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{SECT 0 {PARA 257 "" 0 "" {TEXT 263 1 " " }}{PARA 257 "" 0 "" {TEXT 
265 20 "Geometr\355a Proyectiva" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 257 "" 0 "" {TEXT 264 10 "CURVAS EN " }{XPPEDIT 18 0 "R^2;" "6#*
$%\"RG\"\"#" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" 
{TEXT 262 13 "Gonzalo Comas" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }{TEXT 266 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 
"Lo que sigue es una gu\355a elemental para graficar curvas en " }
{XPPEDIT 18 0 "R^2;" "6#*$%\"RG\"\"#" }{TEXT -1 2 " ." }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 44 "Se comienza colocand
o el cursor en el signo " }{TEXT 257 3 "(+)" }{TEXT -1 155 " en la sec
ci\363n de abajo y cliqueando con el mouse, luego se coloca el cursor \+
en la l\355nea de comandos (que aparecer\341 en letras rojas) y se apr
ieta la tecla " }{TEXT 256 5 "Enter" }{TEXT -1 220 " para ejecutarlos \+
. Si el comando termina con punto y coma, Maple responder\341 con un e
co de la tarea realizada en letras azules y si el comando termina con \+
dos puntos Maple ejecuta el comando pero no muestra el resultado." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 3 "El
 " }{TEXT 258 9 "paquete  " }{TEXT 259 6 "plots." }}{PARA 0 "" 0 "" 
{TEXT 260 32 "Primero vamos a leer el paquete " }{HYPERLNK 17 "plots" 
2 "plots" "" }{TEXT 261 95 " , que contiene distintas rutinas para dib
ujar curvas, superficies y para realizar animaciones." }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "with(plots);" }{TEXT -1 0 "" }}{PARA 7 "" 1 "" {TEXT 
-1 50 "Warning, the name changecoords has been redefined\n" }}{PARA 
12 "" 1 "" {XPPMATH 20 "6#7W%(animateG%*animate3dG%-animatecurveG%&arr
owG%-changecoordsG%,complexplotG%.complexplot3dG%*conformalG%,conforma
l3dG%,contourplotG%.contourplot3dG%*coordplotG%,coordplot3dG%-cylinder
plotG%,densityplotG%(displayG%*display3dG%*fieldplotG%,fieldplot3dG%)g
radplotG%+gradplot3dG%-implicitplotG%/implicitplot3dG%(inequalG%-listc
ontplotG%/listcontplot3dG%0listdensityplotG%)listplotG%+listplot3dG%+l
oglogplotG%(logplotG%+matrixplotG%(odeplotG%'paretoG%*pointplotG%,poin
tplot3dG%*polarplotG%,polygonplotG%.polygonplot3dG%4polyhedra_supporte
dG%.polyhedraplotG%'replotG%*rootlocusG%,semilogplotG%+setoptionsG%-se
toptions3dG%+spacecurveG%1sparsematrixplotG%+sphereplotG%)surfdataG%)t
extplotG%+textplot3dG%)tubeplotG" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 
9 "Ejemplo I" }}{PARA 0 "" 0 "" {TEXT -1 34 "Graficaremos la curva de \+
ecuaci\363n " }{XPPEDIT 18 0 "Y-X^2;" "6#,&%\"YG\"\"\"*$%\"XG\"\"#!\"
\"" }{TEXT -1 79 ". Lo haremos de dos maneras distintas. Primero harem
os el gr\341fico de la funci\363n" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "f;
" "6#%\"fG" }{TEXT -1 17 " , definida como " }{XPPEDIT 18 0 "f(X) = X^
2;" "6#/-%\"fG6#%\"XG*$F'\"\"#" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "plot(X^2,X=-2..2,scaling=con
strained);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 91 "Los \+
par\341metros obligatorios del comando plot son una f\363rmula en una \+
variable (en este caso " }{XPPEDIT 18 0 "X^2;" "6#*$)%\"XG\"\"#\"\"\"
" }{TEXT -1 95 " y el rango de valores que va a tomar la misma, separa
dos estos por \"..\" (en este caso -2 y 2)." }}{PARA 0 "" 0 "" {TEXT 
-1 136 "La opci\363n \"scaling=constrained\" realiza el dibujo a escal
a. Esto es opcional, si no se coloca se usa el default  que es \"uncon
strained\"." }}{PARA 0 "" 0 "" {TEXT -1 112 "Haciendo click en el dibu
jo aparece un nuevo men\372 que permite cambiar las opciones del mismo
 (explorar a gusto)." }}{PARA 0 "" 0 "" {TEXT -1 54 "En segundo t\351r
mino graficaremos el conjunto de pares (" }{XPPEDIT 18 0 "X,Y;" "6$%\"
XG%\"YG" }{TEXT -1 28 ") que verifican la ecuaci\363n " }{XPPEDIT 18 
0 "Y-X^2 = 0;" "6#/,&%\"YG\"\"\"*$%\"XG\"\"#!\"\"\"\"!" }{TEXT -1 1 ".
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "implicitplot(Y-X^2=0,X=
-2..2,Y=0..4);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 217 
"En este caso los par\341metros obligatorios son una f\363rmula en dos
 variables, y los rangos de las mismas. Observar que al no usar la opc
i\363n \"scaling\" se usa el default y en consecuencia la escala en lo
s ejes es distinta." }}{PARA 0 "" 0 "" {TEXT -1 132 "Podemos usar una \+
tercer manera de graficar si conocemos una parametrizaci\363n de la cu
rva. En nuestro caso una parametrizaci\363n ser\355a (" }{XPPEDIT 18 
0 "t,t^2;" "6$%\"tG*$F#\"\"#" }{TEXT -1 2 ")." }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 42 "plot([t,t^2,t=-2..2],scaling=constrained);" }
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 129 "En este caso los
 par\341metros obligatorios son las dos parametrizaciones, y el rango \+
del par\341metro, todo encerrado entre corchetes." }}}}}{SECT 0 {PARA 
3 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT 267 10 "Ejempl
o II" }}{PARA 0 "" 0 "" {TEXT -1 44 "Vamos a graficar ahora la curva d
e ecuaci\363n " }{XPPEDIT 18 0 "X^2+Y^2 = 1" "6#/,&*$%\"XG\"\"#\"\"\"*
$%\"YGF'F(F(" }{TEXT -1 40 ". Lo hacemos primero de forma impl\355cita
:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "implicitplot(X^2+Y^2=1,
X=-1..1,Y=-1..1,scaling=constrained);" }{TEXT -1 0 "" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 31 "Para dibujarla como funci\363n de " }{XPPEDIT 
18 0 "X;" "6#%\"XG" }{TEXT -1 25 " definimos dos funciones " }
{XPPEDIT 18 0 "F_1,F_2;" "6$%$F_1G%$F_2G" }{TEXT -1 1 " " }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "F_1:=X->sqrt(1-X^2);" }{TEXT -1 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "F_2:=X->-sqrt(1-X^2);" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "Y ahora las graficamos juntas:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "plot([F_1(X),F_2(X)],X=
-1..1,scaling=constrained);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 149 "Observar que al estar el par\341metro fuera de los corch
etes el programa realiza el gr\341fico de las dos funciones, y no lo c
onsidera una parametrizaci\363n." }}{PARA 0 "" 0 "" {TEXT -1 71 "Final
mente consideramos la parametrizaci\363n usual de la circunferencia (
" }{XPPEDIT 18 0 "cos(t),sen(t);" "6$-%$cosG6#%\"tG-%$senG6#F&" }
{TEXT -1 2 "):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "plot([cos
(t),sin(t),t=0..2*Pi],scaling=constrained);" }{TEXT -1 0 "" }}}}{SECT 
0 {PARA 3 "" 0 "" {TEXT 268 11 "Ejemplo III" }}{PARA 0 "" 0 "" {TEXT 
-1 42 "Haremos lo mismo con la curva de ecuaci\363n " }{XPPEDIT 18 0 "
Y^2-X^3-X^2;" "6#,(*$%\"YG\"\"#\"\"\"*$%\"XG\"\"$!\"\"*$F)F&F+" }
{TEXT -1 3 "=0." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "implicitp
lot(Y^2-X^3-X^2=0,X=-1..1,Y=-1..1,scaling=constrained);" }{TEXT -1 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "G_1:=X->sqrt(X^3+X^2);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "G_2:=X->-sqrt(X^3+X^2);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "plot(\{G_1(X),G_2(X)\},
X=-1..1,scaling=constrained);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 51 "Una parametrizaci\363n (no trivial) de esta curva es (
" }{XPPEDIT 18 0 "t^2-1,t*(t^2-1);" "6$,&*$%\"tG\"\"#\"\"\"F'!\"\"*&F%
F',&*$F%F&F'F'F(F'" }{TEXT -1 2 ")." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 56 "plot([t^2-1,t*(t^2-1),t=-1.5..1.5],scaling=constraine
d);" }{TEXT -1 0 "" }}}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 12 "Ejercita
ci\363n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 73 "Graficar las curvas da
das por las siguientes ecuaciones (usar el comando " }{HYPERLNK 17 "im
plicitplot" 2 "implicitplot" "" }{TEXT -1 128 ", m\341s adelante verem
os c\363mo encontrar parametrizaciones para las curvas). Sugerencia: p
robar con distintos rangos de valores de " }{XPPEDIT 18 0 "X;" "6#%\"X
G" }{TEXT -1 3 " e " }{XPPEDIT 18 0 "Y;" "6#%\"YG" }{TEXT -1 39 ", has
ta quedar conforme con el gr\341fico." }}}{EXCHG {PARA 0 "" 0 "" 
{XPPEDIT 18 0 "Y^2-X^3+X;" "6#,(*$%\"YG\"\"#\"\"\"*$%\"XG\"\"$!\"\"F)F
'" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 18 0 "Y^2-X^3;" "6#,&*$%\"YG\"\"
#\"\"\"*$%\"XG\"\"$!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 18 0 "(X^
2+Y^2)^2+3*X^2*Y-Y^3;" "6#,(*$,&*$%\"XG\"\"#\"\"\"*$%\"YGF(F)F(F)*(\"
\"$F)*$F'F(F)F+F)F)*$F+F-!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 18 
0 "(X^2+Y^2)^3-4*X^2*Y^2;" "6#,&*$,&*$%\"XG\"\"#\"\"\"*$%\"YGF(F)\"\"$
F)*(\"\"%F)*$F'F(F)F+F(!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 18 0 
"X^3+X^2+Y^2;" "6#,(*$%\"XG\"\"$\"\"\"*$F%\"\"#F'*$%\"YGF)F'" }}}
{EXCHG {PARA 0 "" 0 "" {XPPEDIT 18 0 "2*X^4-3*X^2*Y+Y^2-2*Y^3+Y^4;" "6
#,,*&\"\"#\"\"\"*$%\"XG\"\"%F&F&*(\"\"$F&*$F(F%F&%\"YGF&!\"\"*$F-F%F&*
&F%F&*$F-F+F&F.*$F-F)F&" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 18 0 "X^4+
X^2*Y^2-2*X^2*Y-X*Y^2+Y^2;" "6#,,*$%\"XG\"\"%\"\"\"*&F%\"\"#%\"YGF)F'*
(F)F'*$F%F)F'F*F'!\"\"*&F%F'*$F*F)F'F-*$F*F)F'" }}}{EXCHG {PARA 0 "" 
0 "" {XPPEDIT 18 0 "X^6-X^2*Y^3-Y^5;" "6#,(*$%\"XG\"\"'\"\"\"*&F%\"\"#
%\"YG\"\"$!\"\"*$F*\"\"&F," }}}}{MARK "0 0" 1 }{VIEWOPTS 1 1 0 1 1 
1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }