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{SECT 0 {PARA 3 "" 0 "" {TEXT -1 17 "Polar Coordinates" }}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 189 "NOTE: This worksheet will contain many p
lots (and animations). It is therefore best if you remove (see the Edi
t pulldown menu) all the output from your worksheet before saving it t
o a disk." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart;\nwith(plots):
" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 12 "Introduction" }}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 45 "We can create polar plots in Maple using \+
the " }{HYPERLNK 17 "polarplot" 2 "polarplot" "" }{TEXT -1 18 " comman
d from the " }{HYPERLNK 17 "plots" 2 "plots" "" }{TEXT -1 9 " package.
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 32 "We begin by plotting the circle " }{XPPEDIT 18 
0 "r=2" "6#/%\"rG\"\"#" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 
31 "This is accomplished by giving " }{HYPERLNK 17 "polarplot" 2 "pola
rplot" "" }{TEXT -1 15 " two parameters" }}{PARA 0 "" 0 "" {TEXT -1 
15 "   - the radius" }}{PARA 0 "" 0 "" {TEXT -1 28 "   - the range for
 the angle" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "polarplot(2, theta=0.
.2*Pi);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "The following command \+
does the same (since Maple assumes " }{XPPEDIT 18 0 "theta" "6#%&theta
G" }{TEXT -1 4 " in " }{XPPEDIT 18 0 "[-Pi,Pi]" "6#7$,$%#PiG!\"\"F%" }
{TEXT -1 38 " if we omit the range for the angle): " }}{PARA 0 "> " 0 
"" {MPLTEXT 1 0 13 "polarplot(2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
75 "If we would like to plot only a part of the circle, ie. restrict t
he angle " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 30 ", this ca
n be done as follows:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "polarplot(
2, theta=Pi/4..3*Pi/4);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "A nice
r picture of this curve is given by" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
54 "polarplot(2, theta=Pi/4..3*Pi/4, view=[-2..2, -2..2]);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 72 "Here is one more example of a simple pola
r plot. This curve is called a " }{TEXT 258 8 "cardioid" }{TEXT -1 1 "
." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "polarplot(1-cos(theta), theta=
0..2*Pi);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "Polarplot can also b
e used in conjunction with " }{TEXT 259 17 "function notation" }{TEXT 
-1 34 ".\nThen the cardioid is defined via" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 51 "r := theta -> 1-cos(theta); \npolarplot(r, 0..2*Pi);
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "An even more powerful and fle
xible use of polarplot is as a " }{TEXT 260 44 "combination with a par
ametric representation" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 67 
"To illustrate this, first consider the following Cartesian example:" 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "f := x -> sin(x);\nplot(f, 0..2*P
i);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "We can repeat this example
 using parametric equations:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "x :
= t -> t;\ny := t -> sin(t);\nplot([x, y, 0..2*Pi]);" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 215 "And we have learned that the parametric concep
t is more powerful than the simple function based definition of a curv
e, since we can now also allow the second coordinate (x in our case) t
o vary with t, allowing, e.g." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "x \+
:= t -> (cos(t))^3;\ny := t -> sin(t);\nplot([x, y, 0..2*Pi]);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 171 "Maple allows us to do the analogo
us thing with polar plots - a concept which is not included in our Cal
culus book, but is a natural extension of the ideas we've discussed." 
}}{PARA 0 "" 0 "" {TEXT -1 74 "Let's reconsider the cardioid displayed
 earlier (using function notation)." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
50 "r := theta -> 1-cos(theta);\npolarplot(r, 0..2*Pi);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 110 "Now, let's redo this in a parametric pol
ar representation (note that we are now using an additional parameter \+
" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 2 "):" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 68 "r := t -> 1-cos(t);\ntheta := t -> t;\npolarplot([r, \+
theta, 0..2*Pi]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 125 "Here now is
 a parametric polar plot (taken from Maple's built-in help facility, a
nd included strictly for your entertainment)" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 138 "r := t -> 100/(100+(t-Pi/2)^8)*(2-sin(7*t)-cos(30*t)
/2):\ntheta := t -> t:\npolarplot([r, theta, -Pi/2..3/2*Pi], numpoints
=2000, axes=NONE);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "With some m
ore fancy plotting techniques (taken from the book " }{TEXT 257 21 "In
troduction to Maple" }{TEXT -1 63 " by Andre Heck, and which you need \+
not worry about) one can get" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 431 "wi
th(plottools): \nr := t -> 100/(100+(t-Pi/2)^8)*(2-sin(7*t)-cos(30*t)/
2):\ntheta := t -> t:\nmapleleaf:=polarplot([r, theta, -Pi/2..3/2*Pi],
 numpoints=1000, axes=NONE):\nmapleleaf:= subs(CURVES=POLYGONS, maplel
eaf):\nrectangles:=rectangle([-5,-1],[-3,4],color=red),rectangle([3,-1
],[5,4],color=red):border:=plot(\{-1,4\},-3..3,color=black): flag2d:=d
isplay([mapleleaf,rectangles,border],view=[-5..5,-1..4], scaling=const
rained):\nflag2d;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "Or even" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 156 "flag3d:=transform((x,y,z)->[x,y,1+
1/3*sin(x)])(flag2d): display(flag3d,scaling=constrained,orientation=[
-110,60],axes=none, style=patchnogrid,shading=none);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 12 
"Assignment 7" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 5 "Ex.1:" }}{PARA 0 
"" 0 "" {TEXT -1 42 "Create plots of the following polar cuves:" }}
{PARA 0 "" 0 "" {TEXT -1 3 "a) " }{XPPEDIT 18 0 "r=1+cos(theta/2)" "6#
/%\"rG,&\"\"\"F&-%$cosG6#*&%&thetaGF&\"\"#!\"\"F&" }{TEXT -1 5 ",    \+
" }{XPPEDIT 18 0 "theta " "6#%&thetaG" }{TEXT -1 4 " in " }{XPPEDIT 
18 0 "[0,4*Pi]" "6#7$\"\"!*&\"\"%\"\"\"%#PiGF'" }{TEXT -1 1 "," }}
{PARA 0 "" 0 "" {TEXT -1 3 "b) " }{XPPEDIT 18 0 "r=1-2*sin(3*theta)" "
6#/%\"rG,&\"\"\"F&*&\"\"#F&-%$sinG6#*&\"\"$F&%&thetaGF&F&!\"\"" }
{TEXT -1 5 ",    " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 4 " i
n " }{XPPEDIT 18 0 "[0,2*Pi]" "6#7$\"\"!*&\"\"#\"\"\"%#PiGF'" }{TEXT 
-1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 3 "c) " }{XPPEDIT 18 0 "r = exp(th
eta/10)" "6#/%\"rG-%$expG6#*&%&thetaG\"\"\"\"#5!\"\"" }{TEXT -1 5 ",  \+
  " }{XPPEDIT 18 0 "theta " "6#%&thetaG" }{TEXT -1 4 " in " }{XPPEDIT 
18 0 "[-20,20]" "6#7$,$\"#?!\"\"F%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" 
{TEXT -1 3 "d) " }{XPPEDIT 18 0 "r = exp(cos(theta)) - 2*cos(4*theta)+
(sin(theta/12))^5" "6#/%\"rG,(-%$expG6#-%$cosG6#%&thetaG\"\"\"*&\"\"#F
--F*6#*&\"\"%F-F,F-F-!\"\"*$-%$sinG6#*&F,F-\"#7F4\"\"&F-" }{TEXT -1 5 
",    " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 4 " in " }
{XPPEDIT 18 0 "[0,24*Pi]" "6#7$\"\"!*&\"#C\"\"\"%#PiGF'" }{TEXT -1 1 "
." }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Ex.2:" }}{PARA 0 "" 0 "" 
{TEXT -1 40 "Plot the polar curve\n                   " }{XPPEDIT 18 
0 "r=cos(m*theta)" "6#/%\"rG-%$cosG6#*&%\"mG\"\"\"%&thetaGF*" }{TEXT 
-1 5 ",    " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 5 "  in " }
{XPPEDIT 18 0 "[0,2*Pi]" "6#7$\"\"!*&\"\"#\"\"\"%#PiGF'" }{TEXT -1 7 "
, \nfor " }{XPPEDIT 18 0 "m" "6#%\"mG" }{TEXT -1 55 "=1,2,...,6. \nInv
estigate what is happening and comment." }}}{SECT 1 {PARA 4 "" 0 "" 
{TEXT -1 5 "Ex.3:" }}{PARA 0 "" 0 "" {TEXT -1 24 "Consider the polar c
urve" }}{PARA 0 "" 0 "" {TEXT -1 14 "              " }{XPPEDIT 18 0 "r
=(a+b*cos(m*theta))*(c+d*sin(n*theta))" "6#/%\"rG*&,&%\"aG\"\"\"*&%\"b
GF(-%$cosG6#*&%\"mGF(%&thetaGF(F(F(F(,&%\"cGF(*&%\"dGF(-%$sinG6#*&%\"n
GF(F0F(F(F(F(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 11 "Note tha
t  " }}{PARA 0 "" 0 "" {TEXT -1 9 "         " }{XPPEDIT 18 0 "a=1" "6#
/%\"aG\"\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "b=1" "6#/%\"bG\"\"\"" }
{TEXT -1 2 ", " }{XPPEDIT 18 0 "m=1/2" "6#/%\"mG*&\"\"\"F&\"\"#!\"\"" 
}{TEXT -1 2 ", " }{XPPEDIT 18 0 "c=1" "6#/%\"cG\"\"\"" }{TEXT -1 2 ", \+
" }{XPPEDIT 18 0 "d= 0" "6#/%\"dG\"\"!" }{TEXT -1 21 "   is  1a),\n   \+
      " }{XPPEDIT 18 0 "a=1" "6#/%\"aG\"\"\"" }{TEXT -1 2 ", " }
{XPPEDIT 18 0 "b= 0" "6#/%\"bG\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "
c=1" "6#/%\"cG\"\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "d=-2" "6#/%\"dG
,$\"\"#!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "n=3" "6#/%\"nG\"\"$" }
{TEXT -1 20 "   is 1b),\n         " }{XPPEDIT 18 0 "a=0" "6#/%\"aG\"\"
!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "b=1" "6#/%\"bG\"\"\"" }{TEXT -1 2 
", " }{XPPEDIT 18 0 "c=d" "6#/%\"cG%\"dG" }{TEXT -1 13 " = 0   is 2).
" }}{PARA 0 "" 0 "" {TEXT -1 35 "Explore the role of the parameters." 
}}{PARA 0 "" 0 "" {TEXT -1 28 "Start, e.g., with the cases " }}{PARA 
0 "" 0 "" {TEXT -1 9 "         " }{XPPEDIT 18 0 "a=1" "6#/%\"aG\"\"\"
" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "b=0" "6#/%\"bG\"\"!" }{TEXT -1 15 "
 (with varying " }{XPPEDIT 18 0 "d" "6#%\"dG" }{TEXT -1 2 ", " }
{XPPEDIT 18 0 "c" "6#%\"cG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "n" "6#%\"
nG" }{TEXT -1 3 "), " }}{PARA 0 "" 0 "" {TEXT -1 9 "         " }
{XPPEDIT 18 0 "c=1" "6#/%\"cG\"\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "
d=0" "6#/%\"dG\"\"!" }{TEXT -1 15 " (with varying " }{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 2 ")," }}{PARA 0 "" 0 "" 
{TEXT -1 9 "         " }{XPPEDIT 18 0 "a=c" "6#/%\"aG%\"cG" }{TEXT -1 
19 " = 0 (with varying " }{XPPEDIT 18 0 "b" "6#%\"bG" }{TEXT -1 2 ", \+
" }{XPPEDIT 18 0 "d" "6#%\"dG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "m" "6#
%\"mG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT -1 3 "). \+
" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Ex.4:" }}{PARA 0 "" 0 "" 
{TEXT -1 63 "Create an animation which shows the spiral in 1c) being d
rawn. " }}{PARA 0 "" 0 "" {TEXT -1 8 "    Use " }{TEXT 261 12 "coords=
polar" }{TEXT -1 17 " as an option in " }{HYPERLNK 17 "animate" 2 "ani
mate" "" }{TEXT -1 60 ". (Otherwise this works just like on the previo
us worksheet)" }}}}}{MARK "3 1" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }
{PAGENUMBERS 0 1 2 33 1 1 }