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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 256 30 "Geodesics \+
on the Mylar Balloon" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 
257 357 "This worksheet works for Maple 8. For previous versions of Ma
ple, some problems may arise. If Maple 7 is used, try Edgardo Cheb-Ter
rab's DNA add-on package. The following two commands read in necessary
 packages and show how to access help on the elliptic integral of the \+
first kind. Note that Maple does not use the definition of the ellipti
c integral with " }{XPPEDIT 18 0 "sqrt(1-k^2*sin(theta)^2);" "6#-%%sqr
tG6#,&\"\"\"F'*&%\"kG\"\"#-%$sinG6#%&thetaGF*!\"\"" }{TEXT 258 32 "  ,
 but rather the version with " }{XPPEDIT 18 0 "sqrt(1-k^2*t^2);" "6#-%
%sqrtG6#,&\"\"\"F'*&%\"kG\"\"#%\"tGF*!\"\"" }{TEXT 261 3 " . " }}}
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"" 1 "" {TEXT -1 50 "Warning, the name changecoords has been redefined
\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 267 51 "We begin with the graphs o
f the elliptic functions." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
139 "ellplots:=plot(\{JacobiSN(u,1/sqrt(2)),JacobiCN(u,1/sqrt(2)),Jaco
biDN(u,1/sqrt(2))\},\nu=-1.854074677..3*1.854074677,color=[red,blue,bl
ack]):\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "elliptext:=text
plot([[-0.9,0.3,`cn(u)`],[4,0.3,`sn(u)`],[5.15,0.9,`dn(u)`]],color=bla
ck):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "display(\{ellplots,
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}{TEXT 259 96 "The following is the parametrization of the Mylar ballo
on of radius R and a plot of the balloon." }}}{EXCHG {PARA 0 "> " 0 "
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sin(v),R/sqrt(2)*(EllipticE(JacobiSN(u,kk),\nkk)/kk^2-(1-kk^2)/kk^2*El
lipticF(JacobiSN(u,kk),kk))];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'My
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F(-%$sinGF0F(,$*&#F(\"\"#F(*(F'F(F8F7,&*&-%*EllipticEG6$-%)JacobiSNGF+
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pF`tFjcu7%$!+xSp\"z'F`t$!+_PZd&)F`tFjcu7%$!+L6/K[F`t$!+v-T)z*F`tFjcu7%
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t$!+Z+'[2\"F0Fjcu7%$\"+7)[3=%F`t$!+ohM45F0Fjcu7;7%$\"+V0UMmF+F,$\"+u6q
!*fF07%$\"+1Q*[['F+$\"+?Gg+9F+Fh[v7%$\"+sO0VgF+$\"+(*>2QFF+Fh[v7%$\"+Z
l\")G`F+$\"+.(>@&RF+Fh[v7%$\"+@uPuVF+$\"+k9-))\\F+Fh[v7%$\"+Q'eFA$F+$
\"+>H3*z&F+Fh[v7%$\"+B1(e#>F+$\"+BYu[jF+Fh[v7%$\"+r'><U&F6$\"+&**HAh'F
+Fh[v7%$!+'fc'f')F6$\"+a@mxlF+Fh[v7%$!+v*o]B#F+$\"+%G*fYiF+Fh[v7%$!+4U
U.NF+$\"+\"RkRj&F+Fh[v7%$!+;*eQh%F+$\"+NEPnZF+Fh[v7%$!+(3>j^&F+$\"+^l)
eo$F+Fh[v7%$!+#QD,<'F+$\"+f_DQCF+Fh[v7%$!+&z1ea'F+$\"+^qr!3\"F+Fh[v7%$
!+#4Hki'F+$!+WhNbKF6Fh[v7%$!+P\"e$3kF+$!+?W6<<F+Fh[v7%$!+4PU,fF+$!+)[#
HJIF+Fh[v7%$!+4kZG^F+$!+l=$)3UF+Fh[v7%$!+[wNCTF+$!+CPl'>&F+Fh[v7%$!+N
\"HV$HF+$!+f4B]fF+Fh[v7%$!+aE.7;F+$!+%G&fNkF+Fh[v7%$!+#3@2<#F6$!+$Ro3j
'F+Fh[v7%$\"+\">tw=\"F+$!+W\"[s_'F+Fh[v7%$\"+uG))QDF+$!+I`SHhF+Fh[v-%'
COLOURG6#%*XYSHADINGG-%+LIGHTMODELG6#%(LIGHT_3G-%(SCALINGG6#%,CONSTRAI
NEDG-%+PROJECTIONG6%$!#mF-$\"#pF-\"\"\"" 1 2 0 1 10 0 2 2 5 1 1 
1.000000 69.000000 -66.000000 1 0 "Curve 1" }}}}{EXCHG {PARA 11 "" 1 "
" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 260 
241 "The next set of procedures gives metric coefficients, unit normal
, second fundamental form, Gauss curvature, mean curvature, geodesic e
quations (when F=0) and plots geodesics. The plotting of geodesics on \+
the Mylar balloon takes a long time." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 273 "EFG := proc(X)\nlocal Xu,Xv,E,F,G;\nXu := <diff(X[1]
,u),diff(X[2],u),diff(X[3],u)>;\nXv := <diff(X[1],v),diff(X[2],v),diff
(X[3],v)>;\nE := DotProduct(Xu,Xu,conjugate=false);\nF := DotProduct(X
u,Xv,conjugate=false);\nG := DotProduct(Xv,Xv,conjugate=false);\nsimpl
ify([E,F,G]);\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 244 "UN \+
:= proc(X)\nlocal Xu,Xv,Z,s;\nXu := <diff(X[1],u),diff(X[2],u),diff(X[
3],u)>;\nXv := <diff(X[1],v),diff(X[2],v),diff(X[3],v)>;\nZ := CrossPr
oduct(Xu,Xv);\ns:=VectorNorm(Z,Euclidean,conjugate=false);\nsimplify(<
Z[1]/s|Z[2]/s|Z[3]/s>,symbolic);\nend:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 464 "lmn := proc(X)\nlocal Xu,Xv,Xuu,Xuv,Xvv,U,l,m,n;\nXu
 := <diff(X[1],u),diff(X[2],u),diff(X[3],u)>;\nXv := <diff(X[1],v),dif
f(X[2],v),diff(X[3],v)>;\nXuu := <diff(Xu[1],u),diff(Xu[2],u),diff(Xu[
3],u)>;\nXuv := <diff(Xu[1],v),diff(Xu[2],v),diff(Xu[3],v)>;\nXvv := <
diff(Xv[1],v),diff(Xv[2],v),diff(Xv[3],v)>;\nU := UN(X);\nl := DotProd
uct(U,Xuu,conjugate=false);\nm := DotProduct(U,Xuv,conjugate=false);\n
n := DotProduct(U,Xvv,conjugate=false);\nsimplify([l,m,n],symbolic);\n
end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "Finally we can calculate \+
Gauss curvature K as follows." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 173 "GK := proc(X)\nlocal E,F,G,l,m,n,S,T;\nS := EFG(X);\nT := lmn(X
);\nE := S[1];\nF := S[2];\nG := S[3];\nl := T[1];\nm := T[2];\nn := T
[3];\nsimplify((l*n-m^2)/(E*G-F^2),symbolic);\nend:" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 26 "Mean curvature is given by" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 183 "MK := proc(X)\nlocal E,F,G,l,m,n,S,T;\nS :=
 EFG(X);\nT := lmn(X);\nE := S[1];\nF := S[2];\nG := S[3];\nl := T[1];
\nm := T[2];\nn := T[3];\nsimplify((G*l+E*n-2*F*m)/(2*E*G-2*F^2),symbo
lic);\nend:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "The geodesic equat
ions are obtained by" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 480 "ge
oeq:=proc(X)\nlocal M,eq1,eq2;\nM:=EFG(X);\neq1:=diff(u(t),t$2)+subs(
\{u=u(t),v=v(t)\},diff(M[1],u)/(2*M[1]))*diff(u(t),t)^2+\nsubs(\{u=u(t
),v=v(t)\},diff(M[1],v)/(M[1]))*diff(u(t),t)*diff(v(t),t)- \nsubs(\{u=
u(t),v=v(t)\},diff(M[3],u)/(2*M[1]))*diff(v(t),t)^2=0;\neq2:=diff(v(t)
,t$2)-subs(\{u=u(t),v=v(t)\},diff(M[1],v)/(2*M[3]))*diff(u(t),t)^2+\ns
ubs(\{u=u(t),v=v(t)\},diff(M[3],u)/(M[3]))*diff(u(t),t)*diff(v(t),t)+ \+
\nsubs(\{u=u(t),v=v(t)\},diff(M[3],v)/(2*M[3]))*diff(v(t),t)^2=0;\neq1
,eq2;\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "geoeq(subs(
\{R=1,kk=1/sqrt(2)\},MylarR));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6$/,&-
%%diffG6$-%\"uG6#%\"tG-%\"$G6$F+\"\"#\"\"\"*,F/F0-%)JacobiSNG6$F(,$*&F
/!\"\"F/#F0F/F0F0-%)JacobiCNGF4F0-%)JacobiDNGF4F0)-F&6$-%\"vGF*F+F/F0F
0\"\"!/,&-F&6$F@F,F0*0F/F0F2F0F9F0F;F0,&F0F0*$)F2F/F0F7F7-F&6$F(F+F0F>
F0F7FB" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 487 "The following procedur
e plots a geodesic on a surface. The inputs are: X a parametrization i
n u and v, ustart......vend the range of the parameters, u0,v0 the sta
rting point of the geodesic (i.e. initial condition 1), Du0,Dv0 the st
arting speed of the geodesic (i.e. initial condition 2), n=[a,b,c] whe
re a,b tell what the parameter of the geodesic itself runs from and c \+
denotes how smooth a picture you want, gr=[d,e] whered is the number o
f grid lines for u and e is the number for v." }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 570 "plotgeo:=proc(X,ustart,uend,vstart,vend,u0,v0,D
u0,Dv0,T,N,gr,theta,phi)\nlocal sys,desys,u1,v1,listp,geo,plotX;\nsys:
=geoeq(X);\ndesys:=dsolve(\{sys,u(0)=u0,v(0)=v0,D(u)(0)=Du0,D(v)(0)=Dv
0\},\{u(t),v(t)\},\ntype=numeric, output=listprocedure);\nu1:=subs(des
ys,u(t)); v1:=subs(desys,v(t));\ngeo:=tubeplot(subs(\{u='u1'(t),v='v1'
(t)\},X),t=0..T, radius=0.02,color=black,thickness=2,numpoints=N):\npl
otX:=plot3d(subs(\{u=u(t),v=v(t)\},X),u=ustart..uend,v=vstart..vend,gr
id=[gr[1],gr[2]]):\ndisplay(\{geo,plotX\},shading=XY,lightmodel=light2
,scaling=constrained,orientation=[theta,phi]);\nend:" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 0 "" }{TEXT 262 148 "The following commands compute
 the Gauss and Mean curvatures of the Mylar balloon. Note that Maple i
s not good about simplifying the final results. " }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 32 "GK(subs(\{kk=1/sqrt(2)\},MylarR));" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"#\"\"\",&F&!\"\"*$)-%)JacobiSNG6$%\"
uG,$*&F%F(F%#F&F%F&F%F&F&F&%\"RG!\"#F(" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }{TEXT 264 46 "Of course this simplifies to 2*JacobiCN^2
/R^2." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "MK(subs(\{kk=1/sqr
t(2)\},MylarR));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"$\"\"#\"
\"\"*,%\"RG!\"\",&F'F(*$)-%)JacobiSNG6$%\"uG,$*&F'F+F'#F(F'F(F'F(F+#F+
F'-%)JacobiCNGF1F(-%)JacobiDNGF1F(F'F5F(F(" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }{TEXT 265 36 "This simplifies to 3*JacobiCN/(2*R)." }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 263 125 "The following com
putes the metric coefficients and then the quantity EG-F^2. Then the s
urface area is computed for R=1 using " }{XPPEDIT 18 0 "sqrt(E*G-F^2);
" "6#-%%sqrtG6#,&*&%\"EG\"\"\"%\"GGF)F)*$%\"FG\"\"#!\"\"" }{TEXT 266 
85 "  . Finally, that is compared with the formula for surface area de
rived in the paper." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "EFG
(subs(\{kk=1/sqrt(2)\},MylarR))[1];EFG(subs(\{kk=1/sqrt(2)\},MylarR))[
2];\nEFG(subs(\{kk=1/sqrt(2)\},MylarR))[3];" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,$*&\"\"#!\"\"%\"RGF%\"\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&)%\"RG\"\"
#\"\"\",&F(!\"\"*$)-%)JacobiSNG6$%\"uG,$*&F'F*F'#F(F'F(F'F(F(F(F*" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "lmn(subs(\{kk=1/sqrt(2)\},M
ylarR))[1];lmn(subs(\{kk=1/sqrt(2)\},MylarR))[2];\nlmn(subs(\{kk=1/sqr
t(2)\},MylarR))[3];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*,,&\"\"#\"\"\"
*$)-%)JacobiSNG6$%\"uG,$*&F%!\"\"F%#F&F%F&F%F&F/#F/F%%\"RGF&-%)JacobiC
NGF+F&-%)JacobiDNGF+F&F%F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*.%\"RG\"\"\"-%)JacobiCNG6$%\"uG,$*
&\"\"#!\"\"F-#F&F-F&F&-%)JacobiDNGF)F&,&F&F.*$)-%)JacobiSNGF)F-F&F&F&F
-F/,&F-F&F3F.#F.F-F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "EF
G(subs(\{kk=1/sqrt(2)\},MylarR))[1]*EFG(subs(\{kk=1/sqrt(2)\},MylarR))
[3]-\nEFG(subs(\{kk=1/sqrt(2)\},MylarR))[2]^2;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,$*&#\"\"\"\"\"#F&*&,&F&!\"\"*$)-%)JacobiSNG6$%\"uG,$*&
F'F*F'#F&F'F&F'F&F&F&)%\"RG\"\"%F&F&F*" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 83 "area_element:=sqrt(simplify(-1/2*(-1+JacobiSN(u,1/2*s
qrt(2))^2)*R^4)) assuming R>0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-a
rea_elementG,$*&#\"\"\"\"\"#F(*&)%\"RGF)F(,&F)F(*&F)F()-%)JacobiSNG6$%
\"uG,$*&F)!\"\"F)F'F(F)F(F6F'F(F(" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 
-1 85 "This simplifies to R^2*sqrt(2)/2*JacobiCN(u). Therefore, for R=
1, the surface area is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "S
urf_Area:=evalf(2*Pi*Int(JacobiCN(u,1/sqrt(2))/sqrt(2),u=-1.854074677.
.1.854074677));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*Surf_AreaG$\"+/W
gp)*!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 268 47 "The formula from the
 paper  says this should be" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Pi^2;" "6
#*$%#PiG\"\"#" }{TEXT -1 2 ". " }{TEXT 269 12 "Let's check." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "evalf(Pi^2);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#$\"+/Wgp)*!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
270 182 "The components E,F and G for the metric are functions of u al
one. This means the parametrization is a u-Clairaut parametrization. S
o, every geodesic should obey the Clairaut relation" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "sqrt(G)*sin(phi) = c;" "6#/*&-%%sqrtG6#%\"GG\"\"\"-%$si
nG6#%$phiGF)%\"cG" }{TEXT -1 1 " " }{TEXT 271 5 "where" }{TEXT -1 1 " \+
" }{XPPEDIT 18 0 "phi;" "6#%$phiG" }{TEXT -1 1 " " }{TEXT 272 20 "is t
he angle between" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x_u;" "6#%$x_uG" }
{TEXT -1 1 " " }{TEXT 273 7 "and the" }{TEXT -1 1 " " }{TEXT 274 30 "g
eodesic's tangent vector  and" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c;" "6#
%\"cG" }{TEXT -1 1 " " }{TEXT 275 39 "is a constant. For instance, if \+
we take" }{TEXT -1 1 " " }{XPPEDIT 18 0 "phi = Pi/2;" "6#/%$phiG*&%#Pi
G\"\"\"\"\"#!\"\"" }{TEXT -1 1 " " }{TEXT 276 3 "and" }{TEXT -1 1 " " 
}{XPPEDIT 18 0 "u = 0;" "6#/%\"uG\"\"!" }{TEXT -1 2 ", " }{TEXT 277 
12 "then we have" }{TEXT -1 1 " " }{XPPEDIT 18 0 "JacobiCN(0,k1) = c;
" "6#/-%)JacobiCNG6$\"\"!%#k1G%\"cG" }{TEXT -1 2 ", " }{TEXT 278 2 "so
" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c = 1;" "6#/%\"cG\"\"\"" }{TEXT -1 
2 ". " }{TEXT 279 31 "So, it must always be true that" }{TEXT -1 1 " \+
" }{XPPEDIT 18 0 "JacobiCN(u,k1)*sin(phi) = 1;" "6#/*&-%)JacobiCNG6$%
\"uG%#k1G\"\"\"-%$sinG6#%$phiGF*F*" }{TEXT -1 2 ". " }{TEXT 280 40 "Bu
t the elliptic cosine has values below" }{TEXT -1 1 " " }{XPPEDIT 18 
0 "1;" "6#\"\"\"" }{TEXT -1 1 " " }{TEXT 281 23 "and so does the sine \+
of" }{TEXT -1 1 " " }{XPPEDIT 18 0 "phi;" "6#%$phiG" }{TEXT -1 2 ". " 
}{TEXT 282 5 "Thus," }{TEXT -1 1 " " }{TEXT 283 12 "we must have" }
{TEXT -1 1 " " }{XPPEDIT 18 0 "u = 0;" "6#/%\"uG\"\"!" }{TEXT -1 1 " \+
" }{TEXT 284 3 "and" }{TEXT -1 1 " " }{XPPEDIT 18 0 "phi = Pi/2;" "6#/
%$phiG*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 1 " " }{TEXT 285 57 "througho
ut the life of the geodesic. This is what we see." }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 102 "plotgeo(subs(\{R=1,kk=1/sqrt(2)\},MylarR),-
1.854074677,1.854074677,0,2*Pi,0,0,0,1,15,50,\n[15,20],45,69);" }}
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286 40 "What does the Clairaut relation say when" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "u = .4;" "6#/%\"uG-%&FloatG6$\"\"%!\"\"" }{TEXT -1 1 " \+
" }{TEXT 287 15 "for example and" }{TEXT -1 1 " " }{XPPEDIT 18 0 "phi \+
= Pi/2;" "6#/%$phiG*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 3 " ? " }{TEXT 
288 43 "In general, the Clairaut relation says that" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "c^2 <= G;" "6#1*$%\"cG\"\"#%\"GG" }{TEXT -1 3 " . " }}
{PARA 0 "" 0 "" {TEXT 289 65 "Since the elliptic cosine is decreasing \+
here, we see that we need" }{TEXT -1 1 " " }{XPPEDIT 18 0 "u <= .4;" "
6#1%\"uG-%&FloatG6$\"\"%!\"\"" }{TEXT -1 3 " . " }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 87 "plotgeo(subs(\{R=1,kk=1/sqrt(2)\},MylarR),-0.4
,0.4,0,2*Pi,0.4,0,0,1,15,50,\n[4,30],38,69);" }}{PARA 13 "" 1 "" 
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69.000000 38.000000 1 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 0 "" }{TEXT 290 85 "Here is the plot of a geodesic starting
 in a Northerly direction. We get a longitude." }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 102 "plotgeo(subs(\{R=1,kk=1/sqrt(2)\},MylarR),-1.
854074677,1.854074677,0,2*Pi,0,0,1,0,30,70,\n[15,20],38,69);" }}{PARA 
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\"+8P'yR%F07%$!+*e:)=xF0$\"+s*>1p)Fdhr$\"+9s-tUF07%F`\\il$\"+'HMvJ#Fdh
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INGG-%(SCALINGG6#%,CONSTRAINEDG-%+LIGHTMODELG6#%(LIGHT_2G-%+PROJECTION
G6%$\"#QF-$\"#pF-\"\"\"" 1 2 0 1 10 0 2 2 4 1 1 1.000000 69.000000 
38.000000 1 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
0 "" }{TEXT 291 35 "Now we write a procedure to detect " }{XPPEDIT 18 
0 "u_0;" "6#%$u_0G" }{TEXT -1 1 " " }{TEXT 292 153 "where we get a clo
sed geodesic. We look for positive and negative values and the interme
diate value theorem guarantees a zero somewhere in-between. This " }
{XPPEDIT 18 0 "u_0;" "6#%$u_0G" }{TEXT -1 1 " " }{TEXT 293 24 "gives a
 closed geodesic." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "mylbou
nce:=proc(u0)\nlocal myl,geosystem,uuu,vvv,ttt,uuu0;\nmyl:=subs(\{R=1,
kk=1/sqrt(2)\},MylarR);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "geosyste
m:=dsolve(\{geoeq(myl),u(0)=u0,v(0)=0,D(u)(0)=0,D(v)(0)=1\},\{u(t),v(t
)\}," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "   type=numeric, output=lis
tprocedure);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 198 "uuu:=subs(geosyste
m,u(t)); vvv:=subs(geosystem,v(t));\nttt:=fsolve(vvv(t)=evalf(6*Pi),t)
;\nif type(ttt,float) then \n  uuu0:=eval(uuu(t),t=ttt);\n  evalf(u0-u
uu0);\nelse print(`time error`);\nend if;\nend:" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 75 "for jj from -2 to 2 do \nprint(0.69+jj*0.002,m
ylbounce(0.69+jj*0.002)); \nod;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"
$'o!\"$$\"(o$*y\"!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"$)o!\"$$\"'
q$Q)!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"#p!\"#$\"';KA!#5" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"$#p!\"$$!&82&!#5" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6$$\"$%p!\"$$\"&Y\">!#5" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }{TEXT 294 77 "A change of sign occurs between 0.690 and
 0.692, so we focus attention there." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 79 "for jj from -2 to 2 do \nprint(0.691+jj*0.0005,mylbou
nce(0.691+jj*0.0005)); \nod;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"%+p
!\"%$\"';KA!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"%0p!\"%$\"'vE7!#5
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"" 1 "" {XPPMATH 20 "6$$\"%:p!\"%$!&HW\"!#5" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$$\"%?p!\"%$!&82&!#5" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 93 "for jj from -2 to 2 do \nprint(0.6913565+jj*0.0000002
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{XPPMATH 20 "6$$\"(hN\"p!\"($\"#M!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "
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{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"(pN\"p!\"($!#]!#5" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 0 "" }{TEXT 295 64 "So now we plot an approximation
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#/%$u_0G-%&FloatG6$\"(lN\"p!\"(" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> \+
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854074677,1.854074677,0,2*Pi,0.6913565,0,0,1,\n28,85,[15,15],-4,70);" 
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153 "The symmetry of this geodesic is visible when we remove the surfa
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