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}{TEXT 233 1 " " }{TEXT 231 68 "rolling at a constant \+ speed inside a larger, fixed circle of radius " }{TEXT 232 1 "a" } {TEXT 233 4 ". " }{TEXT 231 198 "As you go through the worksheet, yo u should enter the commands by placing the cursor at the end of the co mmand line (displayed in red) and hitting the return key. The first c ommand loads the Maple " }{TEXT 232 6 "plots " }{TEXT 231 42 "package. The next ones set the values of " }{TEXT 232 1 "a" }{TEXT 233 1 " " }{TEXT 231 4 "and " }{TEXT 232 3 "b, " }{TEXT 231 28 "which you will l ater change." }{TEXT 231 0 "" }}{PARA 200 "" 0 "" {TEXT 231 0 "" }}} {EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 234 13 "with(plots): " } {MPLTEXT 1 234 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 234 15 "a := 3; b := 1;" }{MPLTEXT 1 234 0 "" }}}{EXCHG {PARA 200 "" 0 "" {TEXT 231 0 "" }}{PARA 200 "" 0 "" {TEXT 231 53 "The next two commands define the parametic equations " }{TEXT 232 10 "x(t), y(t)" }{TEXT 231 81 " for the hypocycloid curve. The outer circle is centered at \+ the origin, and at " }{TEXT 232 5 "t = 0" }{TEXT 231 12 ", the point " }{TEXT 232 1 "P" }{TEXT 231 7 " is at " }{TEXT 232 7 "(a,0). " } {TEXT 231 1 "T" }{TEXT 231 71 "he center of the smaller circle rotates once around the origin every 2" }{XPPEDIT 18 0 "Pi;" "6#%#PiG" } {TEXT 231 43 " units of time. Maple uses the notation " }{TEXT 232 9 "x := t ->" }{TEXT 231 1 " " }{TEXT 232 3 "..." }{TEXT 231 12 " to \+ define " }{TEXT 232 2 "x " }{TEXT 231 17 "as a function of " }{TEXT 232 1 "t" }{TEXT 231 41 ", instead of the usual textbook notation " } {TEXT 232 12 "x(t) = ... " }{TEXT 231 0 "" }}{PARA 200 "" 0 "" {TEXT 231 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 234 40 "x := t -> (a -b)*cos(t)+b*cos((a-b)*t/b);" }{MPLTEXT 1 234 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 234 41 "y := t -> (a-b)*sin(t)- b*sin((a-b)*t /b);" }{MPLTEXT 1 234 0 "" }}}{EXCHG {PARA 200 "" 0 "" {TEXT 231 1 " " }{TEXT 231 0 "" }}{PARA 200 "" 0 "" {TEXT 231 123 "Next, we set up a \+ parametric plot of the large outer circle, using the standard parametr ic equations of a circle of radius " }{TEXT 232 1 "a" }{TEXT 231 112 " centered at the origin, as introduced in class. When you enter the t hird command, Maple will give no response." }{TEXT 231 0 "" }}{PARA 200 "" 0 "" {TEXT 231 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 234 22 "Xbig := t -> a*cos(t);" }{MPLTEXT 1 234 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 234 22 "Ybig := t -> a*sin(t);" }{MPLTEXT 1 234 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 234 58 "bigcircle := plot([Xbig(t),Ybig(t),t=0..2*Pi],color=blue):" }{MPLTEXT 1 234 0 "" } }}{EXCHG {PARA 200 "" 0 "" {TEXT 231 0 "" }}{PARA 200 "" 0 "" {TEXT 231 170 "Next we set up parametric equations for the smaller circle. S ince it is rolling around the larger circle, we give its parametric de sciption in terms of another parameter " }{TEXT 232 1 "s" }{TEXT 231 24 ", at a particular time " }{TEXT 232 6 "t = A." }{TEXT 231 0 "" }} {PARA 200 "" 0 "" {TEXT 231 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 234 41 "Xsmall := (s,A) -> b*cos(s)+(a-b)*cos(A);" } {MPLTEXT 1 234 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 234 41 "Y small := (s,A) -> b*sin(s)+(a-b)*sin(A);" }{MPLTEXT 1 234 0 "" }}} {EXCHG {PARA 200 "" 0 "" {TEXT 231 0 "" }}{PARA 200 "" 0 "" {TEXT 231 134 "Finally, for the animation it's helpful to display the line segme nt, or radius, joining the center of the smaller circle to the point " }{TEXT 232 1 "P" }{TEXT 231 112 ". We use the standard parametric fo rm of the line segment joining two points, again in terms of the para meter " }{TEXT 232 1 "s" }{TEXT 231 17 " at a fixed time " }{TEXT 232 1 "t" }{TEXT 231 1 " " }{TEXT 232 3 "= A" }{TEXT 231 39 " in the rotat ion of the smaller circle." }{TEXT 231 0 "" }}{PARA 200 "" 0 "" {TEXT 231 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 234 43 "Xrad := (s,A ) -> s*(a-b)*cos(A)+(1-s)*x(A);" }{MPLTEXT 1 234 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 234 43 "Yrad := (s,A) -> s*(a-b)*sin(A)+(1-s) *y(A);" }{MPLTEXT 1 234 0 "" }}}{EXCHG {PARA 200 "" 0 "" {TEXT 231 0 " " }}{PARA 200 "" 0 "" {TEXT 231 326 "To view the full animation, enter the following command. Once the larger circle appears, click on the \+ inside of the circle. A menu bar should appear at the top of the wind ow; by using the buttons there you can play, stop, speed up and slow d own the animation. You are now ready to answer the questions in Part \+ 1 of the lab." }{TEXT 231 0 "" }}{PARA 200 "" 0 "" {TEXT 231 0 "" }}} {EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 234 109 "animate(plot, [\{[x(t), y(t), t = 0 .. A],[Xsmall(s,A),Ysmall(s,A),s=0..2*Pi],[Xrad(s,A),Yrad (s,A),s=0..1]\}], " }{MPLTEXT 1 234 0 "" }{MPLTEXT 1 234 87 "\nA = 0 . . 2*Pi,background=bigcircle, scaling = constrained, frames = 100, axes = none);" }{MPLTEXT 1 234 0 "" }}}{PARA 202 "" 0 "" {TEXT -1 0 "" }}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }