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0 0 0 1 }1 0 0 0 8 2 1 0 1 0 2 2 -1 1 } {CSTYLE "_cstyle4" -1 213 "Times" 1 14 0 0 0 0 0 1 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle5" -1 214 "Courier" 0 1 0 128 128 1 2 0 1 2 2 2 0 0 0 1 }{CSTYLE "_cstyle6" -1 215 "Times" 0 1 0 0 0 0 0 0 2 2 2 2 0 0 0 1 } {CSTYLE "_cstyle7" -1 216 "Times" 0 1 0 0 0 0 0 1 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle8" -1 217 "Times" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle9" -1 218 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 } {CSTYLE "_cstyle10" -1 219 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 } {CSTYLE "_cstyle11" -1 220 "" 0 1 0 128 128 1 2 0 1 2 2 2 0 0 0 1 } {PSTYLE "_pstyle4" -1 203 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }0 0 0 -1 -1 -1 1 0 1 0 2 2 -1 1 }} {SECT 0 {PARA 200 "" 0 "" {TEXT 210 17 "Polar Coordinates" }}{EXCHG {PARA 201 "" 0 "" {TEXT 211 189 "NOTE: This worksheet will contain man y plots (and animations). It is therefore best if you remove (see the \+ Edit pulldown menu) all the output from your worksheet before saving i t to a disk." }}{PARA 201 "> " 0 "" {MPLTEXT 1 212 8 "restart;" } {MPLTEXT 1 212 13 "\nwith(plots):" }}}{SECT 1 {PARA 202 "" 0 "" {TEXT 213 12 "Introduction" }}{EXCHG {PARA 201 "" 0 "" {TEXT 211 45 "We can \+ create polar plots in Maple using the " }{HYPERLNK 214 "polarplot" 2 " polarplot" "" }{TEXT 211 18 " command from the " }{HYPERLNK 214 "plots " 2 "plots" "" }{TEXT 211 9 " package." }}{PARA 201 "> " 0 "" {MPLTEXT 1 212 12 "with(plots):" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 211 32 "We begin by plotting the circle " }{XPPEDIT 18 0 "r=2" "6#/%\" rG\"\"#" }{TEXT 215 1 " " }{TEXT 211 2 ". " }}{PARA 201 "" 0 "" {TEXT 211 31 "This is accomplished by giving " }{HYPERLNK 214 "polarplot" 2 "polarplot" "" }{TEXT 211 15 " two parameters" }}{PARA 201 "" 0 "" {TEXT 211 15 " - the radius" }}{PARA 201 "" 0 "" {TEXT 211 28 " - \+ the range for the angle" }}{PARA 201 "> " 0 "" {MPLTEXT 1 212 28 "pola rplot(2, theta=0..2*Pi);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 211 57 "Th e following command does the same (since Maple assumes " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT 215 1 " " }{TEXT 211 4 " in " }{XPPEDIT 18 0 "[-Pi,Pi]" "6#7$,$%#PiG!\"\"F%" }{TEXT 215 1 " " }{TEXT 211 38 " \+ if we omit the range for the angle): " }}{PARA 201 "> " 0 "" {MPLTEXT 1 212 13 "polarplot(2);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 211 75 "If \+ we would like to plot only a part of the circle, ie. restrict the angl e " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT 215 1 " " }{TEXT 211 30 ", this can be done as follows:" }}{PARA 201 "> " 0 "" {MPLTEXT 1 212 33 "polarplot(2, theta=Pi/4..3*Pi/4);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 211 41 "A nicer picture of this curve is given by" }}{PARA 201 "> " 0 "" {MPLTEXT 1 212 54 "polarplot(2, theta=Pi/4..3*Pi/4, view=[-2 ..2, -2..2]);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 211 72 "Here is one m ore example of a simple polar plot. This curve is called a " }{TEXT 216 8 "cardioid" }{TEXT 211 1 "." }}{PARA 201 "> " 0 "" {MPLTEXT 1 212 39 "polarplot(1-cos(theta), theta=0..2*Pi);" }}}{EXCHG {PARA 201 " " 0 "" {TEXT 211 47 "Polarplot can also be used in conjunction with " }{TEXT 216 17 "function notation" }{TEXT 211 1 "." }{TEXT 211 33 "\nTh en the cardioid is defined via" }}{PARA 201 "> " 0 "" {MPLTEXT 1 212 28 "r := theta -> 1-cos(theta); " }{MPLTEXT 1 212 23 "\npolarplot(r, 0 ..2*Pi);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 211 60 "An even more power ful and flexible use of polarplot is as a " }{TEXT 216 44 "combination with a parametric representation" }{TEXT 211 1 "." }}{PARA 201 "" 0 " " {TEXT 211 67 "To illustrate this, first consider the following Carte sian example:" }}{PARA 201 "> " 0 "" {MPLTEXT 1 212 17 "f := x -> sin( x);" }{MPLTEXT 1 212 18 "\nplot(f, 0..2*Pi);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 211 54 "We can repeat this example using parametric equation s:" }}{PARA 201 "> " 0 "" {MPLTEXT 1 212 12 "x := t -> t;" }{MPLTEXT 1 212 18 "\ny := t -> sin(t);" }{MPLTEXT 1 212 23 "\nplot([x, y, 0..2* Pi]);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 211 170 "And we have learned \+ that the parametric concept is more powerful than the simple function \+ based definition of a curve, since we can now also allow the second co ordinate (" }{XPPEDIT 2 0 "x" "6#I\"xG6\"" }{TEXT 215 1 " " }{TEXT 211 27 " in our case) to vary with " }{XPPEDIT 2 0 "t" "6#I\"tG6\"" } {TEXT 215 1 " " }{TEXT 211 16 ", allowing, e.g." }}{PARA 201 "> " 0 "" {MPLTEXT 1 212 21 "x := t -> (cos(t))^3;" }{MPLTEXT 1 212 18 "\ny := \+ t -> sin(t);" }{MPLTEXT 1 212 23 "\nplot([x, y, 0..2*Pi]);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 211 171 "Maple allows us to do the analogous t hing with polar plots - a concept which is not included in our Calculu s book, but is a natural extension of the ideas we've discussed." }} {PARA 201 "" 0 "" {TEXT 211 74 "Let's reconsider the cardioid displaye d earlier (using function notation)." }}{PARA 201 "> " 0 "" {MPLTEXT 1 212 27 "r := theta -> 1-cos(theta);" }{MPLTEXT 1 212 23 "\npolarplot (r, 0..2*Pi);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 211 110 "Now, let's r edo this in a parametric polar representation (note that we are now us ing an additional parameter " }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 215 1 " " }{TEXT 211 2 "):" }}{PARA 201 "> " 0 "" {MPLTEXT 1 212 19 "r := \+ t -> 1-cos(t);" }{MPLTEXT 1 212 17 "\ntheta := t -> t;" }{MPLTEXT 1 212 32 "\npolarplot([r, theta, 0..2*Pi]);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 211 125 "Here now is a parametric polar plot (taken from Maple' s built-in help facility, and included strictly for your entertainment )" }}{PARA 201 "> " 0 "" {MPLTEXT 1 212 56 "r := t -> 100/(100+(t-Pi/2 )^8)*(2-sin(7*t)-cos(30*t)/2):" }{MPLTEXT 1 212 17 "\ntheta := t -> t: " }{MPLTEXT 1 212 65 "\npolarplot([r, theta, -Pi/2..3/2*Pi], numpoints =2000, axes=NONE);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 211 62 "With som e more fancy plotting techniques (taken from the book " }{TEXT 217 21 "Introduction to Maple" }{TEXT 211 63 " by Andre Heck, and which you n eed not worry about) one can get" }}{PARA 201 "> " 0 "" {MPLTEXT 1 212 17 "with(plottools): " }{MPLTEXT 1 212 57 "\nr := t -> 100/(100+(t -Pi/2)^8)*(2-sin(7*t)-cos(30*t)/2):" }{MPLTEXT 1 212 17 "\ntheta := t \+ -> t:" }{MPLTEXT 1 212 76 "\nmapleleaf:=polarplot([r, theta, -Pi/2..3/ 2*Pi], numpoints=1000, axes=NONE):" }{MPLTEXT 1 212 46 "\nmapleleaf:= \+ subs(CURVES=POLYGONS, mapleleaf):" }{MPLTEXT 1 212 210 "\nrectangles:= rectangle([-5,-1],[-3,4],color=red),rectangle([3,-1],[5,4],color=red): border:=plot(\{-1,4\},-3..3,color=black): flag2d:=display([mapleleaf,r ectangles,border],view=[-5..5,-1..4], scaling=constrained):" } {MPLTEXT 1 212 8 "\nflag2d;" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 211 7 " Or even" }}{PARA 201 "> " 0 "" {MPLTEXT 1 212 156 "flag3d:=transform(( x,y,z)->[x,y,1+1/3*sin(x)])(flag2d): display(flag3d,scaling=constraine d,orientation=[-110,60],axes=none, style=patchnogrid,shading=none);" } }}{EXCHG {PARA 201 "> " 0 "" }}}{SECT 1 {PARA 202 "" 0 "" {TEXT 213 12 "Assignment 8" }}{SECT 1 {PARA 202 "" 0 "" {TEXT 213 5 "Ex.1:" }} {PARA 201 "" 0 "" {TEXT 211 42 "Create plots of the following polar cu ves:" }}{PARA 201 "" 0 "" {TEXT 211 3 "a) " }{XPPEDIT 18 0 "r=1+cos(th eta/2)" "6#/%\"rG,&\"\"\"F&-%$cosG6#*&%&thetaGF&\"\"#!\"\"F&" }{TEXT 215 1 " " }{TEXT 211 5 ", " }{XPPEDIT 18 0 "theta " "6#%&thetaG" } {TEXT 215 1 " " }{TEXT 211 4 " in " }{XPPEDIT 18 0 "[0,4*Pi]" "6#7$\" \"!*&\"\"%\"\"\"%#PiGF'" }{TEXT 215 1 " " }{TEXT 211 1 "," }}{PARA 201 "" 0 "" {TEXT 211 3 "b) " }{XPPEDIT 18 0 "r=1-2*sin(3*theta)" "6#/ %\"rG,&\"\"\"F&*&\"\"#F&-%$sinG6#*&\"\"$F&%&thetaGF&F&!\"\"" }{TEXT 215 1 " " }{TEXT 211 5 ", " }{XPPEDIT 18 0 "theta" "6#%&thetaG" } {TEXT 215 1 " " }{TEXT 211 4 " in " }{XPPEDIT 18 0 "[0,2*Pi]" "6#7$\" \"!*&\"\"#\"\"\"%#PiGF'" }{TEXT 215 1 " " }{TEXT 211 1 "," }}{PARA 201 "" 0 "" {TEXT 211 3 "c) " }{XPPEDIT 18 0 "r = exp(theta/10)" "6#/% \"rG-%$expG6#*&%&thetaG\"\"\"\"#5!\"\"" }{TEXT 215 1 " " }{TEXT 211 5 ", " }{XPPEDIT 18 0 "theta " "6#%&thetaG" }{TEXT 215 1 " " }{TEXT 211 4 " in " }{XPPEDIT 18 0 "[-20,20]" "6#7$,$\"#?!\"\"F%" }{TEXT 215 1 " " }{TEXT 211 1 "." }}{PARA 201 "" 0 "" {TEXT 211 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 215 1 " " }{TEXT 211 5 ", " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT 215 1 " " }{TEXT 211 4 " in " }{XPPEDIT 18 0 "[0,24*Pi]" "6#7$\"\"!*&\"#C\"\"\"%#PiGF'" }{TEXT 215 1 " " }{TEXT 211 1 "." }}}{SECT 1 {PARA 202 "" 0 "" {TEXT 213 5 "Ex.2:" }}{PARA 201 "" 0 "" {TEXT 211 20 "Plot the polar curve" }{TEXT 211 20 "\n " }{XPPEDIT 18 0 "r=cos(m*theta)" "6#/%\"rG-%$cosG6#*&%\"mG\"\"\"%&thetaGF*" }{TEXT 215 1 " " }{TEXT 211 5 ", " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT 215 1 " " } {TEXT 211 5 " in " }{XPPEDIT 18 0 "[0,2*Pi]" "6#7$\"\"!*&\"\"#\"\"\"% #PiGF'" }{TEXT 215 1 " " }{TEXT 211 2 ", " }{TEXT 211 5 "\nfor " } {XPPEDIT 18 0 "m" "6#%\"mG" }{TEXT 215 1 " " }{TEXT 211 12 "=1,2,...,6 . " }{TEXT 211 43 "\nInvestigate what is happening and comment." }}} {SECT 1 {PARA 202 "" 0 "" {TEXT 213 5 "Ex.3:" }}{PARA 201 "" 0 "" {TEXT 211 24 "Consider the polar curve" }}{PARA 201 "" 0 "" {TEXT 211 14 " " }{XPPEDIT 18 0 "r=(a+b*cos(m*theta))*(c+d*sin(n*th eta))" "6#/%\"rG*&,&%\"aG\"\"\"*&%\"bGF(-%$cosG6#*&%\"mGF(%&thetaGF(F( F(F(,&%\"cGF(*&%\"dGF(-%$sinG6#*&%\"nGF(F0F(F(F(F(" }{TEXT 215 1 " " } {TEXT 211 1 "." }}{PARA 201 "" 0 "" {TEXT 211 11 "Note that " }} {PARA 201 "" 0 "" {TEXT 211 9 " " }{XPPEDIT 18 0 "a=1" "6#/%\" aG\"\"\"" }{TEXT 215 1 " " }{TEXT 211 2 ", " }{XPPEDIT 18 0 "b=1" "6#/ %\"bG\"\"\"" }{TEXT 215 1 " " }{TEXT 211 2 ", " }{XPPEDIT 18 0 "m=1/2" "6#/%\"mG*&\"\"\"F&\"\"#!\"\"" }{TEXT 215 1 " " }{TEXT 211 2 ", " } {XPPEDIT 18 0 "c=1" "6#/%\"cG\"\"\"" }{TEXT 215 1 " " }{TEXT 211 2 ", \+ " }{XPPEDIT 18 0 "d= 0" "6#/%\"dG\"\"!" }{TEXT 215 1 " " }{TEXT 211 11 " is 1a)," }{TEXT 211 10 "\n " }{XPPEDIT 18 0 "a=1" "6#/ %\"aG\"\"\"" }{TEXT 215 1 " " }{TEXT 211 2 ", " }{XPPEDIT 18 0 "b= 0" "6#/%\"bG\"\"!" }{TEXT 215 1 " " }{TEXT 211 2 ", " }{XPPEDIT 18 0 "c=1 " "6#/%\"cG\"\"\"" }{TEXT 215 1 " " }{TEXT 211 2 ", " }{XPPEDIT 18 0 " d=-2" "6#/%\"dG,$\"\"#!\"\"" }{TEXT 215 1 " " }{TEXT 211 2 ", " } {XPPEDIT 18 0 "n=3" "6#/%\"nG\"\"$" }{TEXT 215 1 " " }{TEXT 211 10 " \+ is 1b)," }{TEXT 211 10 "\n " }{XPPEDIT 18 0 "a=0" "6#/%\"aG\" \"!" }{TEXT 215 1 " " }{TEXT 211 2 ", " }{XPPEDIT 18 0 "b=1" "6#/%\"bG \"\"\"" }{TEXT 215 1 " " }{TEXT 211 2 ", " }{XPPEDIT 18 0 "c=d" "6#/% \"cG%\"dG" }{TEXT 215 1 " " }{TEXT 211 13 " = 0 is 2)." }}{PARA 201 "" 0 "" {TEXT 211 35 "Explore the role of the parameters." }}{PARA 201 "" 0 "" {TEXT 211 28 "Start, e.g., with the cases " }}{PARA 201 "" 0 "" {TEXT 211 9 " " }{XPPEDIT 18 0 "a=1" "6#/%\"aG\"\"\"" } {TEXT 215 1 " " }{TEXT 211 2 ", " }{XPPEDIT 18 0 "b=0" "6#/%\"bG\"\"!" }{TEXT 215 1 " " }{TEXT 211 15 " (with varying " }{XPPEDIT 18 0 "d" " 6#%\"dG" }{TEXT 215 1 " " }{TEXT 211 2 ", " }{XPPEDIT 18 0 "c" "6#%\"c G" }{TEXT 215 1 " " }{TEXT 211 2 ", " }{XPPEDIT 18 0 "n" "6#%\"nG" } {TEXT 215 1 " " }{TEXT 211 3 "), " }}{PARA 201 "" 0 "" {TEXT 211 9 " \+ " }{XPPEDIT 18 0 "c=1" "6#/%\"cG\"\"\"" }{TEXT 215 1 " " } {TEXT 211 2 ", " }{XPPEDIT 18 0 "d=0" "6#/%\"dG\"\"!" }{TEXT 215 1 " " }{TEXT 211 15 " (with varying " }{XPPEDIT 18 0 "a" "6#%\"aG" }{TEXT 215 1 " " }{TEXT 211 2 ", " }{XPPEDIT 18 0 "b" "6#%\"bG" }{TEXT 215 1 " " }{TEXT 211 2 ", " }{XPPEDIT 18 0 "m" "6#%\"mG" }{TEXT 215 1 " " } {TEXT 211 2 ")," }}{PARA 201 "" 0 "" {TEXT 211 9 " " } {XPPEDIT 18 0 "a=c" "6#/%\"aG%\"cG" }{TEXT 215 1 " " }{TEXT 211 19 " = 0 (with varying " }{XPPEDIT 18 0 "b" "6#%\"bG" }{TEXT 215 1 " " } {TEXT 211 2 ", " }{XPPEDIT 18 0 "d" "6#%\"dG" }{TEXT 215 1 " " }{TEXT 211 2 ", " }{XPPEDIT 18 0 "m" "6#%\"mG" }{TEXT 215 1 " " }{TEXT 211 2 ", " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT 215 1 " " }{TEXT 211 3 "). " }}}{SECT 1 {PARA 202 "" 0 "" {TEXT 213 5 "Ex.4:" }}{PARA 201 "" 0 "" {TEXT 211 63 "Create an animation which shows the spiral in 1c) being \+ drawn. " }}{PARA 201 "" 0 "" {TEXT 218 8 " Use " }{TEXT 219 12 "coo rds=polar" }{TEXT 218 17 " as an option in " }{HYPERLNK 220 "animate" 2 "animate" "" }{TEXT 218 4 " or " }{HYPERLNK 220 "animatecurve" 2 "an imatecurve" "" }{TEXT 218 61 " . (Otherwise this works just like on th e previous worksheet)" }}}}{PARA 203 "" 0 "" }}{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }