{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 "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{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 13 "" 1 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 544 "You may notice another distinc tion. The even flowers have all of their petals next to each other, bu t the odd flowers have gaps between the pedals large enough to put twi ce as many petals. Try creating some other roses on your own with diff erent numbers of petals to verify that the even/odd relationship holds . If you were to perform the \"She loves me, she loves me not\" proced ure 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 o f 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 =[navy, 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 th e 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 "He re 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*theta)\}, theta = 0..2*Pi, scaling \+ = constrained);" }}}{PARA 0 "" 0 "" {TEXT -1 90 "Cosine flowers have a pedal on the x-axis. Sine flowers have 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, whi ch is the amplitude in effect the graph? Here we let a =1,2,3...,12 an d see how the resulting graphs look. Each different color is a differe nt 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 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 "\nCa n 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. Val entine Curves" }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }{TEXT 294 16 "Valent ine curves" }{TEXT -1 149 " - there is really no such name but it seem ed reasonable when considering cruves which are a hybrid of rings, hea rts(cardioids), and flowers(roses). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "polarplot( 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 = constrained);" }}}{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 i n your algebra class - such as lines, circles, parabolas, and ellipses - can be expressed in polar form. In polar coordinates, the simplest \+ function for r is r = constant, which makes a circle centered at the o rigin. 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 draw s 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);" }}}}{PARA 0 "" 0 "" {TEXT 259 36 "\n \251 2002 Waterloo Maple \+ Inc " }}}{MARK "0 1" 44 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }