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{SECT 0 {PARA 256 "" 0 "" {TEXT -1 9 "Math 4530" }}{PARA 257 "" 0 "" 
{TEXT -1 30 "Homework for Friday February 4" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 495 "1)  Rework exercise 1.4.7 us
ing Maple; this was a messy hand calculation to compute the curvature,
 torsion, T,N,B for a specific curve which was not parameterized by ar
clength. In \"curtor\" are already written procedures to compute curva
ture and torsion.  You can modify these to write procedures which will
 compute T, N, B.  Note, that since it is easy to compute T, and also \+
B, as the normalized cross product of the first and second derivatives
 of the curve, you can get N by taking B cross T." }}{PARA 259 "" 0 "
" {TEXT -1 239 "   After all those computations,  use Maple to draw a \+
nice picture of the curve, say for t between zero and one.  You can de
cide whether you prefer the \"plot3d\" or \"tubeplot\" commands. (See \+
below for an example of what \"tubeplot\" creates.)" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "I've loaded the procedu
res in curtor.  This will let me compute curvature and torsion:" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "assume(t,real); #this will \+
let Maple know that it should\n#look for real solutions to equations i
nvolving t." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 45 "Bcurve:=[exp(t)*cos(t),exp(t)*sin(t),exp(t)];" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'BcurveG7%*&-%$expG6#%#t|irG\"\"\"-%
$cosGF)F+*&F'F+-%$sinGF)F+F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 25 "curv(Bcurve);tor(Bcurve);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&
kappaG,$*&#\"\"\"\"\"$F(*&\"\"##F(F+-%$expG6#,$%#t|irG!\"\"F(F(F(" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$tauG,$*&#\"\"\"\"\"$F(-%$expG6#,$%#
t|irG!\"\"F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 116 "Let's try to write pr
ocedures for the T,N,B frame.  We may assume the user has already load
ed the curtor procedures." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
755 "TNB:=proc(alpha)\nlocal alphap,alphapp,alphappp,num1,den1,\n   nu
m2,den2,num3,den3,\n   T11,T12,T13,T1,N11,N12,N13,N1,\n   B11,B12,B13,
B1;\nalphap:=diff(alpha,t);\nalphapp:=diff(alphap,t);\nalphappp:=diff(
alphapp,t);\nden1:=simplify(nrm(alphap),trig,sqrt,symbolic);\nT11:=sim
plify(alphap[1]/den1);\nT12:=simplify(alphap[2]/den1);\nT13:=simplify(
alphap[3]/den1);\nT1:=[T11,T12,T13];\nnum2:=simplify(xp(alphap,alphapp
),trig);\nden2:=simplify(nrm(num2),trig,sqrt,symbolic);\nB11:=simplify
(num2[1]/den2);\nB12:=simplify(num2[2]/den2);\nB13:=simplify(num2[3]/d
en2);\nB1:=[B11,B12,B13];\nN11:=simplify(xp(B1,T1)[1],trig,sqrt,symbol
ic);\nN12:=simplify(xp(B1,T1)[2],trig,sqrt,symbolic);\nN13:=simplify(x
p(B1,T1)[3],trig,sqrt,symbolic);\nN1:=[N11,N12,N13];\nRETURN(T= T1,N=N
1,B=B1);\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "assume(a>
0);assume(b>0);\nhel:=[a*cos(t),a*sin(t),b*t];\nTNB(hel);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%$helG7%*&%#a|irG\"\"\"-%$cosG6#%#t|irGF(*&F'F
(-%$sinGF+F(*&%#b|irGF(F,F(" }}{PARA 12 "" 1 "" {XPPMATH 20 "6%/%\"TG7
%,$*(%#a|irG\"\"\"-%$sinG6#%#t|irGF),&*$)F(\"\"#F)F)*$)%#b|irGF1F)F)#!
\"\"F1F6*(F(F)-%$cosGF,F)F.F5*&F4F)F.F5/%\"NG7%,$F8F6,$F*F6\"\"!/%\"BG
7%*(F4F)F*F)F.F5,$*(F4F)F8F)F.F5F6*&F(F)F.F5" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 45 "Bcurve:=[exp(t)*cos(t),exp(t)*sin(t),exp(t)];" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'BcurveG7%*&-%$expG6#%#t|irG\"\"\"-
%$cosGF)F+*&F'F+-%$sinGF)F+F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 12 "TNB(Bcurve);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6%/%\"TG7%,$*&#\"
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F*F)*&,&F2F)F-F)F)F*F4F)F),$*&F*F1F*F4F)/%\"NG7%,$*&#F)F5F)*&F:F)F5F4F
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$*&#F)FOF)*&F:F)FOF4F)F1,$*&F*F1FOF4F)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "with(plots):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning,
 the name changecoords has been redefined\n" }}}{EXCHG {PARA 0 "> " 0 
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)z>=/$Ff[v$\"3Gs?>W_VEnFf[v$\"3/XEf[UuDuFf[v7%$!3`W=013hVIFf[v$\"3[oL5
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XESSTYLEG6#%$BOXG-%(SCALINGG6#%,CONSTRAINEDG" 1 2 0 1 10 0 2 1 1 2 1 
1.000000 49.000000 -30.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 173 "2) Exercise 1.7.8, for kap3 \+
and kap4: Use the plane-curve recreation template which I've posted, o
r create your own procedure, to recreate nice pictures of these two cu
rves." }}{PARA 0 "" 0 "" {TEXT -1 129 "I'll try to write a procedure f
rom the template - although this was not required.  I'll steal the Fre
net procedure and modify it:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "with(DEtools):\nwith(plots):
  #load DE tools and plots packages" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 643 "recreate2dview:=proc(kap,a,b,c,d,e,f)\n   #kap=curva
ture function\n   #arclength parameter from a to b\n   #c..d, e..f are
 x-y ranges for plot\nlocal\n   sys,   #the 2-d Frenet system\n   p , \+
   #dummy for ODE solution to Frenet system\n   ics,   #initial condit
ions\n   p1;    #name for ODEplot of p\nsys:=\n   diff(theta(s),s)=kap
(s),\n   diff(a1(s),s)=cos(theta(s)),\n   diff(a2(s),s)=sin(theta(s)):
\n   \nics:=\n   a1(0)=0,a2(0)=0,\n   theta(0)=0;   \np:=dsolve(\{sys,
ics\},\{a1(s),a2(s),theta(s)\},\n   type=numeric);\np1:=odeplot(p,[a1(
s),a2(s)],a..b,\n   numpoints=200,thickness=1,axes=boxed,color=black):
\ndisplay(p1,scaling=constrained,view=[c..d,e..f]);\nend:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "kap3:=t->exp(t); #the first curvatu
re function" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%kap3G%$expG" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "recreate2dview(kap3,0,4,0,1,
0,1);  #curvature(s) = exp(s)" }}{PARA 13 "" 1 "" {GLPLOT2D 274 245 
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{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 155 "3)  Exe
rcises 1.7.7 and 1.5.15:  Again use the plane-curve recreation templat
e which I've posted or your own procedure, to recreate a curve with cu
rvature " }{TEXT -1 1 " " }{XPPEDIT 18 0 "kappa(s) = 1/(1+s^2);" "6#/-
%&kappaG6#%\"sG*&\"\"\"F),&F)F)*$F'\"\"#F)!\"\"" }{TEXT -1 1 " " }
{TEXT 257 465 ", so that at s=0 the curve is horizontal and at the ori
gin. Now, the real fun begins.  Using the plane- Frenet system which w
e talked about in class (see also exercise 1.5.13 on page 41), you can
 find an explict formula for the curve by antidifferentiation (this is
 the exercise 1.5.15 part).  Prove that the curve you've created is ac
tually a catenary (graph of y=cosh(x) or Euclidean motion of such)!  (
We saw the catenary before as the shape of a hanging cable.)" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "ka
pcat:=t->1/(1+t^2):\nrecreate2dview(kapcat,-4,4,-3,3,-1,4); #catenary?
" }}{PARA 13 "" 1 "" {GLPLOT2D 296 265 265 {PLOTDATA 2 "6'-%'CURVESG6%
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SG6#\"\"\"-%+AXESLABELSG6$Q#a1F)Q#a2F)-%(SCALINGG6#%,CONSTRAINEDG-%*AX
ESSTYLEG6#%$BOXG-%%VIEWG6$;$!\"$F.$\"\"$F.;$!\"\"F.$\"\"%F." 1 2 0 1 
10 0 2 9 1 2 1 1.000000 46.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 71 "We integrate the 2-d Frenet system to ver
ify that this is the catenary:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 70 "theta:=s->int(1/(1+r^2),r=0..s);  #of course, I know this is a
rctan(s)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&thetaGf*6#%\"sG6\"6$%)o
peratorG%&arrowGF(-%$intG6$*&\"\"\"F0,&F0F0*$)%\"rG\"\"#F0F0!\"\"/F4;
\"\"!9$F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "theta(s);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'arctanG6#%\"sG" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 64 "cos(theta(s));sin(theta(s)); #the unit ta
ngent vector components" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\"\"F$*
$,&F$F$*$)%\"sG\"\"#F$F$#F$F*!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
*&%\"sG\"\"\",&F%F%*$)F$\"\"#F%F%#!\"\"F)" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 97 "x:=s->int(1/sqrt(1+r^2),r=0..s):\ny:=s->int(r/sqrt(
1+r^2),r=0..s): #components of parametric curve" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 10 "x(s);y(s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
-%(arcsinhG6#%\"sG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$,&\"\"\"F&*$
)%\"sG\"\"#F&F&#F&F*F&F&!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 46 "solve(x=arcsinh(s),s); #can guess this answer!" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#-%%sinhG6#%\"xG" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 66 "sqrt(1+sinh(x)^2)-1;  #y as a function of x - transla
tion of cosh!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$,&\"\"\"F&*$)-%%s
inhG6#%\"xG\"\"#F&F&#F&F-F&F&!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "simplify(%,trig);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
,&*$*$)-%%coshG6#%\"xG\"\"#\"\"\"#F,F+F,F,!\"\"" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 26 "simplify(%,sqrt,symbolic);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#,&-%%coshG6#%\"xG\"\"\"F(!\"\"" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 188 "4)  Use the procedure \+
\"recreate3dview\", which is one of the files posted on our web page, \+
to create a 3-d curve which is beautiful, with your own choice of curv
ature and torsion functions." }}{PARA 0 "" 0 "" {TEXT -1 17 "Really up
 to you!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 258 
83 "5)  Here is a picture of part of an a=2, b=1 helix, sitting on a r
adius 2 cylinder:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(pl
ots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 152 "cylinder:=\nplot3
d([2*cos(theta),2*sin(theta),z],theta=0..2*Pi,z=0..2*Pi,\nstyle=wirefr
ame,color=black):\n#this is a parametric way of drawing the cylinder" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "helix2:=tubeplot([2*cos(
t),2*sin(t),t],\nt=0..2*Pi,radius=0.1,color=black):\n#this is a corres
ponding piece of the helix" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
93 "display(\{cylinder,helix2\},scaling=constrained,axes=boxed);\n#her
e they are displayed together." }}{PARA 13 "" 1 "" {GLPLOT3D 290 290 
290 {PLOTDATA 3 "6&-%%MESHG6%X.%)anythingG6\"6\"[gl'!%\"!!$^`r\":\":\"
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ESSTYLEG6#%$BOXG-%(SCALINGG6#%,CONSTRAINEDG" 1 2 0 1 10 0 2 1 1 2 1 
1.000000 58.000000 56.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 262 "" 
0 "" {TEXT -1 86 "Your job is to find the osculating circle to the hel
ix at the point when t = Pi , at  " }{XPPEDIT 18 0 "P = [-2, 0, Pi]" "
6#/%\"PG7%,$\"\"#!\"\"\"\"!%#PiG" }{TEXT -1 11 " .         " }}{PARA 
263 "" 0 "" {TEXT -1 91 " (Recall exercise 1.3.28 from last week.)  Ad
d this osculating circle to the display above." }}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "hel2:=[2
*cos(t),2*sin(t),t]:\nTNB(hel2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/%
\"TG7%,$*&#\"\"#\"\"&\"\"\"*&-%$sinG6#%#t|irGF+F*#F+F)F+!\"\",$*&#F)F*
F+*&-%$cosGF/F+F*F1F+F+,$*&F*F2F*F1F+/%\"NG7%,$F7F2,$F-F2\"\"!/%\"BG7%
,$*&#F+F*F+F,F+F+,$*&#F+F*F+F6F+F2,$*(F)F+F*F2F*F1F+" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 21 "curv(hel2);tor(hel2);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#/%&kappaG#\"\"#\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#/%$tauG#\"\"\"\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 218 "
v1:=subs(t=Pi,[-2/5*sin(t)*5^(1/2), 2/5*cos(t)*5^(1/2), 1/5*5^(1/2)]);
\n  #unit tangent when t=Pi\nv2:=subs(t=Pi,[-cos(t), -sin(t), 0]);\n  \+
#unit normal when t=Pi\nP:=[-2,0,Pi]+5/2*v2;\n  #center point for oscu
lating circle" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#v1G7%,$*&#\"\"#\"
\"&\"\"\"*&-%$sinG6#%#PiGF+F*#F+F)F+!\"\",$*&#F)F*F+*&-%$cosGF/F+F*F1F
+F+,$*&F*F2F*F1F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#v2G7%,$-%$cosG
6#%#PiG!\"\",$-%$sinGF)F+\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"
PG7%#\"\"\"\"\"#\"\"!%#PiG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
44 "circ2:=evalm(P-5/2*cos(t)*v2+5/2*sin(t)*v1);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%&circ2G-%'vectorG6#7%,&#\"\"\"\"\"#F+*&#\"\"&F,F+-%$c
osG6#%#t|irGF+!\"\",$*&-%$sinGF2F+F/F*F4,&%#PiGF+*&F*F+F6F+F+" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "circ2plot:=tubeplot(circ2,ra
dius=0.05,t=0..2*Pi,\ncolor=black):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 104 "display(\{cylinder,helix2,circ2plot\},\nscaling=cons
trained,axes=boxed);\n#here they are displayed together." }}{PARA 13 "
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