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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 11 "Calculus II" }}{PARA 256 
"" 0 "" {TEXT -1 29 "Lesson 25:  Parametric Curves" }}}{SECT 0 {PARA 
3 "" 0 "" {TEXT -1 12 "Introduction" }}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 673 "Some simple curves in a plane may be described algebraically b
y setting up a suitable co-ordinate system in the plane and describing
 the curve as the graph of a function.  We have seen already how usefu
l such a description is; for example, we used it to obtain a formula f
or the arclength of such a curve.  The description fails, however, for
 even so simple a curve as a circle: there is no choice of co-ordinate
 system for which a circle is the graph of a function.  (Why?)  To des
cribe a general curve, either in the plane or in 3-dimensional space, \+
we must take a different approach.  Instead of relating one co-ordinat
e on the curve to the other by an equation such as " }{XPPEDIT 18 0 "y
 = f(x);" "6#/%\"yG-%\"fG6#%\"xG" }{TEXT -1 45 ", we relate each co-or
dinate separately to a " }{TEXT 256 9 "parameter" }{TEXT -1 23 " by a \+
set of equations " }{XPPEDIT 18 0 "x = f(t);" "6#/%\"xG-%\"fG6#%\"tG" 
}{TEXT -1 2 ", " }{XPPEDIT 18 0 "y = g(t);" "6#/%\"yG-%\"gG6#%\"tG" }
{TEXT -1 6 " (and " }{XPPEDIT 18 0 "z = h(t);" "6#/%\"zG-%\"hG6#%\"tG
" }{TEXT -1 76 " if we are in 3 dimensions).  You can use any letter f
or the parameter, but " }{XPPEDIT 18 0 "t;" "6#%\"tG" }{TEXT -1 187 " \+
is a traditional choice, partly because it is useful to think of the p
arameter as representing time, and to imagine the curve as the path tr
aced out by a particle which is at the point (" }{XPPEDIT 18 0 "x(t);
" "6#-%\"xG6#%\"tG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "y(t);" "6#-%\"yG6
#%\"tG" }{TEXT -1 10 ") at time " }{XPPEDIT 18 0 "t;" "6#%\"tG" }
{TEXT -1 1 "." }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 32 "Plotting Param
etric Curves with " }{TEXT 257 5 "Maple" }{TEXT -1 0 "" }}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 52 "2-dimensional parametric curves can be pl
otted with " }{TEXT 258 7 "Maple's" }{TEXT -1 1 " " }{TEXT 259 4 "plot
" }{TEXT -1 9 " command." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 
"plot([cos(t), sin(t), t=0..2*Pi], scaling=constrained);" }}{PARA 13 "
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d might be clearer if you click on the picture and force " }{TEXT 260 
5 "Maple" }{TEXT -1 108 " to draw it with a 1:1 scale on the axes.)  W
hat is the difference between the last curve and this next one?" }}}
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yntax of the parametric plot is probably not unexpected, except for on
e thing: the range appears " }{TEXT 261 6 "inside" }{TEXT -1 197 " the
 square brackets.  There is a reason for this: it enables you to plot \+
more than one parametric curve on the same set of axes, with different
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ve 1" "Curve 2" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "(Note the use \+
of braces in the last command.)" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 
10 "Question 1" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 " Parametric curv
es of the form " }{XPPEDIT 18 0 "x = a*cos(m*t);" "6#/%\"xG*&%\"aG\"\"
\"-%$cosG6#*&%\"mGF'%\"tGF'F'" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "y = b*
sin(n*t);" "6#/%\"yG*&%\"bG\"\"\"-%$sinG6#*&%\"nGF'%\"tGF'F'" }{TEXT 
-1 7 ", with " }{XPPEDIT 18 0 "t;" "6#%\"tG" }{TEXT -1 4 " in " }
{XPPEDIT 18 0 "[0, 2*Pi];" "6#7$\"\"!*&\"\"#\"\"\"%#PiGF'" }{TEXT -1 
15 ", are known as " }{TEXT 262 17 "Lissajous curves." }{TEXT -1 8 "  \+
Here, " }{XPPEDIT 18 0 "t;" "6#%\"tG" }{TEXT -1 22 " is the parameter \+
and " }{XPPEDIT 18 0 "a;" "6#%\"aG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "b
;" "6#%\"bG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "m;" "6#%\"mG" }{TEXT -1 
5 " and " }{XPPEDIT 18 0 "n;" "6#%\"nG" }{TEXT -1 90 " are constants w
hich determine the particular curve in the family.  Here are two examp
les:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "plot([2*cos(3*t), 7
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BG$\"#5!\"\"F+F+-%+AXESLABELSG6$Q!6\"Fcbo-%%VIEWG6$%(DEFAULTGFhbo" 1 
2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 118 "Trace around these two curves unt
il you understand how they are related to the equations which define t
hem.  Then ask " }{TEXT 263 5 "Maple" }{TEXT -1 134 " to plot one or t
wo other Lissajous curves.  See if you can guess what each one will lo
ok like before you plot it.  (For some reason, " }{TEXT 269 5 "Maple" 
}{TEXT -1 57 " was giving me strange results with curves that involved
 " }{XPPEDIT 18 0 "cos(4*t);" "6#-%$cosG6#*&\"\"%\"\"\"%\"tGF(" }
{TEXT -1 112 "; I suggest you avoid these examples.  You can try them \+
if you like, but don't necessarily believe the results!)" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 272 9 "Sol
ution:" }{TEXT -1 140 "  The easiest way to see what is happening is t
o concentrate on one co-ordinate at a time.  The first curve above is \+
given by the equations " }{XPPEDIT 18 0 "x = 2*cos(3*t);" "6#/%\"xG*&
\"\"#\"\"\"-%$cosG6#*&\"\"$F'%\"tGF'F'" }{TEXT -1 2 ", " }{XPPEDIT 18 
0 "y = 7*sin(2*t);" "6#/%\"yG*&\"\"(\"\"\"-%$sinG6#*&\"\"#F'%\"tGF'F'
" }{TEXT -1 7 " .  As " }{XPPEDIT 18 0 "t;" "6#%\"tG" }{TEXT -1 16 " g
oes from 0 to " }{XPPEDIT 18 0 "2*Pi;" "6#*&\"\"#\"\"\"%#PiGF%" }
{TEXT -1 6 ", the " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 73 " co-or
dinate will oscillate back and forth between 2 and -2 three times (" }
{XPPEDIT 18 0 "3*t;" "6#*&\"\"$\"\"\"%\"tGF%" }{TEXT -1 90 " will pass
 through three complete periods of the cosine function).  At the same \+
time, the " }{XPPEDIT 18 0 "y;" "6#%\"yG" }{TEXT -1 226 " co-ordinate \+
will oscillate between 7 and -7 twice.  You should be able to trace ro
und the curve, concentrating on either the horizontal or the vertical \+
motion, and see these two oscillations.  Similarly, in the second curv
e, " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 39 " goes through 5 compl
ete periods while " }{XPPEDIT 18 0 "y;" "6#%\"yG" }{TEXT -1 16 " goes \+
through 3." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 189 "The oscillations with different frequencies in different direc
tions can sometimes give amusing results.  Consider the effect that an
 apparently small change makes in the following examples." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "plot([cos(32*t), sin(63*t), t=0..2*
Pi]);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURV
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