{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 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 
0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }}
{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 52 "Sample path length on a torus:  alpha=Pi*t, beta=k*t" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "a
lpha:=Pi*t; beta:=k*t;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "x
:=(cos(alpha)+2)*cos(beta);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
28 "y:=(cos(alpha)+2)*sin(beta);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "z:=sin(alpha);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 14 "dx:=diff(x,t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "d
y:=diff(y,t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "dz:=diff(z
,t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "Integrand:=simplify
(dx^2+dy^2+dz^2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "F:=eva
lf(Int(sqrt(Integrand),t=0..1))+(Pi/2 - k);" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 18 "plot(F,k=0..Pi/2);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 37 "Soln:=fsolve(diff(F,k)=0,k,0.6..1.0);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "2*evalf(subs(k=Soln,F));" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "No
w we'll plot the path on the surface of our torus:" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots)
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "P1:=spacecurve([subs(k
=Soln,x),subs(k=Soln,y),subs(k=Soln,z)],t=0..1,color=black,thickness=5
):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "X:=(cos(A)+2)*cos(B);
 Y:=(cos(A)+2)*sin(B); Z:=sin(A);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 61 "Path2:=subs(A=Pi, B=(1-t)*(Pi/2-Soln)+t*(Pi/2+Soln),[
X,Y,Z]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "P2:=spacecurve(
Path2,t=0..1,color=black,thickness=5):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 57 "Path3:=subs(A=(1-t)*Pi,B=(1-t)*(Pi/2+Soln)+t*Pi,[X,Y,
Z]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "P3:=spacecurve(Path
3,t=0..1,color=black,thickness=5):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 90 "T:=plot3d([X,Y,Z],A=0..2*Pi,B=0..2*Pi,scaling=constra
ined,style=patchnogrid,shading=ZHUE):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 24 "display3d(\{T,P1,P2,P3\});" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 0 "" }}}}{MARK "16 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 
1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }