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{SECT 0 {PARA 256 "" 0 "" {TEXT -1 46 "Plane Curves, with prescribed p
lanar curvature" }}{PARA 257 "" 0 "" {TEXT -1 12 "Math 4530-1 " }}
{PARA 258 "" 0 "" {TEXT -1 20 "Wednesday January 26" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 279 "For plane curves we ca
n define the normal vector globally as the rotation by +Pi/2 of the un
it tangent vector, and now there is no restriction that the curvature \+
never be zero.  This leads to a simpler Frenet system than for space c
urves, and you can recreate interesting curves:" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "with(DEtools):with(plots):" 
}}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been
 redefined\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "kap:=t->t:
  #planar curvature function\na:=-5:b:=5: #s-range\nc:=-2:d:=2: #x-ran
ge \nf:=-2:g:=2: #y-range " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
367 "sys:=\n   diff(theta(s),s)=kap(s),\n   diff(b1(s),s)=cos(theta(s)
),\n   diff(b2(s),s)=sin(theta(s)):\n#this is the plane-curve Frenet s
ystem\nics:=\n   theta(0)=0,  #start flat\n   b1(0)=0,     #start at o
rigin\n   b2(0)=0:\np:=dsolve(\{sys,ics\},\{theta(s),b1(s),b2(s)\},\n \+
  type=numeric):\np1:=odeplot(p,[b1(s),b2(s)],a..b,numpoints=400,\n   \+
thickness=1,axes=framed,color=black):\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 49 "display(p1,view=[c..d,f..g],title=`kappa(s)=s`);\n" }
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 " 0 "" {MPLTEXT 1 0 109 "kap:=t->t*sin(t):  #planar curvature functio
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