{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 "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 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 "Highlight" -1 256 "" 0 0 0 255 0 1 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" 256 257 "" 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 256 258 "" 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 256 259 "" 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 256 260 "" 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 256 261 "" 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" 256 262 "" 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }3 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Head ing 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }1 0 0 -1 8 2 0 0 0 0 0 0 -1 0 }{PSTYLE "Dash Item" 0 16 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 3 3 0 0 0 0 0 0 16 3 } } {SECT 0 {PARA 3 "" 0 "" {TEXT -1 17 "Parametric Curves" }}{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" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 47 "An Animation of How a Parametric Curve is Drawn" }}{PARA 0 "" 0 "" {TEXT -1 27 "We will illustrate how the " } {HYPERLNK 17 "animate" 2 "animate" "" }{TEXT -1 117 " command in Maple works, and use it to generate a movie which shows how a parametric cu rve is drawn as the parameter " }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 24 " ranges over its domain." }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "Here is a parametric curve defined by the following two equations:" } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "x := t -> cos(t) + cos(7*t)/2 + si n(17*t)/3;\ny := t -> sin(t) + sin(7*t)/2 + cos(17*t)/3;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "We can display the entire curve using the " }{HYPERLNK 17 "plot" 2 "plot" "" }{TEXT -1 77 " command, where the \+ two parametric equations and the range for the parameter " }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 33 " are enclosed in square brackets:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "plot([x(t), y(t), t=0..2*Pi]);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "In order to create a movie which s hows how the curve is drawn we use " }{HYPERLNK 17 "animate" 2 "animat e" "" }{TEXT -1 17 " (located in the " }{HYPERLNK 17 "plots" 2 "plots " "" }{TEXT -1 61 " package, which we have loaded at the top of this w orksheet)." }}{PARA 0 "" 0 "" {TEXT -1 59 "Note that we need to provid e various pieces of information:" }}{PARA 16 "" 0 "" {TEXT -1 60 "the \+ object to be animated (in this case a parametric curve)," }}{PARA 16 " " 0 "" {TEXT -1 18 "a frame parameter," }}{PARA 16 "" 0 "" {TEXT -1 21 "the number of frames." }}{PARA 0 "" 0 "" {TEXT -1 57 "In our examp le below we have also increased the value of " }{HYPERLNK 17 "numpoint s" 2 "plot,options" "" }{TEXT -1 58 " above the default in order to ge t a smooth looking curve." }}{PARA 0 "" 0 "" {TEXT -1 125 "In a sense, the animation process itself can be viewed as a \"parametric operatio n\" - the parameter being the frame parameter " }{XPPEDIT 18 0 "d" "6# %\"dG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 141 "In our example we create a cartoon consisting of 30 frames, each of which shows an i ncreasingly larger piece of the total curve. (The use of " }{XPPEDIT 18 0 "d=0..1" "6#/%\"dG;\"\"!\"\"\"" }{TEXT -1 76 " accomplishes exact ly this - start with nothing, end with the entire curve.)" }}{PARA 0 " " 0 "" {TEXT -1 105 "Note: To play the movie, click on the plot and us e the VCR controls that appear at the top of the screen." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "animate( [x(d*t), y(d*t), t=0..2*Pi], d=0..1, frames=30, numpoints=200);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "We now offer an alternative approach to creating the same movie. Instead of " }{HYPERLNK 17 "animate" 2 "animate" "" }{TEXT -1 9 ", we use " } {HYPERLNK 17 "display" 2 "plots,display" "" }{TEXT -1 17 " with the op tion " }{HYPERLNK 17 "insequence=true" 2 "plots,display" "" }{TEXT -1 84 ". To mimic the behavior of animate we would have to do (here the o perator || is the " }{HYPERLNK 17 "concatenation" 2 "concatenate" "" } {TEXT -1 69 " operator in Maple, i.e., variables p1, p2, p3,... are cr eated below)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 165 "frames:=30:\nfor i from 1 to frames do\n d := (i-1)/(frames-1);\n p||i := plot([x(d* t), y(d*t), t=0..2*Pi]):\nod:\ndisplay([seq(p||i, i=1..frames)], inseq uence=true);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "Note that we can \+ modify our commands slightly, giving a more natural role to " } {XPPEDIT 18 0 "d" "6#%\"dG" }{TEXT -1 53 " as a multiplier for the upp er bound of the range of " }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 10 " . This is " }{TEXT 257 3 "not" }{TEXT -1 33 " possible with animate (t ry it!)." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 163 "frames:=30:\nfor i fro m 1 to frames do\n d := (i-1)/(frames-1);\n p||i := plot([x(t), y( t), t=0..d*2*Pi]):\nod:\ndisplay([seq(p||i, i=1..frames)], insequence= true);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "Another advantage of us ing " }{HYPERLNK 17 "display" 2 "plots,display" "" }{TEXT -1 12 " inst ead of " }{HYPERLNK 17 "animate" 2 "animate" "" }{TEXT -1 87 " is that we have control over the individual frames of the movie. We can, e.g. , use an " }{HYPERLNK 17 "array" 2 "array" "" }{TEXT -1 59 " to displa y the first few frames of the movie side-by-side." }}{PARA 0 "" 0 "" {TEXT -1 136 "Note: The key ingredient here is the use of the array co ntaining the first few frames, the other arguments serve only cosmetic purposes." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "P:=array(1..4, [seq(p ||i, i=1..4)]):\ndisplay(P, axes=framed, scaling=constrained, tickmark s=[2,2]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "If we are only inter ested in " }{TEXT 258 3 "all" }{TEXT -1 73 " frames of the movie displ ayed side-by-side, we can accomplish this with " }{HYPERLNK 17 "animat e" 2 "animate" "" }{TEXT -1 23 " in the following way (" }{HYPERLNK 17 "insequence=true" 2 "plots,display" "" }{TEXT -1 6 " with " } {HYPERLNK 17 "display" 2 "plots,display" "" }{TEXT -1 28 " will again \+ show the movie)." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 135 "d:='d': frames :='frames':\nmovie := animate( [x(d*t), y(d*t), t=0..2*Pi], d=0..1, f rames=30, numpoints=200, axes=none):\ndisplay(movie);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 " Cycloids" }}{PARA 0 "" 0 "" {TEXT -1 39 "Next we illustrate the generation of a " }{TEXT 259 7 "cycloid" }{TEXT -1 112 ". \nA cycloid is obtained by tracking the position of a fixed point on the circumfe rence of a circle with radius " }{XPPEDIT 18 0 "r" "6#%\"rG" }{TEXT -1 32 " rolling along a straight line. " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "The parametric equations (which we will derive in class l ater this week) are" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "x := t -> r* (t-sin(t));\ny := t -> r*(1-cos(t));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "We will generate the picture in several steps." }}{PARA 0 "" 0 "" {TEXT -1 13 "First we use " }{HYPERLNK 17 "animate" 2 "animate" "" }{TEXT -1 48 " to create an animation of a circle with radius " } {XPPEDIT 18 0 "r" "6#%\"rG" }{TEXT -1 18 " moving along the " } {XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 16 "-axis from 0 to " }{XPPEDIT 18 0 "4*r*Pi;" "6#*(\"\"%\"\"\"%\"rGF%%#PiGF%" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 11 "(we choose " }{XPPEDIT 18 0 "4*r*Pi;" "6# *(\"\"%\"\"\"%\"rGF%%#PiGF%" }{TEXT -1 40 " to see two full arches of \+ the cycloid)." }}{PARA 0 "" 0 "" {TEXT -1 76 "To have a specific value for the radius r of the circle for the plot we set " }{XPPEDIT 18 0 " r=1" "6#/%\"rG\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "r:=1:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "In this example we \+ will let the frame parameter " }{XPPEDIT 18 0 "d" "6#%\"dG" }{TEXT -1 101 " play a different role: It will simulate the movement of the circ le from left to right, i.e., we let " }{XPPEDIT 18 0 "d" "6#%\"dG" } {TEXT -1 17 " range from 0 to " }{XPPEDIT 18 0 "4*r*Pi" "6#*(\"\"%\"\" \"%\"rGF%%#PiGF%" }{TEXT -1 26 " and use it to change the " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 126 "-coordinate of the circle. So, in fa ct, the circle does not roll - it just is shifted to the right from on e frame to the next." }}{PARA 0 "" 0 "" {TEXT -1 43 "Note that the bas ic circle is described by " }{XPPEDIT 18 0 "x(t) = -r*sin(t)" "6#/-%\" xG6#%\"tG,$*&%\"rG\"\"\"-%$sinG6#F'F+!\"\"" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "y(t)=-r*cos(t)+r" "6#/-%\"yG6#%\"tG,&*&%\"rG\"\"\"-%$co sG6#F'F+!\"\"F*F+" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "t" "6#%\"tG" } {TEXT -1 5 " in [" }{XPPEDIT 18 0 "0,2*Pi" "6$\"\"!*&\"\"#\"\"\"%#PiGF &" }{TEXT -1 53 "], since this is not a circle centered at the origin. " }}{PARA 0 "" 0 "" {TEXT -1 72 "Here is the moving circle (we've assi gned the animation to the variable " }{XPPEDIT 18 0 "a1" "6#%#a1G" } {TEXT -1 51 " so we can use it again in the more complex movie):" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 187 "a1 := animate( [-r*sin(t) + d, -r* cos(t) + r, t=0..2*Pi], d=0..4*r*Pi, frames=30, color=magenta, scaling =constrained, axes=frame, view=[-Pi/2..9*Pi/2,0..2]):\ndisplay(a1, ins equence=true);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 155 "Next we create an animation of the moving point on the circumference of the circle b y creating a little blue tickmark which rotates along with the circle. " }}{PARA 0 "" 0 "" {TEXT -1 24 "We save this part in a2." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 157 "a2 := animate( [-r*t*sin(d) + d, -r*t*cos( d)+r, t=0.9..1.1], d=0..4*r*Pi, frames=30, color=blue, scaling=constra ined, axes=frame, view=[-Pi/2..9*Pi/2,0..2]):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "Finally, we create the animation for the cycloid curve , and save it in a3." }}{PARA 0 "" 0 "" {TEXT -1 90 "This time we use \+ the frame parameter again to simulate drawing of the curve, i.e., we l et " }{XPPEDIT 18 0 "d" "6#%\"dG" }{TEXT -1 52 " range from 0 to 1 and multiply the curve parameter " }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 4 " by " }{XPPEDIT 18 0 "d" "6#%\"dG" }{TEXT -1 1 "." }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 139 "a3 := animate( [x(d*t), y(d*t), t=0..4*r*Pi] , d=0..1, frames=30, color=green, scaling=constrained, axes=frame, vie w=[-Pi/2..9*Pi/2,0..2]):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 138 "Now \+ we are ready to display all three animations (the rolling circle, the \+ tickmark, and the curve) together. \nNote: Now that we are using " } {HYPERLNK 17 "display" 2 "plots,display" "" }{TEXT -1 76 " to display \+ three animations together, it is no longer necessary to use the " } {HYPERLNK 17 "insequence=true" 2 "plots,display" "" }{TEXT -1 8 " opti on." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "display(\{a1,a2,a3\});" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 12 "Assignment 6" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Ex. 1:" }}{PARA 0 "" 0 "" {TEXT -1 51 "a) Create a plot of the following p arametric curve:" }}{PARA 0 "" 0 "" {TEXT -1 15 " " } {XPPEDIT 18 0 "x(t) = 2*cos(t) + cos(2*t)" "6#/-%\"xG6#%\"tG,&*&\"\"# \"\"\"-%$cosG6#F'F+F+-F-6#*&F*F+F'F+F+" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 15 " " }{XPPEDIT 18 0 "y(t) = 2*sin(t) - s in(2*t)" "6#/-%\"yG6#%\"tG,&*&\"\"#\"\"\"-%$sinG6#F'F+F+-F-6#*&F*F+F'F +!\"\"" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 15 " \+ " }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 4 " in " }{XPPEDIT 18 0 "[0,2 *Pi]" "6#7$\"\"!*&\"\"#\"\"\"%#PiGF'" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 67 "b) What happens if you replace 2 by -2 in the parametri c equations?" }}{PARA 0 "" 0 "" {TEXT -1 38 "c) What happens if you re place 2 by 3?" }}{PARA 0 "" 0 "" {TEXT -1 55 "d) What happens as you i ncrease the value even further?" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Ex.2:" }}{PARA 0 "" 0 "" {TEXT -1 41 "What is the difference betwee n the curves" }}{PARA 0 "" 0 "" {TEXT -1 14 " " } {XPPEDIT 18 0 "x(t) = 6*cos(t) + 5*cos(3*t)" "6#/-%\"xG6#%\"tG,&*&\"\" '\"\"\"-%$cosG6#F'F+F+*&\"\"&F+-F-6#*&\"\"$F+F'F+F+F+" }{TEXT -1 1 ", " }}{PARA 0 "" 0 "" {TEXT -1 14 " " }{XPPEDIT 18 0 "y(t) \+ = 6*sin(t)-5*sin(3*t)" "6#/-%\"yG6#%\"tG,&*&\"\"'\"\"\"-%$sinG6#F'F+F+ *&\"\"&F+-F-6#*&\"\"$F+F'F+F+!\"\"" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 14 " " }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 4 " in " }{XPPEDIT 18 0 "[0,2*Pi]" "6#7$\"\"!*&\"\"#\"\"\"%#PiGF'" } {TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 4 "and " }}{PARA 0 "" 0 "" {TEXT -1 14 " " }{XPPEDIT 18 0 "x(t) = 6*cos(2*t)+5*cos(6 *t)" "6#/-%\"xG6#%\"tG,&*&\"\"'\"\"\"-%$cosG6#*&\"\"#F+F'F+F+F+*&\"\"& F+-F-6#*&F*F+F'F+F+F+" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 14 " " }{XPPEDIT 18 0 "y(t) = 6*sin(2*t)-5*sin(6*t)" "6#/-%\" yG6#%\"tG,&*&\"\"'\"\"\"-%$sinG6#*&\"\"#F+F'F+F+F+*&\"\"&F+-F-6#*&F*F+ F'F+F+!\"\"" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 14 " \+ " }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 4 " in " }{XPPEDIT 18 0 " [0,Pi]" "6#7$\"\"!%#PiG" }{TEXT -1 1 "?" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Ex.3:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 260 16 "Liss ajous curves" }{TEXT -1 98 " are the type of curves that appear on osc illoscopes in physics or electronics. They are given by " }}{PARA 0 " " 0 "" {TEXT -1 11 " " }{XPPEDIT 18 0 "x(t) = cos(a*t)" "6#/ -%\"xG6#%\"tG-%$cosG6#*&%\"aG\"\"\"F'F-" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 11 " " }{XPPEDIT 18 0 "y(t) = sin(b*t)" "6#/-% \"yG6#%\"tG-%$sinG6#*&%\"bG\"\"\"F'F-" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 17 "a) Plot the case " }{XPPEDIT 18 0 "a=3, b=5" "6$/%\"aG \"\"$/%\"bG\"\"&" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "t" "6#%\"tG" } {TEXT -1 4 " in " }{XPPEDIT 18 0 "[0,2*Pi]" "6#7$\"\"!*&\"\"#\"\"\"%#P iGF'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 20 "b) What happens f or " }{XPPEDIT 18 0 "a=b" "6#/%\"aG%\"bG" }{TEXT -1 45 "? You will hav e to pick a specific value for " }{XPPEDIT 18 0 "a" "6#%\"aG" }{TEXT -1 80 ". Try several different ones. How do your graphs differ with th e value you pick?" }}{PARA 0 "" 0 "" {TEXT -1 69 "c) Create an animati on that shows how the curve in a) is being drawn." }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Ex.4:" }}{PARA 0 "" 0 "" {TEXT -1 3 "Let" }}{PARA 0 "" 0 "" {TEXT -1 24 " " }{XPPEDIT 18 0 "x(t) \+ = a*cos(t) - b*cos(p*t)" "6#/-%\"xG6#%\"tG,&*&%\"aG\"\"\"-%$cosG6#F'F+ F+*&%\"bGF+-F-6#*&%\"pGF+F'F+F+!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 24 " \+ " }{XPPEDIT 18 0 "y(t) = c*sin(t) -d*sin(q*t)" "6#/-%\"yG6#%\"tG,&*&%\"cG\"\"\"-%$sinG6#F'F+F+*&%\"dGF+-F-6#*&%\"qGF+ F'F+F+!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 44 "a) Plot the \"slinky curve \" corresponding to " }{XPPEDIT 18 0 "a=16, b=5, c=12, d=3, p=47/3, q= 44/3" "6(/%\"aG\"#;/%\"bG\"\"&/%\"cG\"#7/%\"dG\"\"$/%\"pG*&\"#Z\"\"\"F .!\"\"/%\"qG*&\"#WF3F.F4" }{TEXT -1 6 " with " }{XPPEDIT 18 0 "t" "6#% \"tG" }{TEXT -1 4 " in " }{XPPEDIT 18 0 "[0,12*Pi]" "6#7$\"\"!*&\"#7\" \"\"%#PiGF'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 93 "b) Play wi th the parameters to create an even prettier (more interesting) curve. Be creative!" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Ex.5:" }}{PARA 0 "" 0 "" {TEXT -1 3 "An " }{TEXT 261 10 "epicycloid" }{TEXT -1 107 " \+ is a curve obtained by following the motion of a fixed point on the ci rcumference of a circle with radius " }{XPPEDIT 18 0 "b" "6#%\"bG" } {TEXT -1 23 " which rolls along the " }{TEXT 262 7 "outside" }{TEXT 263 18 " of another circle" }{TEXT -1 13 " with radius " }{XPPEDIT 18 0 "a" "6#%\"aG" }{TEXT -1 3 " > " }{XPPEDIT 18 0 "b" "6#%\"bG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 47 "Its general parametric equati ons are given by: " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 0 "" 0 " " {TEXT -1 24 " " }{XPPEDIT 18 0 "x(t) = (a+b)* cos(t)-b*cos((a+b)*t/b);" "6#/-%\"xG6#%\"tG,&*&,&%\"aG\"\"\"%\"bGF,F,- %$cosG6#F'F,F,*&F-F,-F/6#*(,&F+F,F-F,F,F'F,F-!\"\"F,F6" }}{PARA 0 "" 0 "" {TEXT -1 40 " \n " } {XPPEDIT 18 0 "x(t) = (a+b)*sin(t)-b*sin((a+b)*t/b)" "6#/-%\"xG6#%\"tG ,&*&,&%\"aG\"\"\"%\"bGF,F,-%$sinG6#F'F,F,*&F-F,-F/6#*(,&F+F,F-F,F,F'F, F-!\"\"F,F6" }{TEXT -1 8 ",\nwhere " }{XPPEDIT 18 0 "t" "6#%\"tG" } {TEXT -1 5 " in [" }{XPPEDIT 18 0 "0,2*Pi" "6$\"\"!*&\"\"#\"\"\"%#PiGF &" }{TEXT -1 2 "]." }}{PARA 0 "" 0 "" {TEXT -1 41 "Create an animation of an epicycloid for " }{XPPEDIT 18 0 "a=4" "6#/%\"aG\"\"%" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "b=1" "6#/%\"bG\"\"\"" }{TEXT -1 1 "." }}} }}{MARK "2" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }