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{SECT 0 {EXCHG {PARA 227 "" 0 "" {TEXT 207 0 "" }}{PARA 228 "" 0 "" 
{TEXT 208 18 "Parametric Curves " }{TEXT 208 0 "" }}{PARA 229 "" 0 "" 
{TEXT 209 0 "" }}{PARA 229 "" 0 "" {TEXT 209 0 "" }}{PARA 229 "" 0 "" 
{TEXT 209 0 "" }}{PARA 229 "" 0 "" {TEXT 209 0 "" }}{PARA 229 "" 0 "" 
{TEXT 209 0 "" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 0 8 "restart:"
 }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 0 12 "wit
h(plots);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 229 "" 0 "" {TEXT 209 0 
"" }}{PARA 231 "" 0 "" {TEXT 210 93 "We begin this section by plotting
 the parametric equation of a curve.  We begin with the case" }{TEXT 
210 0 "" }}{PARA 232 "" 0 "" {TEXT 211 0 "" }}{PARA 233 "" 0 "" {TEXT 
212 40 "               x = f(t)        y = g(t)." }{TEXT 212 0 "" }}
{PARA 234 "" 0 "" {TEXT 213 0 "" }}{PARA 235 "" 0 "" {TEXT 214 0 "" }}
{PARA 236 "" 0 "" {TEXT 215 148 "That is: x is a function of time and \+
y is a function of time.  In this case, time or t is the independent v
ariable.  It is often convenient to write" }{TEXT 215 0 "" }}{PARA 
237 "" 0 "" {TEXT 216 0 "" }}{PARA 238 "" 0 "" {TEXT 217 0 "" }}{PARA 
239 "" 0 "" {TEXT 218 38 "               x = x(t)      y = y(t)." }
{TEXT 218 0 "" }}{PARA 229 "" 0 "" {TEXT 209 0 "" }}{PARA 240 "" 0 "" 
{TEXT 219 124 "Using Maple we can easily graph parametric equations.  \+
Begin by using the help menu and \"Topic Search\"  to look up the word
 " }{TEXT 219 0 "" }}{PARA 241 "" 0 "" {TEXT 220 0 "" }}{PARA 242 "" 0
 "" {TEXT 221 4 "plot" }{TEXT 221 0 "" }}{PARA 243 "" 0 "" {TEXT 222 
0 "" }}{PARA 244 "" 0 "" {TEXT 223 77 "You should find an example of a
 parametric equation similar to the following:" }{TEXT 223 0 "" }}
{PARA 229 "" 0 "" {TEXT 209 0 "" }}{PARA 229 "" 0 "" {TEXT 209 0 "" }}
}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 0 49 "plot([cos(t),sin(t), t=0.
.2*Pi], title=Example1);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 230 "> " 
0 "" {MPLTEXT 1 0 1 ":" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 229 "" 0 ""
 {TEXT 209 0 "" }}{PARA 245 "" 0 "" {TEXT 224 94 "We have obtained a c
ircle.  Why?  If it does not look like a circle, correct the aspect ra
tio." }{TEXT 224 0 "" }}{PARA 246 "" 0 "" {TEXT 225 0 "" }}{PARA 247 "
" 0 "" {TEXT 226 62 "Consider the following case:  Can the aspect rati
o be changed?" }{TEXT 226 0 "" }}{PARA 229 "" 0 "" {TEXT 209 0 "" }}
{PARA 229 "" 0 "" {TEXT 209 0 "" }}}{EXCHG {PARA 230 "> " 0 "" 
{MPLTEXT 1 0 73 "plot([cos(t), 2*sin(t), t=0..2*Pi], scaling=constrain
ed, title=Example2);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 230 "> " 0 ""
 {MPLTEXT 1 0 1 ":" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 229 "" 0 "" 
{TEXT 209 0 "" }}{PARA 248 "" 0 "" {TEXT 227 68 "Note that the scaling
 attribute in the above graph forces the graph " }{TEXT 227 0 "" }}
{PARA 249 "" 0 "" {TEXT 228 44 "to treat the x and y axes distances eq
ually." }{TEXT 228 0 "" }}{PARA 229 "" 0 "" {TEXT 209 0 "" }}{PARA 
250 "" 0 "" {TEXT 229 79 "What kind of graph is obtained in Example 2?
 Prove that your answer is correct." }{TEXT 229 0 "" }}{PARA 229 "" 0 
"" {TEXT 209 0 "" }}{PARA 251 "" 0 "" {TEXT 230 28 "Consider the follo
wing graph" }{TEXT 230 0 "" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 
0 1 ":" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 0 
82 "plot([cos(t) + 0.2*cos(10*t) ,sin(t) + 0.2*sin(10*t), t=0..2*Pi], \+
title=Example1);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 230 "> " 0 "" 
{MPLTEXT 1 0 1 ":" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 252 "" 0 "" 
{TEXT 231 38 "Can you explain what this is graphing?" }{TEXT 231 0 "" 
}}{PARA 229 "" 0 "" {TEXT 209 0 "" }}{PARA 229 "" 0 "" {TEXT 209 0 "" 
}}{PARA 251 "" 0 "" {TEXT 230 65 "A major problem with this graphing t
echnique is that it is often " }{TEXT 230 0 "" }{TEXT 230 66 "\nhelpfu
l to see the graph unfold as a function of time.  In other " }{TEXT 
230 0 "" }{TEXT 230 61 "\nwords we need to animate the parametric grap
h.  We begin by " }{TEXT 230 0 "" }{TEXT 230 46 "\nloading a Maple lib
rary by the name of plots." }{TEXT 230 0 "" }{TEXT 230 1 "\n" }}{PARA 
229 "" 0 "" {TEXT 209 0 "" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 0 
15 " nframes := 50;" }{MPLTEXT 1 0 0 "" }}{PARA 230 "> " 0 "" 
{MPLTEXT 1 0 26 "for i from 1 to nframes do" }{MPLTEXT 1 0 0 "" }}
{PARA 230 "> " 0 "" {MPLTEXT 1 0 127 "   para[i] := plot([cos(t) + 0.2
*cos(10*t) ,sin(t) + 0.2*sin(10*t), t=0..2*i*Pi/nframes], scaling=cons
trained, title=Example3);" }{MPLTEXT 1 0 0 "" }}{PARA 230 "> " 0 "" 
{MPLTEXT 1 0 3 "od:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 229 "" 0 "" 
{TEXT 209 0 "" }}{PARA 253 "" 0 "" {TEXT 232 0 "" }{TEXT 232 82 "\nWe \+
can animate this graph by using a display command.  We will plot the s
equence " }{TEXT 232 0 "" }}{PARA 254 "" 0 "" {TEXT 233 84 "of graphs \+
and use the attribute insequence to assure that they are displayed in \+
the " }{TEXT 233 0 "" }}{PARA 255 "" 0 "" {TEXT 234 44 "same sequence \+
in which they were calculated." }{TEXT 234 0 "" }{TEXT 234 1 "\n" }}
{PARA 229 "" 0 "" {TEXT 209 0 "" }}{PARA 229 "" 0 "" {TEXT 209 0 "" }}
}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 0 53 "display(seq(para[k], k=1.
.nframes), insequence=true);" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 1 "\n" 
}}}{EXCHG {PARA 229 "" 0 "" {TEXT 235 30 "Animate the following curves
: " }}{PARA 229 "" 0 "" {TEXT 235 0 "" }}{PARA 229 "" 0 "" {TEXT 235 
32 "1)  x  = cos(a*t);  y= sin(b*t);" }}{PARA 229 "" 0 "" {TEXT 235 0 
"" }}{PARA 229 "" 0 "" {TEXT 235 29 "2)  x= cos(a*t); y =cos(b*t);" }}
{PARA 229 "" 0 "" {TEXT 235 0 "" }}{PARA 229 "" 0 "" {TEXT 235 53 "3) \+
 x = cos(t) + r*cos(a*t); y = sin(t) + r*sin(a*t);" }}{PARA 229 "" 0 "
" {TEXT 235 0 "" }}{PARA 229 "" 0 "" {TEXT 235 43 "4)  x = a*t;      y
= b*t^n;    t in [-2,2];" }}{PARA 229 "" 0 "" {TEXT 235 0 "" }}{PARA 
229 "" 0 "" {TEXT 235 32 "5)  x = a*cos(t);  y = b*sin(t);" }{TEXT 
209 0 "" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 0 1 ":" }{MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 0 1 ":" }{MPLTEXT 1 0 
0 "" }}}{EXCHG {PARA 230 "> " 0 "" {MPLTEXT 1 0 1 ":" }{MPLTEXT 1 0 0 
"" }}}{PARA 256 "" 0 "" {TEXT -1 0 "" }}}{MARK "0 0 0" 0 }{VIEWOPTS 1 
1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }