{VERSION 6 0 "IBM INTEL NT" "6.0" } {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 0 0 0 0 0 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{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 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 271 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 0 1 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 "Title" -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Author" -1 19 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 8 8 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT 256 25 "Coloring Inside the Line s" }}}{EXCHG {PARA 19 "" 0 "" {TEXT -1 13 "J. A. Ziegler" }}{PARA 256 "" 0 "" {TEXT -1 17 "February 20, 2005" }}{PARA 256 "" 0 "" {TEXT -1 55 "Web Page: http://www2.SPSU.edu/math/ziegler/index.html" }}{PARA 256 "" 0 "" {TEXT -1 26 "E-mail: jziegler@spsu.edu" }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 64 "Coloring The Area Between Two Curves For A First Calculus Course" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 131 "Maple does not provide a command for coloring the a rea between two curves. To color such an area, we may use the followi ng method." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "with(plots):w ith(plottools):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 140 "First, to ill ustrate finding the area between two curves, we define two curves we m ight choose on the interval [0,T]. We choose the curves " }{XPPEDIT 18 0 "y = x^2;" "6#/%\"yG*$%\"xG\"\"#" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y = 2*sqrt(2*x);" "6#/%\"yG*&\"\"#\"\"\"-%%sqrtG6#*&F&F'%\"xGF'F '" }{TEXT -1 232 " , which cross at x = 2, so that we may show that th e method is independent of which curve is the \"upper\" curve. To col or between the curves, it will be convenient to describe the curves pa rametrically. We begin by choosing T = 2." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 5 "T:=2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 " a:=plot([t,t^2,t=0..T],color=blue,thickness=4):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 54 "b:=plot([t,2*sqrt(2*t),t=0..T],color=red,thick ness=4):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "display(a,b,sca ling=constrained);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 242 "Now we col or the area between them by drawing a sufficiently large number of thi ck vertical lines. We define a typical line, then make a collection o f them. If the curves have not been described parametrically before, \+ here, they need to be." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "L :=s->line([s,s^2],[s,2*sqrt(2*s)],color=green,thickness=4):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "S:=n->seq(k/n,k=0..T*n):" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 175 "Here we will choose 40 lines, bu t with other curves some experimentation may be needed to determine ho w many will be sufficient to produce a solid coloring. Our collection is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "pic:=display(seq(L(s ),s=S(40))):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "We display the re sult of our efforts." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "dis play(a,b,pic,scaling=constrained);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 147 "To see that \+ this method need not be changed even though the upper and lower curves exchange places, return to the definition of T and choose T = 3." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 21 "A More Difficult Case" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 506 "In the spring of 2003, my friend and colleague, Dr. Donald You ng, showed me the defininition of a curve he had devised. In part, the purpose was to show his students that the region enclosed by a piecew ise smooth, simple closed curve might be considerably more complex tha n those often used as textbook illustrations. He asked if I could use Maple to color the region enclosed by this curve. This proved to be \+ more challenging than I anticipated, but, as I succeeded, I would like to show you the result." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" } {TEXT 257 24 "Drawing D. Young's Curve" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "restart:with(plots):with(plottools):with(linalg):" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 164 "f:=piecewise(-1/Pi<=t and \+ t<0,[t^3*cos(1/abs(t)),t^3*sin(1/abs(t))],t=0,0,0 " 0 "" {MPLTEXT 1 0 36 "g:=t->[cos(1/abs(t)),sin(1/a bs(t))];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "F1:=t->convert( evalm(t^3*g(t)),list):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "F1 (t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "F2:=t->convert(eval m(g(1/(2*Pi))*t^3),list):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "F2(t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "P1:=plot([op(F1( t)),t=-1/Pi..1/(2*Pi)],color=red,thickness=2,numpoints=2000):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "P2:=plot([op(F2(t)),t=1/(2*P i)..1/Pi],thickness=3,color=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "display(P1,P2,scaling=constrained);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 191 "The reason Dr. Young named this curve a Fool's Cap is now visible. At its end is a very interesting \"whirly bit\" \+ centered at the origin. We would like to be able to examine it more c losely." }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 13 "Magnification" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "Now for z > 1, " }{XPPEDIT 18 0 "-1/pi;" "6#,$*&\"\"\"F% %#piG!\"\"F'" }{TEXT -1 3 " < " }{XPPEDIT 18 0 "-1/(z*pi);" "6#,$*&\" \"\"F%*&%\"zGF%%#piGF%!\"\"F)" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "1/ (2*pi*z);" "6#*&\"\"\"F$*(\"\"#F$%#piGF$%\"zGF$!\"\"" }{TEXT -1 3 " < \+ " }{XPPEDIT 18 0 "1/(2*pi);" "6#*&\"\"\"F$*&\"\"#F$%#piGF$!\"\"" } {TEXT -1 41 " , so reducing the plotting interval to [" }{XPPEDIT 18 0 "-1/(z*pi);" "6#,$*&\"\"\"F%*&%\"zGF%%#piGF%!\"\"F)" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "1/(2*pi*z);" "6#*&\"\"\"F$*(\"\"#F$%#piGF$%\"zGF$! \"\"" }{TEXT -1 75 " ] will produce the desired effect. We exhibit th is via the definition of " }{TEXT 260 5 "pic, " }{TEXT -1 23 "below, a nd its display." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 200 "To zoom in on the \"whirlpool,\" in the display command, below, use successively i ncreasing, odd, positive integers as the argument of \"pic.\" Begin w ith 1, as indicated, and execute the line with the " }{TEXT 258 7 "dis play" }{TEXT -1 55 " command. Now change the 1 to, say, 5 and execute the " }{TEXT 259 7 "display" }{TEXT -1 59 " line, again. Continue in this manner as far as desired. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "pic:=z->plot([op(F1(t)),t=-1/(z*Pi)..1/(z*2*Pi)],colo r=red,thickness=2,numpoints=2000):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "display(pic(5),scaling=constrained);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "Notice the effect of the factor, z. The \+ t-interval is now " }{XPPEDIT 18 0 "[-1/(z*Pi), 1/(2*z*Pi)];" "6#7$,$* &\"\"\"F&*&%\"zGF&%#PiGF&!\"\"F**&F&F&*(\"\"#F&F(F&F)F&F*" }{TEXT -1 577 " , so that the larger z is, the shorter the interval is. Because it always contains 0, we still travel into the whirlpool and then out . (Of course, analytically, we must exclude t = 0, but in drawing the graph, Maple doesn't actually use t = 0, so we will ignore this point .) When z = 1, we come out \"half a turn\" less than we went in, but for larger values of z this is no longer true. Before we can color t he region between the \"in\" and \"out\" curves, we must find a way to guarantee that, for each (odd) value of z, we always come out \"half \+ a turn\" less than we went in." }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 61 "Trying to color the region between the \"in\" and \"out\" curves." }} }{EXCHG {PARA 4 "" 0 "" {TEXT -1 20 "The \"In\" Curve - Red" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "pic1:=z->plot([op(F1(t/z)),t =-1/Pi..0],color=red,thickness=2,numpoints=2000):#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "display(pic1(1),scaling=constrained);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 22 "The \"Out\" Curve - Blue" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "We want to travel outward on the t-interval ( 0, " } {XPPEDIT 18 0 "1/(2*z*Pi)" "6#*&\"\"\"F$*(\"\"#F$%\"zGF$%#PiGF$!\"\"" }{TEXT -1 20 " ] ; that is, (0, " }{XPPEDIT 18 0 "1/(z*pi);" "6#*&\" \"\"F$*&%\"zGF$%#piGF$!\"\"" }{TEXT -1 5 " ] , " }{TEXT 261 19 "less \+ \"half a turn.\"" }{TEXT -1 18 " Again, F1(t) = " }{XPPEDIT 18 0 "t^ 3;" "6#*$%\"tG\"\"$" }{TEXT -1 3 " < " }{XPPEDIT 18 0 "cos(1/abs(t)); " "6#-%$cosG6#*&\"\"\"F'-%$absG6#%\"tG!\"\"" }{TEXT -1 3 " , " } {XPPEDIT 18 0 "sin(1/abs(t));" "6#-%$sinG6#*&\"\"\"F'-%$absG6#%\"tG!\" \"" }{TEXT -1 27 " > , but as t > 0, we have " }{XPPEDIT 18 0 "abs(t); " "6#-%$absG6#%\"tG" }{TEXT -1 17 " = t. If we let " }{XPPEDIT 18 0 " theta = 1/t;" "6#/%&thetaG*&\"\"\"F&%\"tG!\"\"" }{TEXT -1 24 " , then, for t in (0, " }{XPPEDIT 18 0 "1/pi;" "6#*&\"\"\"F$%#piG!\"\"" } {TEXT -1 14 " ] , we have " }{XPPEDIT 18 0 "F1(1/theta);" "6#-%#F1G6# *&\"\"\"F'%&thetaG!\"\"" }{TEXT -1 2 " =" }{XPPEDIT 18 0 "(1/theta)^3; " "6#*$*&\"\"\"F%%&thetaG!\"\"\"\"$" }{TEXT -1 7 " < cos(" }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 7 "), sin(" }{XPPEDIT 18 0 "theta ;" "6#%&thetaG" }{TEXT -1 9 ") > for " }{XPPEDIT 18 0 "theta" "6#%&th etaG" }{TEXT -1 5 " in " }{XPPEDIT 18 0 "[pi, infinity];" "6#7$%#piG% )infinityG" }{TEXT -1 23 " . Note that because " }{XPPEDIT 18 0 "d*t heta/dt;" "6#*(%\"dG\"\"\"%&thetaGF%%#dtG!\"\"" }{TEXT -1 4 " = " } {XPPEDIT 18 0 "-1/(t^2);" "6#,$*&\"\"\"F%*$%\"tG\"\"#!\"\"F)" }{TEXT -1 8 " , then " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 18 " is \+ decreasing on " }{XPPEDIT 18 0 "[pi, infinity];" "6#7$%#piG%)infinityG " }{TEXT -1 14 ". If we want " }{XPPEDIT 18 0 "theta" "6#%&thetaG" } {TEXT -1 21 " to decrease, not to " }{XPPEDIT 18 0 "pi;" "6#%#piG" } {TEXT -1 28 ", but to half a turn before " }{XPPEDIT 18 0 "pi;" "6#%#p iG" }{TEXT -1 20 ", we should define " }{XPPEDIT 18 0 "phi = theta+pi ;" "6#/%$phiG,&%&thetaG\"\"\"%#piGF'" }{TEXT -1 8 " . Then " } {XPPEDIT 18 0 "phi;" "6#%$phiG" }{TEXT -1 7 " is in " }{XPPEDIT 18 0 " [2*pi, infinity];" "6#7$*&\"\"#\"\"\"%#piGF&%)infinityG" }{TEXT -1 19 " and decreasing to " }{XPPEDIT 18 0 "2*pi;" "6#*&\"\"#\"\"\"%#piGF%" }{TEXT -1 8 ". So " }{XPPEDIT 18 0 "F1(1/phi);" "6#-%#F1G6#*&\"\"\" F'%$phiG!\"\"" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "(1/phi)^3;" "6#*$*&\" \"\"F%%$phiG!\"\"\"\"$" }{TEXT -1 6 "< cos(" }{XPPEDIT 18 0 "phi;" "6# %$phiG" }{TEXT -1 7 "), sin(" }{XPPEDIT 18 0 "phi;" "6#%$phiG" }{TEXT -1 11 ") > , for " }{XPPEDIT 18 0 "phi;" "6#%$phiG" }{TEXT -1 4 " in \+ " }{XPPEDIT 18 0 "[2*pi, infinity];" "6#7$*&\"\"#\"\"\"%#piGF&%)infini tyG" }{TEXT -1 56 " and decreasing, describes the \"out\" curve. To s top at " }{XPPEDIT 18 0 "2*z*pi;" "6#*(\"\"#\"\"\"%\"zGF%%#piGF%" } {TEXT -1 35 " instead, we only need to redefine " }{XPPEDIT 18 0 "thet a" "6#%&thetaG" }{TEXT -1 5 " as " }{XPPEDIT 18 0 "z/t;" "6#*&%\"zG\" \"\"%\"tG!\"\"" }{TEXT -1 13 " , then use " }{XPPEDIT 18 0 "phi = the ta+pi;" "6#/%$phiG,&%&thetaG\"\"\"%#piGF'" }{TEXT -1 16 " , again. Th us " }{XPPEDIT 18 0 "phi;" "6#%$phiG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "z/t+pi;" "6#,&*&%\"zG\"\"\"%\"tG!\"\"F&%#piGF&" }{TEXT -1 3 " = " } {XPPEDIT 18 0 "(z+t*pi)/t;" "6#*&,&%\"zG\"\"\"*&%\"tGF&%#piGF&F&F&F(! \"\"" }{TEXT -1 68 " , and, as a function of t, the \"magnified\" ou t curve is given by " }{XPPEDIT 18 0 "F1(t/(t*pi+z));" "6#-%#F1G6#*&% \"tG\"\"\",&*&F'F(%#piGF(F(%\"zGF(!\"\"" }{TEXT -1 14 " for t in (0, \+ " }{XPPEDIT 18 0 "1/pi;" "6#*&\"\"\"F$%#piG!\"\"" }{TEXT -1 4 " ] ." } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 135 "To see how this works, execute \+ the following two lines for z = 1, 3, 5, ... . For comparison, we inc lude the corresponding \"in\" curve." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "pic1b:=z->plot([op(F1(t/(t*Pi+z))),t=0..1/Pi],color=b lue,thickness=2,numpoints=2000):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "display(pic1(1),pic1b(1),scaling=constrained);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 262 27 "Coloring Between the Curves" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "If we add to this picture a radial line, \+ say, " }{TEXT 265 1 "r" }{TEXT -1 21 "(t) = < t, t/2 > for " } {XPPEDIT 18 0 "0 <= t;" "6#1\"\"!%\"tG" }{TEXT -1 73 " , we will see h ow to color between the curves, at least after a fashion." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "r:=plot([t,t/2,t=0..0.021],color=bl ack):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "display(r,pic1(1), pic1b(1),scaling=constrained);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "The idea is to draw a thick, colored line along " }{TEXT 266 1 "r" } {TEXT -1 322 "(t) from the blue curve to the red one and to do this fo r a large number of closely spaced values of t. The result should at \+ least give the illusion that the area is colored. To this end, we nee d to be able to trace the \"out\" curve in the opposite direction. Th is can be done by, first, letting t be in the interval [ " }{XPPEDIT 18 0 "-1/pi;" "6#,$*&\"\"\"F%%#piG!\"\"F'" }{TEXT -1 19 " , 0 ) and, a s now " }{XPPEDIT 18 0 "abs(t) = -t;" "6#/-%$absG6#%\"tG,$F'!\"\"" } {TEXT -1 12 ", defining " }{XPPEDIT 18 0 "phi;" "6#%$phiG" }{TEXT -1 5 " via " }{XPPEDIT 18 0 "phi = z/(-t)+pi;" "6#/%$phiG,&*&%\"zG\"\"\", $%\"tG!\"\"F+F(%#piGF(" }{TEXT -1 8 ". Then " }{XPPEDIT 18 0 "1/phi = t/(Pi*t-z);" "6#/*&\"\"\"F%%$phiG!\"\"*&%\"tGF%,&*&%#PiGF%F)F%F%%\"zG F'F'" }{TEXT -1 40 " and the reversed out-curve is given by " } {XPPEDIT 18 0 "F1(t/(t*pi-z));" "6#-%#F1G6#*&%\"tG\"\"\",&*&F'F(%#piGF (F(%\"zG!\"\"F-" }{TEXT -1 13 " for t in [ " }{XPPEDIT 18 0 "-1/pi;" "6#,$*&\"\"\"F%%#piG!\"\"F'" }{TEXT -1 116 " , 0 ). To check up on t his, we can redraw our last picture with the out-curve replaced by our reversed out-curve." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "pic 1c:=z->plot([op(F1(t/(t*Pi-z))),t=-1/Pi..0],color=blue,thickness=2,num points=2000):#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "display(r ,pic1(1),pic1c(1),scaling=constrained);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "We may draw our colored line segment via" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "LS:=(z,t)->line(F1(t/z),F1(t/(t*Pi- z)),color=green,thickness=3):#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "Notice that " }{XPPEDIT 18 0 "d*phi/dt = z/(t^2);" "6#/*(%\"dG\"\"\"% $phiGF&%#dtG!\"\"*&%\"zGF&*$%\"tG\"\"#F)" }{TEXT -1 14 " > 0, so that \+ " }{XPPEDIT 18 0 "phi;" "6#%$phiG" }{TEXT -1 67 " is increasing with t on our reversed out-curve. On the in-curve, " }{XPPEDIT 18 0 "theta \+ = z/abs(t);" "6#/%&thetaG*&%\"zG\"\"\"-%$absG6#%\"tG!\"\"" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "z/(-t);" "6#*&%\"zG\"\"\",$%\"tG!\"\"F(" } {TEXT -1 19 " since t is in [ " }{XPPEDIT 18 0 "-1/pi;" "6#,$*&\"\" \"F%%#piG!\"\"F'" }{TEXT -1 13 " , 0 ) . So " }{XPPEDIT 18 0 "d*phi/d t = d*theta/dt;" "6#/*(%\"dG\"\"\"%$phiGF&%#dtG!\"\"*(F%F&%&thetaGF&F( F)" }{TEXT -1 148 " and our radial line from the origin through the t wo curves will always cut them for the same value of t. This allows u s to proceed as we planned." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 213 "W e now define a sequence of n values for t, so that we can change n if \+ we like. Because the distance of either curve from the origin decreas es so rapidly as t approaches 0, it suffices to select values of t fro m " }{XPPEDIT 18 0 "[-1/pi, -1/(5*pi)];" "6#7$,$*&\"\"\"F&%#piG!\"\"F( ,$*&F&F&*&\"\"&F&F'F&F(F(" }{TEXT -1 102 " . We then fill the region \+ between the two unmagnified curves to see that our procedure really wo rks." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "S:=n->seq(-1/Pi+4*k /(5*n*Pi)+0.0001,k=0..n):#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "fill:=seq(LS(1,s),s=S(300)):#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 263 18 "The Original Curve" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "display(P2,pic1c(1),pic1(1),fill,scaling=constrained, axes=none);#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 96 "Here we have chos en to use 300 lines (see S(300), in \"fill\"). The result is quite sa tisfactory." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 264 10 "Zo oming In" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 171 "To fill the region b etween two magnified curves with color, we use our \"fill\" sequece to define a \"Fill\" function depending on the \"magnification\", m; i.e ., the \"z-value.\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "Fill :=m->seq(LS(m,s),s=S(300)):#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 133 " Now choose a \"magnification\", m, and display the result. To see tha t we really are \"zooming in\", a coordinate system has been added." } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "m:=3;#Use m=1 to 5." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "display(pic1c(2*m-1),pic1(2* m-1),Fill(2*m-1),scaling=constrained,axes=normal);#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 326 "If this is enlarged, we can easily see the green lines. In \"Fill\", changing the number of \+ lines from 300 to 600 will almost, but not quite, restore the illusion that the region has been fully colored. For larger magnifications, t he number of lines must be increased even further. The reason is that for a magnification, z, " }{XPPEDIT 18 0 "phi;" "6#%$phiG" }{TEXT -1 25 " increases by the amount " }{XPPEDIT 18 0 "4*z*pi;" "6#*(\"\"%\"\" \"%\"zGF%%#piGF%" }{TEXT -1 266 " or 2z revolutions, so that the color ed lines are spread out over a larger number of revolutions. The incr eased time of computation makes increasing the number of lines less an d less acceptable. However, on the whole, the coloring scheme is reaso nably satisfactory." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 3 "" 0 "" {TEXT -1 20 "A Different Approach" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 391 "I was not fully satisfied with the previ ous method. It occured to me that Maple could color surfaces automati cally as part of its \"plot3d\" command, so I ought to be able to colo r between curves in the x,y-plane by describing the area as a flat, 3- dimensional surface confined to that plane. To see most easily how th is should be done, I returned to my original example, drew the curves \+ as " }{TEXT 269 11 "spacecurves" }{TEXT -1 44 " and the \"fill\" as a \+ surface, colored green." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 " restart:with(plots):with(plottools):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "T:=3;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "U: =spacecurve([x,2*sqrt(2*x),0],x=0..T,color=red,thickness=3):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "L:=spacecurve([x,x^2,0],x=0. .T,color=blue,thickness=3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "display(U,L,scaling=constrained,color=green,style=patchnogrid,orie ntation=[-90,0],axes=normal);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "F:=plot3d([x,y,0],x=0..T,y=x^2..2*sqrt(2*x)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "display(U,L,F,scaling=constrained,color=g reen,style=patchnogrid,orientation=[-90,0],axes=normal);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "To apply this idea to the Fool's Cap curv e, let us define it again." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "restart:with(plots):with(plottools):with(linalg):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "g:=t->[cos(1/abs(t)),sin(1/abs(t))] ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "F1:=t->convert(evalm(t ^3*g(t)),list):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "Then" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "F1(t);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 44 "F2:=t->convert(evalm(g(1/(2*Pi))*t^3),list):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "F2(t);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "We define the \"upper\" and \"lower\" curves, as \+ well as the \"linear bit\", as spacecurves." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "Up:=z->spacecurve([op(F1(t/z)),0],t=-1/Pi..0,color =red,thickness=3,numpoints=2000):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "Lw:=z->spacecurve([op(F1(t/(t*Pi-z))),0],t=-1/Pi..0,c olor=blue,thickness=3,numpoints=2000):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "C:=spacecurve([op(F2(t)),0],t=1/(2*Pi)..1/Pi,color=bl ue,thickness=3):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "We display th ese, using the " }{TEXT 267 4 "view" }{TEXT -1 54 " option to ensure t he same scaling in both directions." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "display(Up(1),Lw(1),C,axes=none,orientation=[-90,0], view=[-0.005..0.033,-0.005..0.033,0..0.0001],axes=normal);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 204 "The definition of the Fool's Cap curve m akes the use of cylindrical (polar) coordinates especially convenient. The surface bounded by these curves, for two and one half \"revoluti ons\", may be described as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 155 "G:=z->plot3d([r,1/abs(t)+Pi,0],r=abs((t/(t*Pi-1))^3)..abs(t^3),t= -1/(z*Pi)..-1/((z+5)*Pi),coords=cylindrical,color=green,style=patchnog rid,numpoints=50^2):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "Using thi s to color between the curves yields" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 115 "display(Up(1),Lw(1),C,G(1),axes=none,orientation=[-9 0,0],view=[-0.005..0.033,-0.005..0.033,0..0.0001],axes=normal);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 111 "But is this really better than be fore? To see, we examine the \"whirlpool\", again with a \"magnificat ion\" m = 3." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "m:=3;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 127 "display(Up(2*m-1),Lw(2*m-1) ,G(2*m-1),orientation=[-90,0],view=[-2.6e-04..2.6e-04,-2.6e-04..2.6e-0 4,-1e-06..1e-06],axes=normal);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "Can we actually color all the way to the origin? To see, we let " } {TEXT 270 1 "t" }{TEXT -1 10 " run from " }{XPPEDIT 18 0 "-1/(z*Pi);" "6#,$*&\"\"\"F%*&%\"zGF%%#PiGF%!\"\"F)" }{TEXT -1 6 " to 0." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 144 "G1:=z->plot3d([r,1/abs(t)+P i,0],r=abs((t/(t*Pi-1))^3)..abs(t^3),t=-1/(z*Pi)..0,coords=cylindrical ,color=green,style=patchnogrid,numpoints=14^4):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 128 "display(Up(2*m-1),Lw(2*m-1),G1(2*m-1),orienta tion=[-90,0],view=[-2.6e-04..2.6e-04,-2.6e-04..2.6e-04,-1e-06..1e-06], axes=normal);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "Using the " } {TEXT 268 4 "view" }{TEXT -1 228 " option to force suitable 1:1 scalin g, we may, at last, begin with the full Fool's Cap, then zoom in on th e origin several times, correctly \"coloring inside the lines\" each t ime . The results seem to me completely satisfactory." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 274 8 "Remark: " }{TEXT -1 295 "A t this point, if one is using the \"new\" interface for Maple 9.5, ins tead of the \"classic interface,\" one can click on the picture, then \+ use the \"Transformation \" button on the contextbar to choose \"scale \", then \"zoom in\" on the origin by moving the mouse pointer appropr iately on the picture. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 245 "W:=[[-0.005..0.033,-0.005..0.033,0..0.0001],[-1.2e-03..1.2e-03,-1 .2e-03..1.2e-03,-1e-06..1e-06],[-2.6e-04..2.6e-04,-2.6e-04..2.6e-04,-1 e-06..1e-06],[-1e-04..1e-04,-1e-04..1e-04,-1e-06..1e-06],[-4.5e-05..4. 5e-05,-4.5e-05..4.5e-05,-1e-06..1e-06]]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "V:=z->op(z,W):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Choose m = 1, 2, ... , 5." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "m:=1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "display(Up(2* m-1),Lw(2*m-1),G1(2*m-1),orientation=[-90,0],axes=normal,view=V(m));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 4 "" 0 " " {TEXT -1 9 "Exercises" }{TEXT 272 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 445 "As we said before, Dr. Young invented the Fool's Cap cur ve, in part, to show that a piecewise continuous, simple closed curve, and the region it contains, may be more complicated than those normal ly used as textbook illustrations. As an exercise, he asked his stude nts to find the arclength of the Fool's Cap and the area of the region it contains. You might like to try this, too. The answers, by hand, \+ are (not forgetting the \"linear bit\") " }}{PARA 0 "" 0 "" {TEXT 271 2 "L " }{TEXT -1 2 "= " }{XPPEDIT 18 0 "(1/27)(-2+(9/(Pi^2)+1)^(3/ 2)+(9/(4*Pi^2)+1)^(3/2))+7/((2*Pi)^3);" "6#,&-*&\"\"\"F&\"#F!\"\"6#,( \"\"#F(),&*&\"\"*F&*$%#PiGF+F(F&F&F&*&\"\"$F&F+F(F&),&*&F/F&*&\"\"%F&* $F1F+F&F(F&F&F&*&F3F&F+F(F&F&*&\"\"(F&*$*&F+F&F1F&F3F(F&" }{TEXT -1 6 " and " }{TEXT 273 1 "A" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "31/(320*Pi ^5);" "6#*&\"#J\"\"\"*&\"$?$F%*$%#PiG\"\"&F%!\"\"" }{TEXT -1 64 " . \+ You might also like to use Maple to answer these questions." }}}} {MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }