{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 128 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 128 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 128 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 128 128 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 262 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 0 } {CSTYLE "" 18 272 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 274 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 276 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 277 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 278 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 280 "" 0 1 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 281 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 282 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 283 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 284 "" 0 1 128 0 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 285 "" 0 1 128 0 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 286 "" 0 1 128 0 0 1 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 287 "" 0 1 128 128 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 288 "" 0 1 128 128 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 289 " " 0 1 128 128 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 290 "" 0 1 128 0 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 291 "" 0 1 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 292 "" 0 1 128 0 0 1 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 293 "" 0 1 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 294 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times " 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 } {PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 257 54 "High School Modul es > Precalculus by Gregory A. Moore\n" }}{PARA 3 "" 0 "" {TEXT -1 4 " " }{TEXT 256 48 "Polar Graphs - Roses, Rings, Bracelets, & Hearts " }}{PARA 0 "" 0 "" {TEXT -1 60 "\nAn exploration of the usual and som e unusual polar graphs.\n" }}{PARA 0 "" 0 "" {TEXT 258 153 "[Direction s : Execute the Code Resource section first. Although there will be no output immediately, these definitions are used later in this workshee t.]" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 260 7 "0. Code" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "restart; with(plots): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "fade := proc( a1,a2,a3,k,n)\nlocal c;\nc := COLOR(RGB, a1*k/n, a 2*k/n, a3*k/n); \nreturn c;\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 229 "PolarFadePlot := proc( f)\n local n;\n n := 10; \n display( [seq( \n polarplot( k*f(x)/n, theta = 0..2*Pi, scaling = constrained,\n filled = true, \n color = fa de(.9,.6,.7,k,n)), k = 1..n )] );\n\nend proc:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " } {TEXT 261 23 "1. Cardioids & Limacons" }}{PARA 0 "" 0 "" {TEXT -1 1 " \n" }{TEXT 267 8 "Limacons" }{TEXT -1 83 " are a family of polar graph s which are defined by the polar equations of the form:" }}{PARA 0 "" 0 "" {TEXT -1 21 " " }{XPPEDIT 18 0 "y = a + b*sin (theta)" "6#/%\"yG,&%\"aG\"\"\"*&%\"bGF'-%$sinG6#%&thetaGF'F'" }{TEXT -1 13 " " }{XPPEDIT 18 0 "y = a - b*sin(theta)" "6#/%\"yG, &%\"aG\"\"\"*&%\"bGF'-%$sinG6#%&thetaGF'!\"\"" }{TEXT -1 23 " \n \+ " }{XPPEDIT 18 0 "y = a + b*cos(theta)" "6#/%\"yG,&%\"a G\"\"\"*&%\"bGF'-%$cosG6#%&thetaGF'F'" }{TEXT -1 12 " " } {XPPEDIT 18 0 "y = a - b*cos(theta) " "6#/%\"yG,&%\"aG\"\"\"*&%\"bGF'- %$cosG6#%&thetaGF'!\"\"" }{TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 90 "Although there appear to be four different forms, they can all be \+ expressed in the form : " }{XPPEDIT 18 0 "y = a+b*sin(theta-C);" "6#/% \"yG,&%\"aG\"\"\"*&%\"bGF'-%$sinG6#,&%&thetaGF'%\"CG!\"\"F'F'" }{TEXT -1 14 "\nwhere C = 0, " }{XPPEDIT 18 0 "Pi/2" "6#*&%#PiG\"\"\"\"\"#!\" \"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 5 ", or \+ " }{XPPEDIT 18 0 "3*Pi/2" "6#*(\"\"$\"\"\"%#PiGF%\"\"#!\"\"" }{TEXT -1 97 ". Thus they all represent similar graphs - with differing amoun ts of right angle rotation.\n\n\n " }{TEXT 270 15 "Comparing a & b " }{TEXT -1 41 "\n\nThe relationship between the constants " }{TEXT 268 1 "a" }{TEXT -1 5 " and " }{TEXT 269 1 "b" }{TEXT -1 221 " determi nes the shape of the graph. In particular, there are three cases : \n \+ I. |a| = |b|\n II. |a| > |b|, and \n III | a| < |b|. \nEach of these cases creates a distinctive version of the l imacon." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "\n " }{TEXT 284 8 "Case I ." }{TEXT -1 161 " \nWhen \+ |a| = |b|, the graph passes through the origin. This shape is known as a Cardioid - \"heart\" shaped curves . Note the reference circles of \+ radius 1 and 2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 209 "display(\n polarplot( \{1,2\}, theta = 0.. 2*Pi, \n scaling = constrained, color = blue, linestyle = 2),\n \+ polarplot( 1+sin(theta), theta = 0..2*Pi, \n scaling = constrained, color = red, thickness = 3));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "`This is a rough indication only`;\nPolarFadePlot( 3 + 3*sin( theta));" }}}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 14 " " }{TEXT 286 9 "Case II ." }{TEXT -1 117 " \nWhe n |a| > |b|, the graph maintains some distance between it and the orig in, resulting in a rounder, puffier plot." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 215 "display(\n polarp lot( \{1,3,5\}, theta = 0..2*Pi, \n scaling = constrained, color = \+ blue, linestyle = 2),\n polarplot( 3+2*sin(theta), theta = 0..2*Pi, \n scaling = constrained, color = gold, thickness = 3));\n" }}} {PARA 0 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 14 " \+ " }{TEXT 285 10 "Case III ." }{TEXT -1 105 " \nWhen |a| < |b|, the graph not only passes through the origin, but also part of it fol ds inside itself." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 216 "display(\n polarplot( \{2,3,8\}, theta = \+ 0..2*Pi, \n scaling = constrained, color = blue, linestyle = 2),\n \+ polarplot( 3+5*sin(theta), theta = 0..2*Pi, \n scaling = constra ined, color = green, thickness = 3));\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "Lets see all of these varieties in one glance. The green curve is the cardioid " }{TEXT 288 20 "(case I \+ : |a| = |b|)" }{TEXT -1 29 ". The red curves are flatter " }{TEXT 287 21 "(case II : |a| > |b|)" }{TEXT -1 48 ", and the blue curves bend in side of themselves " }{TEXT 289 22 "(case III : |a| < |b|)" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 290 "display( \n polarplot( 8 + 8*cos(theta) \+ , theta = 0..2*Pi, \n scaling = constrained, color = green, thickne ss = 3), \n polarplot(\{8 + a*cos(theta) $ a = 9..15\}, \n th eta = 0..2*Pi, color = blue), \n polarplot(\{ 8 + a*cos(theta) $ a \+ = 1..7\}, \n theta = 0..2*Pi, color = red));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 83 "Here's the same phenomenon shown as an animation. H ere, we see the polar graph of " }{XPPEDIT 18 0 "5+a*cos(theta)" "6#,& \"\"&\"\"\"*&%\"aGF%-%$cosG6#%&thetaGF%F%" }{TEXT -1 13 " changing as \+ " }{XPPEDIT 18 0 "a" "6#%\"aG" }{TEXT -1 35 " increases continuously f rom 0 to 8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "for i from 1 to 30 do\n a := i/30 * 8;\n p[i]:=polarplot(5+a*cos(theta), numpoin ts=500);\nend do:\ndisplay(seq(p[i],i=1..30), insequence=true, scaling =constrained);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 5 "\n " }{TEXT 280 37 "Comparing the Four Different Versio ns" }{TEXT 271 58 "\n\nAlthough there appear to be four different form s - with " }{TEXT -1 32 "sine, cosine, -sine, and -cosine" }{TEXT 281 52 " - actually they can all be expressed in the form : " }{XPPEDIT 272 0 "y = a+b*sin(theta-C);" "6#/%\"yG,&%\"aG\"\"\"*&%\"bGF'-%$sinG6# ,&%&thetaGF'%\"CG!\"\"F'F'" }{TEXT 273 16 ", where C = 0, " } {XPPEDIT 274 0 "Pi/2" "6#*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT 275 2 ", " } {XPPEDIT 276 0 "Pi" "6#%#PiG" }{TEXT 277 5 ", or " }{XPPEDIT 278 0 "3* Pi/2" "6#*(\"\"$\"\"\"%#PiGF%\"\"#!\"\"" }{TEXT 279 91 ". Thus they al l represent similar graphs - with differing amounts of right angle rot ation.\n" }{TEXT -1 1 "\n" }{TEXT 282 37 "Lets take a look at all four at once!" }{TEXT -1 1 " " }{TEXT 283 110 "Can you decide which graph \+ belongs to which? Think about what values of theta make the sine and c osine maxima!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 173 "polarplot(\{ 8 + 7*sin(theta), 8 + 7*cos(theta) , 8 - 7*sin(theta), 8 -7*cos(theta)\}, theta = 0..2*Pi, scaling = cons trained, thickness = 2, color = [red, blue, gold, green]);" }}}{PARA 0 "" 0 "" {TEXT -1 113 "\n\nHere is another variation. This \"band of \+ gold\" was created by making the angle multiplier an irrational number ." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "polarplot( 10 + sin(2*P i*theta), theta = 0..20*Pi, \n color = coral, scaling = constrained); " }}}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 262 18 "2. The Rose Garden" }}{PARA 0 "" 0 "" {TEXT -1 55 "\nWe're going to look at polar functions of the form : " } {XPPEDIT 18 0 "r = a*sin(n*theta)" "6#/%\"rG*&%\"aG\"\"\"-%$sinG6#*&% \"nGF'%&thetaGF'F'" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "r = a*cos(n*th eta)" "6#/%\"rG*&%\"aG\"\"\"-%$cosG6#*&%\"nGF'%&thetaGF'F'" }{TEXT -1 52 " - which are sometimes called multi-petaled roses.\n\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 290 8 " " }{TEXT 291 26 "Even & Odd Numbered Petals" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 141 "The first distinction to be made is betw een when n is an even or odd number. When n is an odd number, the resu lting rose has exactly n petals" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 147 "polarplot( [9, 9*sin(3* the ta)], theta = 0..2*Pi, \n scaling = constrained, linestyle = [2, 1], \n thickness = [1,3], color =[navy, red]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 149 "polarplot( [20, 20*sin(5* theta)], theta = 0..2*Pi, \n scaling = constrained, linestyle = [2,1], \n \+ thickness = [1,3], color =[navy, red]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "\nHowever, when n is even, the \+ rose has 2n petals." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 150 "polarplot( [5, 5*sin(2*theta)], theta = 0. .2*Pi, \n scaling = constrained, linestyle = [2,1], \n thi ckness = [1,3], color =[navy, magenta]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 150 "polarplot( [5, 5*cos(4*theta)], theta = 0..2*Pi, \n \+ scaling = constrained, linestyle = [2,1], \n thickness = [ 1,3], color =[navy, magenta]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 150 "polarplot( [5, 5*sin(6*theta)], theta = 0..2*Pi, \n sca ling = constrained, linestyle = [2,1], \n thickness = [1,3], col or =[navy, magenta]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 181 "You mig ht be wondering, how does the transition between n petals and 2n petal s happen? Does it jump or evolve \"smoothly\"? An animation is a goo d way to find out. Here we look at " }{XPPEDIT 18 0 "cos(a*theta)" "6 #-%$cosG6#*&%\"aG\"\"\"%&thetaGF(" }{TEXT -1 4 " as " }{XPPEDIT 18 0 " a" "6#%\"aG" }{TEXT -1 86 " increases continuously from 0 to 4. As yo u watch, can you identify the moments when " }{XPPEDIT 18 0 "a " "6#% \"aG" }{TEXT -1 15 " is an integer?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 126 "for i from 0 to 60 do\n a := i/60 * 4;\n p[i] := p olarplot(cos(a*theta));\nend do:\ndisplay(seq(p[i],i=0..60), insequenc e=true);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 544 "You may notice another distinction. The even flowers have all of \+ their petals next to each other, but the odd flowers have gaps between the pedals large enough to put twice as many petals. Try creating som e other roses on your own with different numbers of petals to verify t hat the even/odd relationship holds. If you were to perform the \"She \+ loves me, she loves me not\" procedure on once of these petals, would \+ you prefer that n be even or odd?\n\nWhat about a single-petaled rose? Do you recognize the inner shape of the \"single petaled rose\"?" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 144 "polarplot( [9, 9*sin(theta)], theta = 0..2*Pi, \n scaling = constrained, linestyle = [2,1], \n thickness = [1,3], color =[n avy, red]);" }}}{PARA 0 "" 0 "" {TEXT -1 9 "\n " }{TEXT 292 27 "Sine & Cosine Powered Roses" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 263 "Although r = sin(x) and r = cos(x) will create n-petaled roses inscribed in the unit circle, what is the difference \+ between them? The graph with the sine appears tangent to the positive \+ x axis, while the cosine version has a petal centered at the positive \+ x axis." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "polarplot( \{sin(3*theta), cos(3*theta)\}, theta = 0. .2*Pi, scaling = constrained);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 63 "Here is an illustration of the same idea \+ with even more petals." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "polarplot(\{sin(6*theta),cos(6*thet a)\}, theta = 0..2*Pi, scaling = constrained);" }}}{PARA 0 "" 0 "" {TEXT -1 90 "Cosine flowers have a pedal on the x-axis. Sine flowers h ave a gap at the x-axis.\n\n " }{TEXT 293 9 "Amplitude" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 268 "In the formula above, how does the number a, which is the amplitude in effect the gr aph? Here we let a =1,2,3...,12 and see how the resulting graphs look. Each different color is a different graph. You can see that they are \+ inscribed in circles of radius 1,2,3,...,12." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "a := 'a':" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "polarplot( \{a*cos(6*theta) \+ $ a = 1..12\}, theta = 0..2*Pi, scaling = constrained);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "polarplot( \{a*cos(11*theta) $ a = \+ 1..12\}, theta = 0..2*Pi, scaling = constrained);" }}}{PARA 0 "" 0 "" {TEXT -1 114 "\nCan you explain why this graph looks like a flower too , although its not in the same form as the functions above?" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "polarplot( sin(theta)^2 - cos(theta )^2, theta = 0..2*Pi,scaling = constrained, color = gold);" }}}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " } {TEXT 263 19 "3. Valentine Curves" }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" } {TEXT 294 16 "Valentine curves" }{TEXT -1 149 " - there is really no s uch name but it seemed reasonable when considering cruves which are a \+ hybrid of rings, hearts(cardioids), and flowers(roses). " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "polar plot( 4 + cos(6*theta) , theta = 0..2*Pi, scaling = constrained);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "polarplot( 4 + 3*sin(7*theta ), theta = 0..2*Pi, scaling = constrained);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "This one wraps in on itself" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "polarplot( 3 + 7*sin(3*theta), theta = 0..2*Pi, scaling = constrai ned);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "Here are whole families of similar curves" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "polarplot( \{ 6 + \+ a*cos(6*theta) $ a = 1..11\}, theta = 0..2*Pi, scaling = constrained); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "polarplot( \{12 + a*sin (7*theta) $ a = 1..12\}, theta = 0..2*Pi, scaling = constrained);" }}} {PARA 0 "" 0 "" {TEXT -1 2 "\n " }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 264 45 "4. Familiar Shapes Disguised in Polar Clothes" }} {PARA 0 "" 0 "" {TEXT -1 316 "\nMany familiar shapes that you came to \+ love in your algebra class - such as lines, circles, parabolas, and el lipses - can be expressed in polar form. In polar coordinates, the sim plest function for r is r = constant, which makes a circle centered at the origin. Lets look at the graphs of r = 1, r = 2, ... , r = 20." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "This dra ws concentric circles of radius r = 1, 2, 3, ... , 20" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "polarplot ( \{k $ k = 1..20\}, theta = 0..2*Pi, scaling = constrained);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "We can al so draw circles not centered at the origin." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "polarplot( cos(the ta), theta = 0..2*Pi, scaling = constrained);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "polarplot( cos(theta - Pi/4), theta = 0..2*Pi, s caling = constrained);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "notice that these pass through the origin\n\nWe can \+ also draw ellipses and parabolas...." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "polarplot( 1/(8 - 7*cos(th eta)), theta = 0..2*Pi, scaling = constrained);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 48 "polarplot( 1/(1 - cos(theta)), theta = 0..2*Pi );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "polarplot( 1/(3 + 2*s in(theta)), theta = 0..2*Pi, scaling = constrained);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "...even horizontal and vertical lines" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 46 "polarplot( 2*csc(theta), theta = -2*Pi..2*Pi); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "polarplot(2*sec(theta), theta = -2*Pi..2*Pi);" }}}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 " " 0 "" {TEXT -1 2 "\n " }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " } {TEXT 265 27 "5. Spiraling Out of Control" }}{PARA 0 "" 0 "" {TEXT -1 31 "\nA basic spiral is of the form " }{XPPEDIT 18 0 "r = theta" "6#/% \"rG%&thetaG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "polarplot(theta,theta = 0..4 *Pi, scaling = constrained);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "polarplot(theta, theta = 0..40*Pi, scaling = constrained);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "A larger \+ range of values for theta gives more chance for the graph to wrap arou nd." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 156 "E ven more interesting graphs can be created using the product of theta \+ and a trigonometric function. As theta increases there is a sort of sp iraling effect." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 69 "polarplot( theta*sin(theta), theta = 0..3*Pi, \+ scaling = constrained);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 " polarplot( theta*sin(theta), theta = 0..25*Pi, scaling = constrained, \+ numpoints = 500);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "As we increase the range of values for theta, we get even more of the same." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "Here is another variation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "polarplot( 2*cos(theta ) + sqrt( abs( 4*cos(theta)^2 -3)), theta = 0..2*Pi, scaling = constra ined, numpoints = 1000);" }}}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 266 29 "6. How to Build a Bett er Rose" }}{PARA 0 "" 0 "" {TEXT -1 129 "\nThe so-called 'roses' above , really bore more of a resemblance to daisies. Here is something that looks a little more rose-like." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "polarplot( theta + 2*sin(2*P i*theta), theta = 0..12*Pi,color = red, thickness = 2 );" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "Here are some othe r beautiful botanicals." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "polarplot( theta + 3*sin(4*theta) \+ - 5*cos(4*theta), theta = 0..12*Pi,color = violet, thickness = 2 );" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 124 "polarplot( theta + 2*sin( 2*Pi*theta) + 4*cos(2*Pi*theta), theta = 0..12*Pi,color = green, thick ness = 2 , numpoints = 1000);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "polarplot( sin(theta) + 2*cos(2*theta), theta = 0..2*Pi,scaling \+ = constrained );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 150 "polarp lot( sin(theta) + 2*cos(2*theta) + 3*sin(3*theta) + 4*cos(4*theta), \n theta = 0..2*Pi,scaling = constrained, thickness = 2, color = gr een);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 152 "polarplot( 4*sin( theta) + 3*cos(2*theta) + 2*sin(3*theta) + 1*cos(4*theta), \n the ta = 0..2*Pi,scaling = constrained, thickness = 3, color = coral);" }} }{PARA 0 "" 0 "" {TEXT -1 32 "\n\nMaybe more chrysanthanum-like." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "polarplot( cos(.95*theta), t heta = 0..40*Pi,scaling = constrained, color = brown);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "polarplot( cos(.2*theta) - sin(.5*t heta), theta = 0..40*Pi,scaling = constrained, color = gold);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "polarplot( sin(theta)^5 - co s(theta)^6, theta = 0..2*Pi,scaling = constrained, color = gold);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "polarplot( sin(theta)^3 - co s(theta)^9, theta = 0..2*Pi,scaling = constrained, color = gold);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "polarplot( sin(theta)^3 - co s(2*theta)^3, theta = 0..12*Pi,scaling = constrained, color = gold);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT 259 36 "\n \251 2002 Waterloo Maple Inc " }}}{MARK "0 1 " 25 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }