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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 " . " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 25 "with(plots):with(linalg):" }}{PARA 7 "" 1 " " {TEXT -1 50 "Warning, the name changecoords has been redefined\n" }} {PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names norm and tra ce have been redefined and unprotected\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 267 51 "We begin with the graphs of the elliptic functions." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 139 "ellplots:=plot(\{JacobiSN(u ,1/sqrt(2)),JacobiCN(u,1/sqrt(2)),JacobiDN(u,1/sqrt(2))\},\nu=-1.85407 4677..3*1.854074677,color=[red,blue,black]):\n" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 89 "elliptext:=textplot([[-0.9,0.3,`cn(u)`],[4,0.3 ,`sn(u)`],[5.15,0.9,`dn(u)`]],color=black):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "display(\{ellplots,elliptext\});" }}{PARA 13 "" 1 "" {GLPLOT2D 477 411 411 {PLOTDATA 2 "6*-%'CURVESG6$7gr7$$!+xY2a=!\"*$ \"+^:\"F*$\"+$Hy$yyF-7$$!+'z$)e2\"F*$\"+1%)*Q0)F-7$$!+-)3$f)F-7$$!*? 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\"+?ej_aF-7$Fd[l$\"+wxvbVF-7$F^\\l$\"+@>z!H$F-7$Fc\\l$\"+MWWq@F-7$F]]l $\"+]*f*Q6F-7$Fa^l$\"+O(R%)e#Feim7$F_`l$!+J(*GI6F-7$Fi`l$!+S=LO@F-7$F^ al$!+e^(*)F-7$Fh[m$!+>yX^#)F-7$Fb\\m$!+k67utF-7$F\\]m$!+\"=@bS'F-7$Ff]m$ !+$HqyX&F-7$F`^m$!+W<1JVF-7$Fj^m$!+*4tNI$F-7$F__m$!+Y*>#*>#F-7$Fi_m$!+ qnTR6F-7$Fgam$\"+V0UMmF`\\n-Fjam6&F\\bmFabmFabmFabm-%%TEXTG6%7$$F*!\" \"$\"\"$F`gnQ&cn(u)6\"Fifn-F\\gn6%7$$\"\"%FabmFagnQ&sn(u)FdgnFifn-F\\g n6%7$$\"$:&!\"#$\"\"*F`gnQ&dn(u)FdgnFifn-%+AXESLABELSG6$Q\"uFdgnQ!Fdgn -%%VIEWG6$;F(Fgam%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 259 96 " The following is the parametrization of the Mylar balloon of radius R \+ and a plot of the balloon." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "MylarR:=[R*JacobiCN(u,kk)*cos(v),R*JacobiCN(u,kk)*sin(v),R/sqrt(2 )*(EllipticE(JacobiSN(u,kk),\nkk)/kk^2-(1-kk^2)/kk^2*EllipticF(JacobiS N(u,kk),kk))];" 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The plotting of geodesics on the Mylar balloon takes a long time. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "dp := proc(X,Y)\n X[1] *Y[1]+X[2]*Y[2]+X[3]*Y[3];\n end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "nrm := proc(X)\n local ans;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 69 " ans:=sqrt(dp(X,X));\n simplify(combine(ans),radica l,symbolic,trig);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "xp := proc(X,Y)" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 16 " local a,b,c;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 " a := X[2]*Y[3]-X[3]*Y[2];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 " b := X[3]*Y[1]-X[1]*Y[3];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 " c := X[1]*Y[2]-X[2]*Y[1];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 " [a,b,c];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 256 " " 0 "" {TEXT 268 86 "The next procedure computes the metric E=x_u dot \+ x_u, F=x_u dot x_v and G=x_v dot x_v." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "EFG := proc(X)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 " local Xu,Xv,E,F,G;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 107 " \+ Xu := [diff(X[1],u),diff(X[2],u),diff(X[3],u)];\n Xv := [diff(X[1 ],v),diff(X[2],v),diff(X[3],v)];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 " E := dp(Xu,Xu);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 " F : = dp(Xu,Xv);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 " G := dp(Xv,Xv );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 " simplify([E,F,G],symbol ic);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 257 " " 0 "" {TEXT -1 177 "The following procedure gives the unit normal to \+ the input surface X. If you wish to take UN(X), then X must be given i n parametrized form in square brackets as done down below." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "UN := proc(X)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 " local Xu,Xv,Z,s;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 103 " Xu := [diff(X[1],u),diff(X[2],u),diff(X[3],u)];\n Xv := \+ [diff(X[1],v),diff(X[2],v),diff(X[3],v)];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 " Z := xp(Xu,Xv);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 " s := nrm(Z);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 " simpl ify([Z[1]/s,Z[2]/s,Z[3]/s],symbolic);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 " end:" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 141 "The formulas fo r Gauss and Mean Curvature requires dotting certain second partial der ivatives with the unit normal. This is done as follows. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "lmn := proc(X)" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 36 " local Xu,Xv,Xuu,Xuv,Xvv,U,l,m,n;" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 51 " Xu := [diff(X[1],u),diff(X[2],u),diff(X[3] ,u)];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 " Xv := [diff(X[1],v),di ff(X[2],v),diff(X[3],v)];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 " Xu u := [diff(Xu[1],u),diff(Xu[2],u),diff(Xu[3],u)];" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 55 " Xuv := [diff(Xu[1],v),diff(Xu[2],v),diff(Xu[3], v)];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 " Xvv := [diff(Xv[1],v),d iff(Xv[2],v),diff(Xv[3],v)];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 " \+ U := UN(X);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 " l := dp(U,Xuu) ;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 " m := dp(U,Xuv);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 " n := dp(U,Xvv);" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 32 " simplify([l,m,n],symbolic);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 " GK := proc(X)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 " local E,F,G,l, m,n,S,T;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 " S := EFG(X);" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 " T := lmn(X);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 " E := S[1];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 " F := S[2];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 " G := S[3] ;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 " l := T[1];" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 14 " m := T[2];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 " n := T[3];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "simplify( (l*n-m^2)/(E*G-F^2),symbolic);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "en d:" }}}{EXCHG {PARA 256 "" 0 "" {TEXT 269 49 "The Mean Curvature can b e calculated as follows. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "MK := proc(X)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 " local E,F, G,l,m,n,S,T;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 " S := EFG(X);" } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 " T := lmn(X);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 " E := S[1];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 " F := S[2];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 " G := S[ 3];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 " l := T[1];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 " m := T[2];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 " n := T[3];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "simplify( (E*n+G*l-2*F*m)/(2*E*G-2*F^2),symbolic);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{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 is not good about simplifying th e final results. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "GK(sub s(\{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 cour se this simplifies to 2*JacobiCN^2/R^2." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "MK(subs(\{kk=1/sqrt(2)\},MylarR));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,$*&#\"\"$\"\"#\"\"\"*,%\"RG!\"\"-%)JacobiDNG6$%\"uG, $*&F'F+F'#F(F'F(F(F'F2-%)JacobiCNGF.F(,&F'F(*$)-%)JacobiSNGF.F'F(F+#F+ F'F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 265 36 "This si mplifies to 3*JacobiCN/(2*R)." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 " " }{TEXT 263 125 "The following computes the metric coefficients and t hen the quantity EG-F^2. Then the surface area is computed for R=1 usi ng " }{XPPEDIT 18 0 "sqrt(E*G-F^2);" "6#-%%sqrtG6#,&*&%\"EG\"\"\"%\"GG F)F)*$%\"FG\"\"#!\"\"" }{TEXT 266 85 " . Finally, that is compared wi th the formula for surface area derived 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)\},MylarR))[1];lmn(subs(\{kk=1/sqrt(2)\},M ylarR))[2];\nlmn(subs(\{kk=1/sqrt(2)\},MylarR))[3];" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#*,,&\"\"#\"\"\"*$)-%)JacobiSNG6$%\"uG,$*&F%!\"\"F%#F& F%F&F%F&F/#F/F%-%)JacobiDNGF+F&%\"RGF&-%)JacobiCNGF+F&F%F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$* .-%)JacobiDNG6$%\"uG,$*&\"\"#!\"\"F+#\"\"\"F+F.F.,&F.F,*$)-%)JacobiSNG F'F+F.F.F.F+F-%\"RGF.-%)JacobiCNGF'F.,&F+F.F0F,#F,F+F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "EFG(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#,$*&,&!\"\"\"\"\"*$) -%)JacobiSNG6$%\"uG,$*$-%%sqrtG6#\"\"#F'#F'F3F3F'F'F')%\"RG\"\"%F'#F&F 3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "area_element:=sqrt(sim plify(-1/2*(-1+JacobiSN(u,1/2*sqrt(2))^2)*R^4)) assuming R>0;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%-area_elementG,$*&#\"\"\"\"\"#F(*&)% \"RGF)F(,&F)F(*&F)F()-%)JacobiSNG6$%\"uG,$*&F)!\"\"F)F'F(F)F(F6F'F(F( " }}}{EXCHG {PARA 259 "" 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 "Surf_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$\"+/Wgp)*!\"*" }}}{EXCHG {PARA 0 "" 0 " " {TEXT 270 47 "The formula from the paper says this should be" } {TEXT -1 1 " " }{XPPEDIT 18 0 "Pi^2;" "6#*$)%#PiG\"\"#\"\"\"" }{TEXT -1 2 ". " }{TEXT 271 12 "Let's check." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "evalf(Pi^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+/ Wgp)*!\"*" }}}{EXCHG {PARA 260 "" 0 "" {TEXT -1 86 "Now we write proce dures for the geodesic equations and plots of geodesics on surfaces." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "geoeq:=proc(X)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 " local M,eq1,eq2;" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 14 " M:=EFG(X);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 82 " 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" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 78 " +su bs(\{u=u(t),v=v(t)\},diff(M[1],v)/(M[1]))*diff(u(t),t)*diff(v(t),t)" } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 73 " - subs(\{u=u(t),v=v(t)\} ,diff(M[3],u)/(2*M[1]))*diff(v(t),t)^2=0;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 82 " 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" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 78 " +subs(\{u=u(t),v=v(t)\},diff(M[3],u)/(M[3]))*diff(u(t),t)*di ff(v(t),t)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 73 " + subs(\{u= u(t),v=v(t)\},diff(M[3],v)/(2*M[3]))*diff(v(t),t)^2=0;" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 12 " eq1,eq2;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "end: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "plotgeo2:=pro c(X,ustart,uend,vstart,vend,u0,v0,Du0,Dv0,T,N,gr,theta,phi)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 " local sys,desys,u1,v1,listp,geo,plotX; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 " sys:=geoeq(X);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 75 " desys:=dsolve(\{sys,u(0)=u0,v(0)=v0,D(u)(0 )=Du0,D(v)(0)=Dv0\},\{u(t),v(t)\}," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 " type=numeric, output=listprocedure);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 " u1:=subs(desys,u(t)); v1:=subs(desys,v(t));" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 " geo:=tubeplot(subs(u='u1'(t),v=' v1'(t),X),t=0..T,radius=0.018, color=black,\n numpoints=N):" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 133 " plotX:=plot3d(subs(\{u=u(t),v=v (t)\},X),u=ustart..uend,v=vstart..vend,\n grid=[gr[1],gr[2]],shadin g=ZGRAYSCALE,lightmodel=light3):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 80 " display(\{geo,plotX\},style=patch,scaling=constrained,orientation= [theta,phi]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 272 182 "The components E,F and G for the metric are functions of u alone. This means the parametrization is a u-Clair aut parametrization. So, every geodesic should obey the Clairaut relat ion" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(G)*sin(phi) = c;" "6#/*&-%%s qrtG6#%\"GG\"\"\"-%$sinG6#%$phiGF)%\"cG" }{TEXT -1 1 " " }{TEXT 273 5 "where" }{TEXT -1 1 " " }{XPPEDIT 18 0 "phi;" "6#%$phiG" }{TEXT -1 1 " " }{TEXT 274 20 "is the angle between" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x_u;" "6#%$x_uG" }{TEXT -1 1 " " }{TEXT 275 7 "and the" }{TEXT -1 1 " " }{TEXT 276 30 "geodesic's tangent vector and" }{TEXT -1 1 " " } {XPPEDIT 18 0 "c;" "6#%\"cG" }{TEXT -1 1 " " }{TEXT 277 39 "is a const ant. For instance, if we take" }{TEXT -1 1 " " }{XPPEDIT 18 0 "phi = P i/2;" "6#/%$phiG*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 1 " " }{TEXT 278 3 "and" }{TEXT -1 1 " " }{XPPEDIT 18 0 "u = 0;" "6#/%\"uG\"\"!" }{TEXT -1 2 ", " }{TEXT 279 12 "then we have" }{TEXT -1 1 " " }{XPPEDIT 18 0 "JacobiCN(0,k1) = c;" "6#/-%)JacobiCNG6$\"\"!%#k1G%\"cG" }{TEXT -1 2 " , " }{TEXT 280 2 "so" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c = 1;" "6#/%\"c G\"\"\"" }{TEXT -1 2 ". " }{TEXT 281 31 "So, it must always be true th at" }{TEXT -1 1 " " }{XPPEDIT 18 0 "JacobiCN(u,k1)*sin(phi) = 1;" "6#/ *&-%)JacobiCNG6$%\"uG%#k1G\"\"\"-%$sinG6#%$phiGF*F*" }{TEXT -1 2 ". " }{TEXT 282 40 "But the elliptic cosine has values below" }{TEXT -1 1 " " }{XPPEDIT 18 0 "1;" "6#\"\"\"" }{TEXT -1 1 " " }{TEXT 283 23 "and s o does the sine of" }{TEXT -1 1 " " }{XPPEDIT 18 0 "phi;" "6#%$phiG" } {TEXT -1 2 ". " }{TEXT 284 5 "Thus," }{TEXT -1 1 " " }{TEXT 285 12 "we must have" }{TEXT -1 1 " " }{XPPEDIT 18 0 "u = 0;" "6#/%\"uG\"\"!" } {TEXT -1 1 " " }{TEXT 286 3 "and" }{TEXT -1 1 " " }{XPPEDIT 18 0 "phi \+ = Pi/2;" "6#/%$phiG*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 1 " " }{TEXT 287 57 "throughout the life of the geodesic. This is what we see." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "plotgeo2(subs(\{R=1,kk=1/sq rt(2)\},MylarR),-1.854074677,1.854074677,0,2*Pi,0,0,0,1,15,50,\n[15,20 ],45,69);" }}{PARA 13 "" 1 "" {GLPLOT3D 452 380 380 {PLOTDATA 3 "6(-%% MESHG6$7T7,7%$\"+i&*y<5!\"*$\"++qNWF!#7$\"\"!F07%$\"+&fFO,\"F+$\"+J%*H -@F.$!+(p_5\"**!# 5$!+*\\y@P\"F.$!+Fd%)e:F87%$\"+-G$G$)*FD$!+B?&)yDF.$!+nDOchF.7%FJ$!+@? &)yDF.$\"+zDOchF.7%$\"+\">_5\"**FD$!+-&y@P\"F.$\"+Ed%)e:F87%F:$\"+Kf_l ZF>$\"+bRlsbIFDF67%$\"+2k*[c*FD$\"+Gf0BI FDF?7%$\"+WqF\\%*FD$\"+xN^')HFDFG7%$\"+k=\"QP*FD$\"+_AmiHFD$!+mDOchF.7 %$\"+l=\"QP*FDF\\q$\"+yDOchF.7%FepFgpFZ7%F`pFbpFin7%F[pF]pF^o7%FfoFhoF ao7,7%$\"+KS!4L)FD$\"+*\\20&eFDF/7%$\"+f7W'H)FD$\"+yaIEeFDF67%$\"+a%y \"4#)FD$\"+5Q-ldFD$!+bRls&)\\cFD$!+lDOchF.7%FasFcs$\"+xDOchF.7%F\\sF^sFZ7%FerFgr$\"+aRl sk/='FD$\"+CZ9*3)FDF/7%$\"+Gs*[:'FD $\"+Y?ob!)FD$!+'pyr,\"F+F/7%$\"+A4ssSF8$\"+(RqH,\"F+F67%$\"+LO) )HSF8$\"+seJ-5F+F?7%$\"+k3<\")RF8$\"+Q***>!**FDFG7%$\"+jeP\\RF8$\"+e\" >H#)*FDFN7%FdzFfzFS7%F_zFazFZ7%FjyF\\zFin7%FeyFgyF^o7%F`yFbyFao7,7%$!+ 'Haan#FD$\"+Ea8A)*FDF/7%$!+TmQkEFD$\"+=Q]\"y*FDF67%$!+aBOOEFD$\"+v4iy' *FDF?7%$!+5V\\/EFD$\"+Apih&*FDFG7%$!+ARp$e#FD$\"+yWE&[*FDF^q7%Fc\\lFe \\lFcq7%F^\\lF`\\lFZ7%Fi[lF[\\lFin7%Fd[lFf[lF^o7%F_[lFa[lFao7,7%$!+6\\ 56bFD$\"+OF@f&)FDF/7%$!+5pI)[&FD$\"+4b!Q_)FDF67%$!+W.eIaFD$\"+k7:M%)FD Fir7%$!+!yFD$\"+F;btkFDF67%$!+U_&)>xFD$\"+f @Y0kFDF?7%$!+B&Qli(FD$\"+sO.GjFDFG7%$!+h.jlvFD$\"+2(yAFDF/7%$!+1Yk!))*FD$!+$y!GpAFDF67%$!+!**=nx*FD$!+fATXAFD F?7%$!+S\"R&e'*FD$!+v*p#=AFDFG7%$!+@FS\"e*FD$!+2Tb+AFDF^q7%FfflFhflFcq 7%FaflFcflFZ7%F\\flF^flFin7%FgelFielF^o7%FbelFdelFao7,7%$!+AAqt()FD$!+ 0W!G;&FDF/7%$!+3xSP()FD$!+KsWT^FDF[u7%$!+tn]X')FD$!+s*ot3&FDFir7%$!+') 3+T&)FD$!+VM(e-&FDFgu7%$!+7&*ys%)FD$!+)3Nd)\\FD$!+jDOchF.7%$!+8&*ys%)F DFghl$\"+vDOchF.7%F`hlFbhlFdv7%F[hlF]hlF\\t7%FfglFhglFhv7%FaglFcglF[w7 ,7%$!+!3D*4oFD$!+1n&oc(FDF/7%$!+*>a%FD$!+j`**G#*FDF67%$!+*e%R^TFD$!+gQ #>8*FDF?7%$!+6I@,TFD$!+>#Q:-*FDFG7%$!+7$f%oSFD$!+4\"*[\\*)FDF^q7%F]]mF _]mFcq7%Fh\\mFj\\mFZ7%Fc\\mFe\\mFin7%F^\\mF`\\mF^o7%Fi[mF[\\mFao7,7%$! +8:DC7FD$!+P7FD$!++6V15F+F[u7%$!+sNO17FD$!+jNXe**FDFi r7%$!+)=\"y\">\"FD$!+Ep2Q)*FDFgu7%$!+8KE#=\"FD$!+dm]f(*FDFihl7%F\\_mF^ _mF^il7%Fg^mFi^mFdv7%Fb^mFd^mF\\t7%F]^mF_^mFhv7%Fh]mFj]mF[w7,7%$\"+I5G y=FD$!+m@_+5F+F/7%$\"+s5^q=FD$!+oE$Q'**FDF67%$\"+[o$3&=FD$!+s?.f)*FDF? 7%$\"+YTYG=FD$!+Js&)R(*FDFG7%$\"+-9'Q\"=FD$!+y82i'*FDFN7%F[amF]amFS7%F f`m$!+Is&)R(*FDFZ7%Fa`mFc`mFin7%F\\`mF^`mF^o7%Fg_mFi_mFao7,7%$\"+G$oh! 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We loo k for positive and negative values and the intermediate value theorem \+ guarantees a zero somewhere in-between. This " }{XPPEDIT 18 0 "u_0;" " 6#%$u_0G" }{TEXT -1 1 " " }{TEXT 295 24 "gives a closed geodesic." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "mylbounce:=proc(u0)\nlocal m yl,geosystem,uuu,vvv,ttt,uuu0;\nmyl:=subs(\{R=1,kk=1/sqrt(2)\},MylarR) ;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "geosystem:=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=listprocedure);" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 198 "uuu:=subs(geosystem,u(t)); vvv:=su bs(geosystem,v(t));\nttt:=fsolve(vvv(t)=evalf(6*Pi),t);\nif type(ttt,f loat) then \n uuu0:=eval(uuu(t),t=ttt);\n evalf(u0-uuu0);\nelse prin t(`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,mylbounce(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 296 77 "A change of sign occurs between 0.690 and 0.692, so we f ocus attention there." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "fo r jj from -2 to 2 do \nprint(0.691+jj*0.0005,mylbounce(0.691+jj*0.0005 )); \nod;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"%+p!\"%$\"';KA!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$$\"%0p!\"%$\"'vE7!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"$\"p!\"$$\"&TM%!#5" }}{PARA 11 "" 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,mylbounce(0.6913565+jj*0.00 00002)); \nod;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"(hN\"p!\"($\"#M!# 5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"(jN\"p!\"($\"#:!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"(lN\"p!\"($F%!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"(nN\"p!\"($!#H!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 $$\"(pN\"p!\"($!#]!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 297 64 "So now we plot an approximation to the closed geodesic. 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