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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);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 85 "(The output from this command might be cl
earer 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.)  What is the \+
difference between the last curve and this next one?" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 59 "plot([cos(2*t), sin(2*t), t=0..2*Pi], sca
ling=constrained);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "plot(
[sin(t), sin(t)^2, t=-2..2]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 102 
"The syntax of the parametric plot is probably not unexpected, except \+
for one 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 parameter ranges.  The following examples show this." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "f1 := t-> 3*cos(2*t) ; g1 :=
 t-> sin(4*t) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "plot([f1
(t), g1(t), t=0..2*Pi]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 
"f2 := t-> cos(t)/ln(t) ; g2 := t-> sin(t)/ln(t) ;" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 30 "plot([f2(t), g2(t), t=3..20]);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "plot(\{[f1(t), g1(t), t=0..2*Pi], [
f2(t), g2(t), t=3..20]\});" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "(No
te 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 " Para
metric curves 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$\"\"!*&\"\"#\"\"\"%#PiGF'" }
{TEXT -1 15 ", are known as " }{TEXT 262 17 "Lissajous curves." }
{TEXT -1 8 "  Here, " }{XPPEDIT 18 0 "t;" "6#%\"tG" }{TEXT -1 22 " is \+
the parameter and " }{XPPEDIT 18 0 "a;" "6#%\"aG" }{TEXT -1 2 ", " }
{XPPEDIT 18 0 "b;" "6#%\"bG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "m;" "6#%
\"mG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "n;" "6#%\"nG" }{TEXT -1 90 "
 are constants which determine the particular curve in the family.  He
re are two examples:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "plo
t([2*cos(3*t), 7*sin(2*t), t=0..2*Pi]);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 40 "plot([cos(5*t), 2*sin(3*t), t=0..2*Pi]);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 118 "Trace around these two curves until you \+
understand how they are related to the equations which define them.  T
hen ask " }{TEXT 263 5 "Maple" }{TEXT -1 134 " to plot one or two othe
r Lissajous curves.  See if you can guess what each one will look like
 before you plot it.  (For some reason, " }{TEXT 269 5 "Maple" }{TEXT 
-1 57 " was giving me strange results with curves that involved " }
{XPPEDIT 18 0 "cos(4*t);" "6#-%$cosG6#*&\"\"%\"\"\"%\"tGF(" }{TEXT -1 
112 "; I suggest you avoid these examples.  You can try them if you li
ke, but don't necessarily believe the results!)" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 272 9 "Solution:
" }{TEXT -1 140 "  The easiest way to see what is happening is to conc
entrate 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*&\"\"(\"\"\"-%$sinG6#*&\"\"#F'%\"tGF'F'" }
{TEXT -1 7 " .  As " }{XPPEDIT 18 0 "t;" "6#%\"tG" }{TEXT -1 16 " goes
 from 0 to " }{XPPEDIT 18 0 "2*Pi;" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT 
-1 6 ", the " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 73 " co-ordinate
 will oscillate back and forth between 2 and -2 three times (" }
{XPPEDIT 18 0 "3*t;" "6#*&\"\"$\"\"\"%\"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]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "plot([cos(32*t), s
in(64*t), t=0..2*Pi]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Do you \+
see what happened in this last curve?" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 152 "Finally, execute the following command, and explain what
 it does.  (You will have to click on the picture which is produced to
 get additional controls.)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "animate
([2*cos(3*t), b*sin(2*t), t=0..2*Pi, numpoints=500], b=2..10);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 98 "Here is another version of the sam
e command.  What , if anything, is the difference in the result?" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "animate([2*cos(3*t), (6 + 4*
sin(b))*sin(2*t), t=0..2*Pi], b=0..2*Pi);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }{TEXT 273 9 "Solution:" }{TEXT -1 289 " The factors in \+
front of the sine or cosine functions in Lissajopus curves are simply \+
scale factors: they stretch the curve in one or other of the co-ordina
te directions.  Here we have a family of Lissajous curves with a scale
 factor in the vertical direction which depends on a parameter " }
{XPPEDIT 18 0 "b;" "6#%\"bG" }{TEXT -1 7 ".  The " }{TEXT 274 7 "anima
te" }{TEXT -1 60 " command draws the curves for a certain number of va
lues of " }{XPPEDIT 18 0 "b;" "6#%\"bG" }{TEXT -1 15 " between 0 and \+
" }{XPPEDIT 18 0 "2*Pi;" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT -1 68 ", and \+
plays them as a movie.  Since the scale factor is periodic in " }
{XPPEDIT 18 0 "b;" "6#%\"bG" }{TEXT -1 92 ", the movie can be played c
ontinuously, and its last frame joins up smoothly with the first." }}}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }{TEXT 264 5 "Maple" }{TEXT -1 84 " can plot 3-dimension
al parametric curves, too.  For this, you must use the command " }
{TEXT 265 10 "spacecurve" }{TEXT -1 23 ", which resides in the " }
{TEXT 266 5 "plots" }{TEXT -1 58 " package.  If you did Question 1, yo
u have already loaded " }{TEXT 267 5 "plots" }{TEXT -1 12 " to use the
 " }{TEXT 268 7 "animate" }{TEXT -1 51 " command, but there is no harm
 in reloading it now." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "wi
th(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "spacecurve([t
, sin(t), cos(t)], t=0..2*Pi);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 
"You will almost always want to use this command with axes and labels:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "spacecurve([t, sin(t), \+
cos(t)], t=0..2*Pi, axes=normal, labels=[x,y,z]);" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 0 "" }{TEXT 270 10 "Spacecurve" }{TEXT -1 210 " will p
lot more than one curve at a time, but this is not as useful as it is \+
with 2-dimensional curves: the picture gets cluttered very quickly.  Y
ou can look at the help page to find the syntax if you need it." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 4 "" 0 "" 
{TEXT -1 10 "Question 2" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "Use " }
{TEXT 271 10 "spacecurve" }{TEXT -1 42 " to plot spirals whose axes li
e along the " }{XPPEDIT 18 0 "y;" "6#%\"yG" }{TEXT -1 5 " and " }
{XPPEDIT 18 0 "z;" "6#%\"zG" }{TEXT -1 6 " axes." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 275 9 "Solution:
" }{TEXT -1 12 "  Along the " }{XPPEDIT 18 0 "y;" "6#%\"yG" }{TEXT -1 
22 "-axis first, then the " }{XPPEDIT 18 0 "z;" "6#%\"zG" }{TEXT -1 6 
"-axis." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "spacecurve([sin(
t),t, cos(t)], t=0..2*Pi, axes=normal, labels=[x,y,z]);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "spacecurve([sin(t), cos(t),t], t=0.
.2*Pi, axes=normal, labels=[x,y,z]);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0 0" 4 }
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