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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" 
{TEXT -1 8 "Cycloids" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 
0 "" {TEXT -1 197 "Here is an example to show how to use Maple to calc
ulate and draw cycloids,\nincluding slipping cycloids.\n\nFirst we des
cribe the motion of the wheel's center.  (We assume the wheel\nhas rad
ius 1.) \n\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "p1 := a*t; p2 := 1;
  a := 1;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 45 "plot( [t,1,t=0..6*Pi] , scaling=constrained);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 103 "\nNot very interesting so far ...
\n\nNow we describe the wheel's motion, as though the center were stil
l:\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 40 "q1 := cos(b*t); q2 := -sin(b*t); b := 1;" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
46 "plot( [q1,q2,t=0..6*Pi], scaling=constrained);" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 86 "\nThat's not too interesting either.    Things ge
t better when we add the two\nmotions. " }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "r1 := p1+q1; r2 := p2+q2; " }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "plot( \{[p1,p2,t=0..6*Pi],
[q1,q2,t=0..6*Pi],[r1,r2,t=0..6*Pi]\}, scaling=constrained, title = `A
 cycloid`);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 151 "\nWe can introduc
e some slippage into the picture by playing with\nthe multiplicative c
onstants    a and/or   b.   Let's put the slippage\non the wheel\n \n
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "b := 2; q1 := sin(b*t); q2 := -
cos(b*t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "plot( [ p1+q1,
 p2+q2, t=0..6*Pi], scaling=constrained);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 195 "Let's do something with epicycloids.    The general form
 is   \n\n   P(t) = (R+1)( cos(at), sin(at) ) + ( cos(bt), sin(bt) ), \+
for appropriate values\n\nof the constants  R, a, and b.    For exampl
e:\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "R := 3; a := 1; p1 := (R+1)
* cos(a*t); p2 := (R+1)*sin(a*t); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 41 "Now let's parametrize the inner circle:\n\n" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 36 "c1 := R* cos(a*t); c2 := R*sin(a*t);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "b=2; q1 := cos(b*t); q2 := sin(b*t)
;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "Now we'll plot everything:\n
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "plot( \{[p1+q1,p2+q2,t=0..2*Pi]
,[c1,c2,t=0..2*Pi]\}, scaling=constrained );" }}}}{MARK "13" 0 }
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