\documentclass[11pt,leqno]{article}
\usepackage[latin1]{inputenc}
\usepackage[spanish]{babel}
\usepackage{amssymb}
\usepackage{graphicx}
\setlength{\topmargin}{-1.5cm}  \setlength{\textheight}{24cm}
\setlength{\oddsidemargin}{0.0cm} \setlength{\evensidemargin}{0.0cm}
\setlength{\textwidth}{16cm} \thispagestyle{empty}
\NeedsTeXFormat{LaTeX2e}[1995/12/01]
\usepackage{maple2e}
 \def\emptyline{\vspace{12pt}}
\DefineParaStyle{Left Justified Maple Output}
\DefineParaStyle{Maple Output}
\DefineParaStyle{Maple Plot}
\DefineParaStyle{Warning}
\DefineParaStyle{_pstyle1}
\DefineParaStyle{_pstyle2}
\DefineParaStyle{_pstyle3}
\DefineParaStyle{_pstyle4}
\DefineParaStyle{_pstyle5}
\DefineParaStyle{_pstyle6}
\DefineParaStyle{_pstyle7}
\DefineParaStyle{_pstyle8}
\DefineParaStyle{_pstyle9}
\DefineCharStyle{2D Output}
\DefineCharStyle{_cstyle1}
\DefineCharStyle{_cstyle10}
\DefineCharStyle{_cstyle11}
\DefineCharStyle{_cstyle2}
\DefineCharStyle{_cstyle3}
\DefineCharStyle{_cstyle4}
\DefineCharStyle{_cstyle5}
\DefineCharStyle{_cstyle6}
\DefineCharStyle{_cstyle7}
\DefineCharStyle{_cstyle8}
\DefineCharStyle{_cstyle9}

\begin{document}

\begin{center}
\textsc{Universidad de los Andes \hfill Departamento de Matemáticas}
\newline
\huge{Geometría Diferencial MATE2410\\
Laboratorios de Maple\\
Lab4: Geodésicas}
\author{Prof. José Ricardo ARTEAGA B.}
\end{center}

\section{\QTR{_cstyle3}{Objetivos:}\QTR{_cstyle4}{ }}

\begin{maplegroup}
\begin{_pstyle3}
\QTR{_cstyle5}{Usar }\QTR{_cstyle6}{Maple}\QTR{_cstyle5}{ para hacer
muchos cálculos algebraicos necesarios para hallar geodésicas.}
\end{_pstyle3}

\end{maplegroup}

\section{\QTR{_cstyle7}{Geodésicas}}

\subsection{\QTR{_cstyle8}{Cálculos básicos}}

\begin{maplegroup}
\begin{_pstyle6}
\begin{mapleinput}
\mapleinline{active}{1d}{restart: with(plots): with(LinearAlgebra):}{%
}
\mapleinline{active}{1d}{
EFG:=proc(X)# PRIMERA FORMA FUNDAMENTAL}{%
}
\mapleinline{active}{1d}{
local Xu,Xv,E,F,G;}{%
}
\mapleinline{active}{1d}{
Xu:=<diff(X[1],u),diff(X[2],u),diff(X[3],u)>;}{%
}
\mapleinline{active}{1d}{
Xv:=<diff(X[1],v),diff(X[2],v),diff(X[3],v)>;}{%
}
\mapleinline{active}{1d}{
E:=DotProduct(Xu,Xu,conjugate=false);}{%
}
\mapleinline{active}{1d}{
F:=DotProduct(Xu,Xv,conjugate=false);}{%
}
\mapleinline{active}{1d}{
G:=DotProduct(Xv,Xv,conjugate=false);}{%
}
\mapleinline{active}{1d}{
simplify([E,F,G]);}{%
}
\mapleinline{active}{1d}{
end:}{%
}
\end{mapleinput}
\end{_pstyle6}

\begin{_pstyle6}
\begin{mapleinput}
\mapleinline{active}{1d}{UN:=proc(X) # VECTOR NORMAL}{%
}
\mapleinline{active}{1d}{
local Xu,Xv,Z,s;}{%
}
\mapleinline{active}{1d}{
Xu:=<diff(X[1],u),diff(X[2],u),diff(X[3],u)>;}{%
}
\mapleinline{active}{1d}{
Xv:=<diff(X[1],v),diff(X[2],v),diff(X[3],v)>;}{%
}
\mapleinline{active}{1d}{
Z:=CrossProduct(Xu,Xv);}{%
}
\mapleinline{active}{1d}{
s:=VectorNorm(Z,Euclidean,conjugate=false);}{%
}
\mapleinline{active}{1d}{
simplify(<Z[1]/s|Z[2]/s|Z[3]/s>,sqrt,trig,simbolic);}{%
}
\mapleinline{active}{1d}{
end:}{%
}
\end{mapleinput}
\end{_pstyle6}

\begin{_pstyle6}
\begin{mapleinput}
\mapleinline{active}{1d}{lmn:=proc(X) # SEGUNDA FORMA FUNDAMENTAL}{%
}
\mapleinline{active}{1d}{
local Xu,Xv,Xuu,Xuv,Xvv,U,l,m,n;}{%
}
\mapleinline{active}{1d}{
Xu:=<diff(X[1],u),diff(X[2],u),diff(X[3],u)>;}{%
}
\mapleinline{active}{1d}{
Xv:=<diff(X[1],v),diff(X[2],v),diff(X[3],v)>;}{%
}
\mapleinline{active}{1d}{
Xuu:=<diff(Xu[1],u),diff(Xu[2],u),diff(Xu[3],u)>;}{%
}
\mapleinline{active}{1d}{
Xuv:=<diff(Xu[1],v),diff(Xu[2],v),diff(Xu[3],v)>;}{%
}
\mapleinline{active}{1d}{
Xvv:=<diff(Xv[1],v),diff(Xv[2],v),diff(Xv[3],v)>;}{%
}
\mapleinline{active}{1d}{
U:=UN(X);}{%
}
\mapleinline{active}{1d}{
l:=DotProduct(U,Xuu,conjugate=false);}{%
}
\mapleinline{active}{1d}{
m:=DotProduct(U,Xuv,conjugate=false);}{%
}
\mapleinline{active}{1d}{
n:=DotProduct(U,Xvv,conjugate=false);}{%
}
\mapleinline{active}{1d}{
simplify([l,m,n],sqrt,trig,symbolic);}{%
}
\mapleinline{active}{1d}{
end:}{%
}
\end{mapleinput}
\end{_pstyle6}

\begin{_pstyle6}
\begin{mapleinput}
\mapleinline{active}{1d}{MK:=proc(X)# CURVATURA MEDIA}{%
}
\mapleinline{active}{1d}{
local E,F,G,l,m,n,S,T;}{%
}
\mapleinline{active}{1d}{
S:=EFG(X); T:=lmn(X);}{%
}
\mapleinline{active}{1d}{
E:=S[1]; F:=S[2]; G:=S[3];}{%
}
\mapleinline{active}{1d}{
l:=T[1]; m:=T[2]; n:=T[3];}{%
}
\mapleinline{active}{1d}{
simplify((G*l+E*n-2*F*m)/(2*E*G-2*F^2),sqrt,trig,symbolic);}{%
}
\mapleinline{active}{1d}{
end:}{%
}
\end{mapleinput}
\end{_pstyle6}

\begin{_pstyle6}
\begin{mapleinput}
\mapleinline{active}{1d}{GK:=proc(X)# CURVATURA DE GAUSS}{%
}
\mapleinline{active}{1d}{
local E,F,G,l,m,n,S,T;}{%
}
\mapleinline{active}{1d}{
S:=EFG(X); T:=lmn(X);}{%
}
\mapleinline{active}{1d}{
E:=S[1]; F:=S[2]; G:=S[3];}{%
}
\mapleinline{active}{1d}{
l:=T[1]; m:=T[2]; n:=T[3];}{%
}
\mapleinline{active}{1d}{
simplify((l*n-m^2)/(E*G-F^2),sqrt,trig,symbolic);}{%
}
\mapleinline{active}{1d}{
end:}{%
}
\end{mapleinput}
\end{_pstyle6}

\begin{_pstyle6}
\begin{mapleinput}
\mapleinline{active}{1d}{shape:=proc(X)# OPERADOR FORMA}{%
}
\mapleinline{active}{1d}{
local Y,Z,a,b,c,d;}{%
}
\mapleinline{active}{1d}{
Y:=EFG(X);}{%
}
\mapleinline{active}{1d}{
Z:=lmn(X);}{%
}
\mapleinline{active}{1d}{
a:=simplify((Z[1]*Y[3]-Z[2]*Y[2])/(Y[1]*Y[3]-Y[2]^2));}{%
}
\mapleinline{active}{1d}{
b:=simplify((Z[2]*Y[1]-Z[1]*Y[2])/(Y[1]*Y[3]-Y[2]^2));}{%
}
\mapleinline{active}{1d}{
c:=simplify((Z[2]*Y[3]-Z[3]*Y[2])/(Y[1]*Y[3]-Y[2]^2));}{%
}
\mapleinline{active}{1d}{
d:=simplify((Z[3]*Y[1]-Z[2]*Y[2])/(Y[1]*Y[3]-Y[2]^2));}{%
}
\mapleinline{active}{1d}{
[S(x_u)=a*x_u+b*x_v,S(x_v)=c*x_u+d*x_v];}{%
}
\mapleinline{active}{1d}{
end:}{%
}
\end{mapleinput}
\end{_pstyle6}

\begin{_pstyle6}
\begin{mapleinput}
\mapleinline{active}{1d}{shape_matrix:=proc(X)# MATRIZ OPERADOR FORMA}{%
}
\mapleinline{active}{1d}{
local Y,Z,a,b,c,d;}{%
}
\mapleinline{active}{1d}{
Y:=EFG(X);}{%
}
\mapleinline{active}{1d}{
Z:=lmn(X);}{%
}
\mapleinline{active}{1d}{
a:=simplify((Z[1]*Y[3]-Z[2]*Y[2])/(Y[1]*Y[3]-Y[2]^2));}{%
}
\mapleinline{active}{1d}{
b:=simplify((Z[2]*Y[1]-Z[1]*Y[2])/(Y[1]*Y[3]-Y[2]^2));}{%
}
\mapleinline{active}{1d}{
c:=simplify((Z[2]*Y[3]-Z[3]*Y[2])/(Y[1]*Y[3]-Y[2]^2));}{%
}
\mapleinline{active}{1d}{
d:=simplify((Z[3]*Y[1]-Z[2]*Y[2])/(Y[1]*Y[3]-Y[2]^2));}{%
}
\mapleinline{active}{1d}{
Matrix([[a,c],[c,d]]);}{%
}
\mapleinline{active}{1d}{
end:}{%
}
\end{mapleinput}
\end{_pstyle6}

\mapleresult
\begin{maplettyout}
Warning, the name changecoords has been redefined
\end{maplettyout}

\end{maplegroup}

\subsection{\QTR{_cstyle8}{Ecuaciones de las geodésicas}}

\begin{maplegroup}
\begin{_pstyle6}
\begin{mapleinput}
\mapleinline{active}{1d}{geoeq:=proc(X)}{%
}
\mapleinline{active}{1d}{
local M,eq1,eq2;}{%
}
\mapleinline{active}{1d}{
M:=EFG(X);}{%
}
\mapleinline{active}{1d}{
eq1:=diff(u(t),t$2)+subs(\{u=u(t),v=v(t)
\},diff(M[1],u)/(2*M[1]))*diff(u(t),t)^2}{%
}
\mapleinline{active}{1d}{
+subs(\{u=u(t),v=v(t)\},diff(M[1],v)/(M[1]))*diff(u(t),t)*diff(v(t),t)
}{%
}
\mapleinline{active}{1d}{
-subs(\{u=u(t),v=v(t)\},diff(M[3],u)/(2*M[1]))*diff(v(t),t)^2=0;}{%
}
\mapleinline{active}{1d}{
eq2:=diff(v(t),t$2)-subs(\{u=u(t),v=v(t)
\},diff(M[1],v)/(2*M[3]))*diff(u(t),t)^2}{%
}
\mapleinline{active}{1d}{
+subs(\{u=u(t),v=v(t)\},diff(M[3],u)/(M[3]))*diff(u(t),t)*diff(v(t),t)
}{%
}
\mapleinline{active}{1d}{
+ subs(\{u=u(t),v=v(t)\},diff(M[3],v)/(2*M[3]))*diff(v(t),t)^2=0;}{%
}
\mapleinline{active}{1d}{
eq1,eq2;}{%
}
\mapleinline{active}{1d}{
end: }{%
}
\end{mapleinput}
\end{_pstyle6}

\end{maplegroup}

\subsection{\QTR{_cstyle8}{Ejemplos}}

\begin{maplegroup}
\begin{_pstyle6}
\begin{mapleinput}
\mapleinline{active}{1d}{sphere:=<cos(u)*cos(v)|sin(u)*cos(v)|sin(v)>;}{%
}
\end{mapleinput}
\end{_pstyle6}

\begin{_pstyle6}
\begin{mapleinput}
\mapleinline{active}{1d}{EFG(sphere);}{%
}
\end{mapleinput}
\end{_pstyle6}

\begin{_pstyle6}
\begin{mapleinput}
\mapleinline{active}{1d}{geoeq(sphere);}{%
}
\end{mapleinput}
\end{_pstyle6}

\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{sphere := _rtable[2723872];}{%
\[
\mathit{sphere} := [\mathrm{cos}(u)\,\mathrm{cos}(v), \,\mathrm{
sin}(u)\,\mathrm{cos}(v), \,\mathrm{sin}(v)]
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{[cos(v)^2, 0, 1];}{%
\[
[\mathrm{cos}(v)^{2}, \,0, \,1]
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{diff(u(t),`$`(t,2))-2/cos(v(t))*sin(v(t))*diff(u(t),t)*diff(v(t),t) =
0, diff(v(t),`$`(t,2))+cos(v(t))*sin(v(t))*diff(u(t),t)^2 = 0;}{%
\[
({\frac {\partial ^{2}}{\partial t^{2}}}\,\mathrm{u}(t)) -
{\displaystyle \frac {2\,\mathrm{sin}(\mathrm{v}(t))\,({\frac {
\partial }{\partial t}}\,\mathrm{u}(t))\,({\frac {\partial }{
\partial t}}\,\mathrm{v}(t))}{\mathrm{cos}(\mathrm{v}(t))}} =0,
\,({\frac {\partial ^{2}}{\partial t^{2}}}\,\mathrm{v}(t)) +
\mathrm{cos}(\mathrm{v}(t))\,\mathrm{sin}(\mathrm{v}(t))\,(
{\frac {\partial }{\partial t}}\,\mathrm{u}(t))^{2}=0
\]
%
}
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
\begin{_pstyle6}
\begin{mapleinput}
\mapleinline{active}{1d}{torus:=<(4+cos(u))*cos(v)|(4+cos(u))*sin(v)|sin(u)>;}{%
}
\end{mapleinput}
\end{_pstyle6}

\begin{_pstyle6}
\begin{mapleinput}
\mapleinline{active}{1d}{EFG(torus);}{%
}
\end{mapleinput}
\end{_pstyle6}

\begin{_pstyle6}
\begin{mapleinput}
\mapleinline{active}{1d}{geoeq(torus);}{%
}
\end{mapleinput}
\end{_pstyle6}

\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{torus := _rtable[2749748];}{%
\[
\mathit{torus} := [(4 + \mathrm{cos}(u))\,\mathrm{cos}(v), \,(4
 + \mathrm{cos}(u))\,\mathrm{sin}(v), \,\mathrm{sin}(u)]
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{[1, 0, 16+8*cos(u)+cos(u)^2];}{%
\[
[1, \,0, \,16 + 8\,\mathrm{cos}(u) + \mathrm{cos}(u)^{2}]
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{diff(u(t),`$`(t,2))-(-4*sin(u(t))-cos(u(t))*sin(u(t)))*diff(v(t),t)^2
= 0,
diff(v(t),`$`(t,2))+(-8*sin(u(t))-2*cos(u(t))*sin(u(t)))/(16+8*cos(u(t
))+cos(u(t))^2)*diff(u(t),t)*diff(v(t),t) = 0;}{%
\maplemultiline{
({\frac {\partial ^{2}}{\partial t^{2}}}\,\mathrm{u}(t)) - ( - 4
\,\mathrm{sin}(\mathrm{u}(t)) - \mathrm{cos}(\mathrm{u}(t))\,
\mathrm{sin}(\mathrm{u}(t)))\,({\frac {\partial }{\partial t}}\,
\mathrm{v}(t))^{2}=0,  \\
({\frac {\partial ^{2}}{\partial t^{2}}}\,\mathrm{v}(t)) +
{\displaystyle \frac {( - 8\,\mathrm{sin}(\mathrm{u}(t)) - 2\,
\mathrm{cos}(\mathrm{u}(t))\,\mathrm{sin}(\mathrm{u}(t)))\,(
{\frac {\partial }{\partial t}}\,\mathrm{u}(t))\,({\frac {
\partial }{\partial t}}\,\mathrm{v}(t))}{16 + 8\,\mathrm{cos}(
\mathrm{u}(t)) + \mathrm{cos}(\mathrm{u}(t))^{2}}} =0 }
%
}
\end{maplelatex}

\end{maplegroup}
\begin{maplegroup}
\begin{_pstyle6}
\begin{mapleinput}
\mapleinline{active}{1d}{witch:=<2*tan(u)|2*cos(u)^2*cos(v)|2*cos(u)^2*sin(v)>;}{%
}
\end{mapleinput}
\end{_pstyle6}

\begin{_pstyle6}
\begin{mapleinput}
\mapleinline{active}{1d}{EFG(witch);}{%
}
\end{mapleinput}
\end{_pstyle6}

\begin{_pstyle6}
\begin{mapleinput}
\mapleinline{active}{1d}{geoeq(witch);}{%
}
\end{mapleinput}
\end{_pstyle6}

\mapleresult
\begin{maplelatex}
\mapleinline{inert}{2d}{witch := _rtable[3218368];}{%
\[
\mathit{witch} :=  \left[  \! 2\,\mathrm{tan}(u), \,2\,\mathrm{
cos}(u)^{2}\,\mathrm{cos}(v), \,2\,\mathrm{cos}(u)^{2}\,\mathrm{
sin}(v) \!  \right]
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{[-4*(-1-4*cos(u)^6+4*cos(u)^8)/cos(u)^4, 0, 4*cos(u)^4];}{%
\[
[ - 4\,{\displaystyle \frac { - 1 - 4\,\mathrm{cos}(u)^{6} + 4\,
\mathrm{cos}(u)^{8}}{\mathrm{cos}(u)^{4}}} , \,0, \,4\,\mathrm{
cos}(u)^{4}]
\]
%
}
\end{maplelatex}

\begin{maplelatex}
\mapleinline{inert}{2d}{diff(u(t),`$`(t,2))-1/8*(-4*(24*cos(u(t))^5*sin(u(t))-32*cos(u(t))^7*
sin(u(t)))/cos(u(t))^4-16*(-1-4*cos(u(t))^6+4*cos(u(t))^8)/cos(u(t))^5
*sin(u(t)))/(-1-4*cos(u(t))^6+4*cos(u(t))^8)*cos(u(t))^4*diff(u(t),t)^
2-2*cos(u(t))^7*sin(u(t))/(-1-4*cos(u(t))^6+4*cos(u(t))^8)*diff(v(t),t
)^2 = 0,
diff(v(t),`$`(t,2))-4*sin(u(t))/cos(u(t))*diff(u(t),t)*diff(v(t),t) =
0;}{%
\maplemultiline{
({\frac {\partial ^{2}}{\partial t^{2}}}\,\mathrm{u}(t)) -
{\displaystyle \frac {1}{8}} \,{\displaystyle \frac {( - 4\,
{\displaystyle \frac {24\,\mathrm{cos}(\mathrm{u}(t))^{5}\,
\mathrm{sin}(\mathrm{u}(t)) - 32\,\mathrm{cos}(\mathrm{u}(t))^{7}
\,\mathrm{sin}(\mathrm{u}(t))}{\mathrm{cos}(\mathrm{u}(t))^{4}}}
 - {\displaystyle \frac {16\,\mathrm{\%1}\,\mathrm{sin}(\mathrm{u
}(t))}{\mathrm{cos}(\mathrm{u}(t))^{5}}} )\,\mathrm{cos}(\mathrm{
u}(t))^{4}\,({\frac {\partial }{\partial t}}\,\mathrm{u}(t))^{2}
}{\mathrm{\%1}}}  \\
\mbox{} - {\displaystyle \frac {2\,\mathrm{cos}(\mathrm{u}(t))^{7
}\,\mathrm{sin}(\mathrm{u}(t))\,({\frac {\partial }{\partial t}}
\,\mathrm{v}(t))^{2}}{\mathrm{\%1}}} =0, \,({\frac {\partial ^{2}
}{\partial t^{2}}}\,\mathrm{v}(t)) - {\displaystyle \frac {4\,
\mathrm{sin}(\mathrm{u}(t))\,({\frac {\partial }{\partial t}}\,
\mathrm{u}(t))\,({\frac {\partial }{\partial t}}\,\mathrm{v}(t))
}{\mathrm{cos}(\mathrm{u}(t))}} =0 \\
\mathrm{\%1} :=  - 1 - 4\,\mathrm{cos}(\mathrm{u}(t))^{6} + 4\,
\mathrm{cos}(\mathrm{u}(t))^{8} }
%
}
\end{maplelatex}

\end{maplegroup}

\subsection{\QTR{_cstyle8}{Gráficos de las geodésicas}}

\begin{maplegroup}
\begin{_pstyle6}
\begin{mapleinput}
\mapleinline{active}{1d}{plotgeo:=proc(X,ustart,uend,vstart,vend,u0,v0,Du0,Dv0,T,N,gr,theta,ph
i)}{%
}
\mapleinline{active}{1d}{
local sys,desys,u1,v1,listp,geo,plotX;}{%
}
\mapleinline{active}{1d}{
sys:=geoeq(X);}{%
}
\mapleinline{active}{1d}{
desys:=dsolve(\{sys,u(0)=u0,v(0)=v0,D(u)(0)=Du0,D(v)(0)=Dv0\},}{%
}
\mapleinline{active}{1d}{
\{u(t),v(t)\},type=numeric, output=listprocedure);}{%
}
\mapleinline{active}{1d}{
u1:=subs(desys,u(t)); v1:=subs(desys,v(t));}{%
}
\mapleinline{active}{1d}{
geo:=tubeplot(convert(subs(u='u1'(t),v='v1'(t),X),list),}{%
}
\mapleinline{active}{1d}{
t=0..T,radius=0.05, color=black,numpoints=N):}{%
}
\mapleinline{active}{1d}{
plotX:=plot3d(subs(\{u=u(t),v=v(t)
\},X),u=ustart..uend,v=vstart..vend,grid=[gr[1],}{%
}
\mapleinline{active}{1d}{
gr[2]],shading=XY,lightmodel=light3):}{%
}
\mapleinline{active}{1d}{
display(\{geo,plotX
\},style=patch,scaling=constrained,orientation=[theta,phi]);}{%
}
\mapleinline{active}{1d}{
end:}{%
}
\end{mapleinput}
\end{_pstyle6}

\end{maplegroup}

\subsubsection{\QTR{_cstyle10}{Ejemplos}}

\begin{maplegroup}
\begin{_pstyle6}
\begin{mapleinput}
\mapleinline{active}{1d}{plotgeo(torus,0,2*Pi,0,2*Pi,Pi/2,0,0,1,30,150,[15,30],105,33);}{%
}
\end{mapleinput}
\end{_pstyle6}

\mapleresult
\begin{center}
\mapleplot{lab401.eps}
\end{center}

\end{maplegroup}
\begin{maplegroup}
\begin{_pstyle6}
\begin{mapleinput}
\mapleinline{active}{1d}{plotgeo(torus,0,2*Pi,0,2*Pi,0,0,0,1,15,75,[15,30],177,68);}{%
}
\end{mapleinput}
\end{_pstyle6}

\mapleresult
\begin{center}
\mapleplot{lab402.eps}
\end{center}

\end{maplegroup}
\begin{maplegroup}
\begin{_pstyle6}
\begin{mapleinput}
\mapleinline{active}{1d}{plotgeo(torus,0,2*Pi,0,2*Pi,Pi,0,0,1,15,165,[15,30],177,68);}{%
}
\end{mapleinput}
\end{_pstyle6}

\mapleresult
\begin{center}
\mapleplot{lab403.eps}
\end{center}

\end{maplegroup}
\begin{maplegroup}
\begin{_pstyle6}
\begin{mapleinput}
\mapleinline{active}{1d}{plotgeo(torus,0,2*Pi,0,2*Pi,Pi,0,0,1,35,165,[15,30],177,68);}{%
}
\end{mapleinput}
\end{_pstyle6}

\mapleresult
\begin{center}
\mapleplot{lab404.eps}
\end{center}

\end{maplegroup}
\begin{maplegroup}
\begin{_pstyle6}
\begin{mapleinput}
\mapleinline{active}{1d}{plotgeo(sphere,0,2*Pi,0,2*Pi,0,0,1,1,5,35,[15,30],177,68);}{%
}
\end{mapleinput}
\end{_pstyle6}

\mapleresult
\begin{center}
\mapleplot{lab405.eps}
\end{center}

\end{maplegroup}
\begin{maplegroup}
\begin{_pstyle6}
\begin{mapleinput}
\mapleinline{active}{1d}{plotgeo(witch,-1.1,1.1,0,2*Pi,Pi/3,0,0,1,100,200,[15,30],92,-113);}{%
}
\end{mapleinput}
\end{_pstyle6}

\mapleresult
\begin{center}
\mapleplot{lab406.eps}
\end{center}

\end{maplegroup}
\begin{maplegroup}
\begin{mapleinput}
\end{mapleinput}

\end{maplegroup}
\begin{maplegroup}
\emptyline
\end{maplegroup}

\subsection{\QTR{_cstyle8}{Ejercicios}}

\begin{_pstyle2}
\QTR{_cstyle4}{1. Considere el hiperboloide de un solo manto, como superficie de
revolución sobre el eje z. Halle la métrica, las ecuaciones
diferenciales que determinan  las geodésicas y grafique diferentes
geodésicas, entre otras grafique una geodésica que sea un meridiano y
una geodésica que compruebe que un paralelo en particular no es
geodésica. }\QTR{_cstyle3}{Ayuda:}\QTR{_cstyle4}{ }\QTR{_cstyle11}{Vea la Figura 5.19 de la página 206, la cual está hecha para un
cono.}
\end{_pstyle2}

\begin{_pstyle2}
\QTR{_cstyle4}{2. Considere un cilindro vertical. En este caso los
paralelos y los meridianos si son geodésicas. Compruébelo usando
Maple. Además haga dibujos de geodésicas que no sean ni paralelos ni
meridianos. Cómo son estas curvas?. Compruébelo resolviendo las
ecuaciones.}
\end{_pstyle2}

\emptyline
\emptyline

\section{Manejo del comando \texttt{plotgeo}}

El objetivo es dibujar una superficie $M$ con una parametrización
ortogonal $X$, es decir tal que $F=0$, y sobre ella una geodésica
$\alpha$.

Para resolver el sistema de ecuaciones diferenciales ordinarias de
segundo orden debemos dar las condiciones iniciales: un punto $p\in
M$ y un vector $\textbf{v}\in T_pM$. El punto $p=X(u_0,v_0)$, la
curva geodésica $\alpha(t)=X(u(t),v(t))$ y el vector
$\textbf{v}=\alpha'(t)=X_u u'+X_vv'$ el cual da la dirección inicial
se establece dando

\texttt{plotgeo(X,ustart,uend,vstart,vend,u0,v0,Du0,Dv0,T,N,gr,theta,phi);}

\texttt{plotgeo(witch,-1.1,1.1,0,2*Pi,Pi/3,0,0,1,100,200,[15,30],92,-113);}



Este comando tiene 14 entradas a saber:
\begin{enumerate}
\item Nombre de la superficie a graficar: $X=\texttt{witch}$
\item $u_{min}\leq u \leq u_{max}$ $u_{min}= \texttt{ustart}= -1.1$
\item $u_{max}= \texttt{uend}= 1.1$
\item $v_{min}\leq v \leq v_{max}$ $v_{min}= \texttt{vstart}= 0$
\item $v_{max}= \texttt{vend}= \texttt{2*Pi}$
\item Primera coordenada del punto $p$, $u_0=u0=\texttt{Pi/3}$
\item Segunda coordenada del punto $p$, $v_0=v0=0$
\item Primera componente del vector $\textbf{v}$, $\texttt{Du0=0}$
\item Segunda componente del vector $\textbf{v}$, $\texttt{Dv0=1}$
\item El valor de $T$ es el valor numérico máximo del parámetro $t$
de la geodésica $\alpha (t)$, $0\leq t \leq T$. $\texttt{T=100}$
\item El valor de $N$ es el número de puntos cuando se plotea la
geodésica, hace que se vea más suave cuando este valor es grande,
pero exige máquina, pues debe hacer muchos cálculos.
$\texttt{N=200}$
\item Lo que está dentro del corchete es la grilla en el dominio $D$
de $(u,v)$. Entre más grande mejor pero exige máquina para hacer
tanto cálculo. $\texttt{gr}=[15,30]$
\item Para visualizar la gráfica se da un punto de vista (donde esta
nuestro ojo), en coordenadas esféricas $\theta = 92$
\item Es el valor de $\phi$ en coordenadas esféricas donde está
nuestro ojo. $\phi = -113$
\end{enumerate}
Esos últimos valores no son tan importantes aunque siempre se deben
colocar. Recuerde que usted puede mover y rotar el gráfico. Cuando
hace una traducción a otro formato por ejemplo \texttt{.pdf}
aparecerá tal cual aparece en primera instancia.

\subsection{Ecuaciones}

\begin{enumerate}
\item Hiperboloide de un solo manto
\begin{displaymath}
\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1 \qquad  X(u,v)=(a
\cosh u \cos v, b \cosh u \sin v, c \sinh u)
\end{displaymath}

\item Cilindro vertical
\begin{displaymath}
\frac{x^2}{a^2}+\frac{y^2}{b^2}=1 \qquad  X(u,v)=(a  \cos v, b  \sin
v,  u)
\end{displaymath}
\end{enumerate}
\end{document}
%% End of Maple 7.00 Output
