contenga el plano osculador y la esfera \+
osculadora (observe que la intersecci\363n de ambas superficies dar
\341 el circulo osculador)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 75 "El plano osculador es el plano generado por la \+
tangente y la normal, o sea " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "oscplan:=evalm(s*T+u*N+a);" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "Con el comando  " }{HYPERLNK 
17 "animate3d" 2 "animate3d" "" }{TEXT -1 92 "  vamos a dibujar en for
ma consecutiva esta familia de planos y la guardamos en la variable " 
}{TEXT 270 1 "P" }{TEXT -1 2 ". " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 79 "P:=animate3d(oscplan,s=-3..3,u=-3..6,t=0..15,scaling=
constrained,style=hidden):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "El \+
centro de curvatura  de la curva esta dado por " }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 23 "Evol:=evalm(a+N/kappa);" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 43 "y la  animaci\363n de la esfera osculadora es " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 180 "C:=animate3d([Evol[1]+1/kap
pa*sin(theta)*cos(phi),Evol[2]+1/kappa*sin(theta)*sin(phi),Evol[3]+1/k
appa*cos(theta)],phi=0..2*Pi,theta=0..Pi, t=0..15,scaling=constrained,
color=green):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "Generamos la ani
maci\363n de la curva de la siguiente manera" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 20 "a1:=subs(t=t*u*s,a);" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 80 "G:=animate3d(a1,t=-2..15,u=1..1.0001,s=1..1.0001,co
lor=red,scaling=constrained):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "
Usamos ahora el comando " }{HYPERLNK 17 "display3d" 2 "display3d" "" }
{TEXT -1 84 "  para graficar la animaci\363n de la curva, el plano osc
ulador y la esfera osculadora " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 19 "display3d([G,P,C]);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 73 "Cliqueando en la figura aparecer\341 una barra de herrami
enta de animaci\363n. " }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 278 11 "Ejem
plo III" }}{PARA 0 "" 0 "" {TEXT -1 130 "Considere nuevamente la curva
 del ejemplo I y dibuje una animaci\363n de una figura tubular de radi
o constante alrededor de la misma." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 47 "La superficie tubular de radio 1 est\341 \+
dada por " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "tubecurv:=evalm
(a+(-cos(theta)*N+sin(theta)*B));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
28 "El disco generado para cada " }{TEXT 267 1 "t" }{TEXT -1 3 " es" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "circ:=evalm(a+(-cos(theta)
*N*s+sin(theta)*B*s));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 81 "Con est
a animaci\363n mostramos los c\355rculos que van a generar la superfic
ie tubular" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "F:=animate3d(
circ,theta=0..2*Pi,s=0..1,t=0..15,style=hidden):" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 62 "Generamos la animaci\363n de la curva como en el e
jemplo anterior" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "a1:=subs
(t=t*theta*s,a);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "G:=anim
ate3d(a1,t=0..15,theta=1..1.001,s=1..1.001,color=red,scaling=constrain
ed):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "display3d([F,G]);" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 91 "Generaremos ahora la funci\363n
 que nos permitar\341 hacer  la animaci\363n de la superficie tubular
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "tubecurv1:=subs(t=s*t,e
valm(a+(-cos(theta)*N+sin(theta)*B)));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 66 "H:=animate3d(tubecurv1,t=0..15,theta=0..2*Pi,s=0..1,s
tyle=hidden):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "display3d(
[F,G,H]);" }}}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 12 "Ejercitaci\363n" 
}}{PARA 0 "" 0 "" {TEXT -1 19 "Considere la curva " }{XPPEDIT 18 0 "a \+
= [1+cos(t), sin(t), 2*sin(1/2*t)];" "6#/%\"aG7%,&\"\"\"F'-%$cosG6#%\"
tGF'-%$sinG6#F+*&\"\"#F'-F-6#*(F'F'F0!\"\"F+F'F'" }{TEXT -1 109 " y re
alice la misma tarea que en los ejemplos I, II y III. Realice  una ani
maci\363n tubular con radio variable." }}}}{MARK "1 0" 0 }{VIEWOPTS 1 
1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }