{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 "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Blue Emphasis" -1 256 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Green Emphasis" -1 257 "Times" 1 12 0 128 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Maroon Emphasis" -1 258 "Times" 1 12 128 0 128 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Purple Emphasis" -1 259 "Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "Red Emphasis" -1 261 "Times" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Dark Red Emphasis " -1 262 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 0 1 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 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 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 O utput" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Time s" 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 } {PSTYLE "Bullet Item" -1 15 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 3 3 1 0 1 0 2 2 15 2 }{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 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "T imes" 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 {PARA 3 "" 0 "" {TEXT -1 16 "Drawing Polygons" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 28.3.2007" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 17 "P lotting polygons" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 29 "Maple has a procedure called " }{TEXT 0 11 "polygonplot" }{TEXT -1 82 " for plotting polygons. This is not a b uilt-in procedure, but it is included in a " }{TEXT 259 7 "package" } {TEXT -1 43 " of extra plotting procedure with the name " }{TEXT 0 5 " plots" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "This package is loaded by means of the command " } {TEXT 0 11 "with(plots)" }{TEXT -1 73 ", after which all the procedure s in the package, including the procedure " }{TEXT 0 11 "polygonplot" }{TEXT -1 95 ", are available for use. Alternatively, the procedure ca n be accessed directly via the command " }{TEXT 0 18 "plots[polygonplo t]" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 84 "We illustrate this procedure by drawing the unit square with vertices at the points " }{XPPEDIT 18 0 "``(0,0)" "6#-%!G6$\"\"! F&" }{TEXT -1 1 "," }{XPPEDIT 18 0 "``(0,1)" "6#-%!G6$\"\"!\"\"\"" } {XPPEDIT 18 0 "``(1,1)" "6#-%!G6$\"\"\"F&" }{TEXT -1 1 "," }{XPPEDIT 18 0 "``(1,0)" "6#-%!G6$\"\"\"\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "This polygon is defined by means of a list of its vertices: " }{XPPEDIT 18 0 "[[0, 0], [0, 1] , [1, 1], [1, 0]];" "6#7&7$\"\"!F%7$F%\"\"\"7$F'F'7$F'F%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "The following sequence of commands:" }}{PARA 15 "" 0 "" {TEXT -1 45 "defi nes the square by a list of its vertices," }}{PARA 15 "" 0 "" {TEXT -1 19 "uses the procedure " }{TEXT 0 11 "polygonplot" }{TEXT -1 20 " t o draw the square." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 96 "square1 := [[0,0],[0,1],[1,1],[1,0]];\nplots [polygonplot](square1,color=red,scaling=constrained);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "A number of polygons can be plotted together by forming a list." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 144 "square1 := [[0,0] ,[0,1],[1,1],[1,0]];\ntriangle1 := [[0,1],[0.5,2],[1,1]];\nplots[polyg onplot]([square1,triangle1],color=red,scaling=constrained);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "Suppose we want to plot a regul ar polygon with " }{TEXT 264 1 "n" }{TEXT -1 44 " sides, and having i ts centre at the origin." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 96 "The vertices of such a polygon are equally spaced \+ around a circle with its centre at the origin." }}{PARA 0 "" 0 "" {TEXT -1 32 "Each side subtends an angle of " }{XPPEDIT 18 0 "360^o/n ;" "6#*&)\"$g$%\"oG\"\"\"%\"nG!\"\"" }{TEXT -1 6 " or " }{XPPEDIT 18 0 "2*Pi/n;" "6#*(\"\"#\"\"\"%#PiGF%%\"nG!\"\"" }{TEXT -1 24 " radi ans at the origin." }}{PARA 0 "" 0 "" {TEXT -1 109 "For simplicity, we make the radius of this circle equal to 1, and we suppose that one ve rtex is at the point " }{XPPEDIT 18 0 "``(1,0)" "6#-%!G6$\"\"\"\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 88 "Going in an anti-clockwise direction around the circle, t he next vertex is at the point " }{XPPEDIT 18 0 "``(cos(2*Pi/n),sin(2* Pi/n));" "6#-%!G6$-%$cosG6#*(\"\"#\"\"\"%#PiGF+%\"nG!\"\"-%$sinG6#*(F* F+F,F+F-F." }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 99 "For example , if we want to draw a regular hexagon, which has 6 sides, the second \+ point around from " }{XPPEDIT 18 0 "``(1,0)" "6#-%!G6$\"\"\"\"\"!" } {TEXT -1 16 " is the point " }{XPPEDIT 18 0 "``(cos(Pi/3),sin(Pi/3)) ;" "6#-%!G6$-%$cosG6#*&%#PiG\"\"\"\"\"$!\"\"-%$sinG6#*&F*F+F,F-" } {TEXT -1 6 " or " }{XPPEDIT 18 0 "``(cos(60^o),sin(60^o));" "6#-%!G6 $-%$cosG6#)\"#g%\"oG-%$sinG6#)F*F+" }{TEXT -1 2 " ." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Now " }{XPPEDIT 18 0 "c os(60^o) = 1/2;" "6#/-%$cosG6#)\"#g%\"oG*&\"\"\"F+\"\"#!\"\"" }{TEXT -1 13 " = 0.5 and " }{XPPEDIT 18 0 "sin(60^o) = sqrt(3)/2;" "6#/-%$s inG6#)\"#g%\"oG*&-%%sqrtG6#\"\"$\"\"\"\"\"#!\"\"" }{TEXT -1 1 " " } {TEXT 265 1 "~" }{TEXT -1 9 " 0.8660. " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 312 286 286 {PLOTDATA 2 "68-%'CURVESG6%7S7$$\"\"\"\"\"!$F*F*7$$\"3w\"4hRPij!**!#=$\"3Ikwb#=y _O\"F/7$$\"3E8J#))4-Qn*F/$\"3%[#\\ff*)GLDF/7$$\"3-N5')yke[#*F/$\"3gLj& [K5J!QF/7$$\"3?goz=42`')F/$\"3j9NXR4U7]F/7$$\"3Sb](G._U!zF/$\"3t_H-qce DhF/7$$\"3_\\$R'oTd#3(F/$\"3!GO#*3qU&fqF/7$$\"3?H$>jubk6'F/$\"3!\\@@& \\!>8\"zF/7$$\"3VGuIc:x5]F/$\"3-$>p(Qh-a')F/7$$\"37C3nW'R,#QF/$\"3aK+1 6bcT#*F/7$$\"3$oY@iF'QDDF/$\"3s&Q\"[M\"oen*F/7$$\"3OB^hAo8X8F/$\"33)fV PO<\"4**F/7$$!3+qB/u(p5(f!#@$\"2%HhJ<#)******!#<7$$!33)\\T#fB[i8F/$\"3 ap%>wGZn!**F/7$$!3)e:d.:#=YEF/$\"3PA>=\\D`V'*F/7$$!3%*yV1J*3Jx$F/$\"3I zF#e`m3E*F/7$$!3V(\\AOF:B/&F/$\"3vU'yI2&oN')F/7$$!3[igNw%*[RgF/$\"3!G; )RQ+BqzF/7$$!3/XUwYjJ*3(F/$\"3AfvOg?x_qF/7$$!3&=e_b?/U!zF/$\"3u3HvXQ F/7$$!3!p!RS4Nij'*F/$\"3o$p9m#Q%=d#F/7$$!3#p(pT>+.2**F/$\"31V?00]Ug8F/ 7$$!3/gKG4>&*****F/$\"3vCA\\O%485$!#?7$$!3__Is=!Q%3**F/$!3q#4*>c=8]8F/ 7$$!3)>b`sYlSn*F/$!3UNbFGIGKDF/7$$!37_J.8AEj#*F/$!3q4&=j`Bsw$F/7$$!3% \\F)4Kh=u')F/$!3kENX')3zv\\F/7$$!3EB*z7xng%zF/$!37W(H!pUCrgF/7$$!3XD84 Pfc5rF/$!3Y?#o?\"yMJqF/7$$!3!Q%y6Ike[gF/$!3)4j2yeGL'zF/7$$!3g@UD=%)o!* \\F/$!3I_Mh6Mil')F/7$$!3o+1s\"[\"ysPF/$!3#\\IN,%***4E*F/7$$!30yCH\"el' 3EF/$!3=V>(fp[Pl*F/7$$!3g67Cvnc\"H\"F/$!3;$RoI*>C;**F/7$$!3EzyOHMQdIFa s$!3W1O#>E`*****F/7$$\"3GIAj`Ks(G\"F/$!3lYZ3X=u;**F/7$$\"3EKRqX8/bDF/$ !3U$[o%H'z!o'*F/7$$\"3@%)yb&Q)zNQF/$!36.2<#>x]B*F/7$$\"3O7w4w0I.]F/$!3 wHgb5wMe')F/7$$\"3@\\#)[*R]$4hF/$!3/a*[=H2o\"zF/7$$\"33/wY'Q+$*4(F/$!3 )4d)eg?sUqF/7$$\"3[f=s$Ry,!zF/$!3D1$H) zD(p_v\\F/7$$\"3!G$e@`4+D#*F/$!3i&>2ndo*fQF/7$$\"31mGAp))oa'*F/$!3%)f] nRQ=0EF/7$$\"3@zb'f`bp!**F/$!3/up91t'4O\"F/7$F($\"36YKhSr8/#)!#F-%'COL OURG6&%$RGBG$\"#5!\"\"F+F+-%*THICKNESSG6#\"\"#-F$6&7%7$F+F+F'7$$\"3+++ +++++]F/$\"3a+++SSDg')F/-Fjz6&F\\[lF*F*F*-%'SYMBOLG6$%'CIRCLEG\"#:-%&S TYLEG6#%&POINTG-F$6&Ff[l-Fjz6&F\\[l$\"*++++\"!\")F+F\\]l-F`\\l6$Fb\\lF ^[lFd\\l-F$6&Ff[lFj\\l-F`\\l6$%(DIAMONDGF^[lFd\\l-F$6&Ff[lFj\\l-F`\\l6 $%&CROSSGF^[lFd\\l-F$6%7$Fg[lFh[lF]\\l-%*LINESTYLEGFb[l-F$6%7$Fh[l7$Fi [lF+F]\\lF^^l-F$6%7$F'Fh[lF]\\lF^^l-%%TEXTG6%7$$\"#&*!\"#$\"#$*F]_lQ:( cos(~~~~~~),sin(~~~~~~))6\"-%%FONTG6$%*HELVETICAG\"\"*-Fh^l6%7$$\"#DF] _l$!\"(F]_lQ$1/2Fa_lFb_l-Fh^l6%7$$\"#vF]_lF\\`lF^`lFb_l-Fh^l6%7$$\"#BF ]_l$\"#`F]_lQ\"1Fa_lFb_l-Fh^l6%7$$\"#xF]_lFi`lF[alFb_l-Fh^l6%7$$\"\"%F _[l$\"#OF]_lQ$/~3Fa_lFb_l-Fh^l6%7$Fdal$\"#MF]_lQ#__Fa_lFb_l-Fh^l6%7$Fd alFj_lQ\"2Fa_lFb_l-Fh^l6%7$$\"#NF]_lF\\blQ\"vFa_l-Fc_l6$Fe_l\"\"(-Fh^l 6%7$$\"$/\"F]_lF^_lQ1p/3~~~~~~~~~~p/3Fa_l-Fc_l6$F`\\lFf_l-%*AXESTICKSG 6$Fc[lFc[l-%+AXESLABELSG6%%!GFjcl-Fc_l6#%(DEFAULTG-%(SCALINGG6#%,CONST RAINEDG-%%VIEWG6$F]dlF]dl" 1 2 0 1 10 0 2 9 1 4 1 1.000000 46.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Cu rve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 18" }} {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 94 "Trigonometric functions are available in Maple, but we mu st use radian measure for the angles." }}{PARA 0 "" 0 "" {TEXT -1 43 " The number Pi is represented by the synbol " }{TEXT 0 2 "Pi" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "cos(Pi/3);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+++++]!#5" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "sin(Pi/3);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$-%% sqrtG6#\"\"$\"\"\"#F)\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+SSDg ')!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 89 "To see how to plot a simpler version of the picture, above open the f ollowing subsection." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 29 "First par t of regular hexagon" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 259 5 "Notes" }{TEXT -1 1 ":" }}{PARA 15 "" 0 "" {TEXT -1 64 " The circle can be plotted using the equation with the procedure " } {TEXT 0 12 "implicitplot" }{TEXT -1 18 ", which is in the " }{TEXT 0 5 "plots" }{TEXT -1 9 " package." }}{PARA 15 "" 0 "" {TEXT -1 31 "Line s can be plotted using the " }{TEXT 0 5 "plot " }{TEXT -1 10 "procedur e." }}{PARA 15 "" 0 "" {TEXT -1 11 "The option " }{TEXT 0 11 "linestyl e=2" }{TEXT -1 31 " causes the lines to be dotted." }}{PARA 15 "" 0 " " {TEXT -1 14 "The procedure " }{TEXT 0 7 "display" }{TEXT -1 18 ", wh ich is in the " }{TEXT 0 5 "plots" }{TEXT -1 70 " package, allows the \+ lines and circle to be shown in the same diagram." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 246 "C := [0,0]: \np0 := [1,0]:\np1 := [cos(Pi/3),sin(Pi/3)]:\nlines := plot([[C,p1],[p 0,p1]],linestyle=2,color=black):\ncrcle := plots[implicitplot](x^2+y^2 =1,x=-1..1,y=-1..1,thickness=2):\nplots[display]([crcle,lines],scaling =constrained,tickmarks=[3,3]);" }}{PARA 13 "" 1 "" {GLPLOT2D 337 276 276 {PLOTDATA 2 "6)-%'CURVESG6du7$7$$!3S+++++++#*!#=$!3-************** QF*7$$!3&))**********\\K*F*$!33-++++++OF*7$F-7$$!3i)*********\\(Q*F*$! 3J.++++]7MF*7$7$$!3Anmmmmm\"f*F*$!3#>++++++!GF*F37$7$$!35mmmmmm\"f*F*F <7$$!3*4AAAAAAs*F*$!3A!yxxxxxF#F*7$7$$!3Immmmmm\"z*F*$!3w,++++++?F*FB7 $FH7$$!3#*************4**F*$!3q,+++++!H\"F*7$7$$!3\\++++++D**F*$!3g,++ ++++7F*FN7$FT7$$!3I!444444***F*$!3-6\"444444%!#>7$7$$!3Ymmmmmm\"***F*$ !3W<++++++SFinFZ7$F[o7$F\\o$\"3zYmmmmm;RFin7$7$F\\o$\"3u#)************ RFinFao7$Feo7$$!3!3Bp2Bp2$**F*$\"3AH#p2Bp28\"F*7$7$FU$\"3G)*********** *>\"F*Fio7$7$FU$\"3U)************>\"F*7$$!3&4dG9dG9#)*F*$\"3op&G9dG9#= F*7$7$FI$\"3W)*************>F*Ffp7$7$$!3Snmmmmm\"z*F*F]q7$$!3yLLLLLLt' *F*$\"3%=LLLLLLZ#F*7$7$F:$\"3/)************z#F*Fcq7$7$F:$\"3e)******** ****z#F*7$$!3y+++++v$\\*F*$\"3+*********\\P4$F*7$7$$!32,+++++D$*F*$\"3 u)************f$F*F`r7$Ffr7$$!39Zw6%HN#)G*F*$\"31Yw6%HN#)o$F*7$7$$!3], ++++++#*F*$\"3-**************QF*F\\s7$7$$!3!>++++++S)F*$!3q8dG9dG9aF*7 $$!3FNOOOOOO&)F*$!3Q-++++++_F*7$F]t7$$!3i'************p)F*$!372++++++ \\F*7$7$$!35rssssss*)F*$!3o,++++++WF*Fct7$FitF'7$Fbs7$$!3S+++++vV!*F*$ \"3\"*)********\\PC%F*7$7$$!3Asssssss*)F*$\"3!*)************R%F*F`u7$F fu7$$!3!*=THN#)eq()F*$\"3Y;THN#)eqZF*7$7$$!3\\POOOOOO&)F*$\"31******** *****>&F*F\\v7$7$$!3QOOOOOOO&)F*Fev7$$!3%\\LLLLLL[)F*$\"3?LLLLLL$G&F*7 $7$$!3!3++++++S)F*$\"3q8dG9dG9aF*F[w7$7$$!3?,++++++wF*$!3G)*********\\ (['F*7$$!3?)************)zF*$!3)>+++++++'F*7$F\\x7$FbwF[t7$Faw7$$!39m< THN#)e\")F*$\"3!Hw6%HN#)edF*7$7$$!3a,+++++!*zF*$\"3m)*************fF*F dx7$Fjx7$$!3Vommmmm;yF*$\"3!fmmmmmm@'F*7$7$$!33+++++++wF*$\"3;(******* **\\(['F*F`y7$7$$!3g,++++++oF*$!3(*>AAAAAAtF*7$F_zF]z7$7$$!33@AAAAAAtF *F]zFgw7$7$Fhw$\"3G)*********\\(['F*7$$!3%))eqk*F*Faal7$Fgal7$$!3]]w6%HN#)o$F*$\"3 #\\kxxxxxxF#F*7$7$FbdlF^rFagm7$Fegm7$F[[mFijl7$7$F_[mFirFgg m7$7$F_[mFf\\mF^dm-%'COLOURG6&%$RGBG\"\"\"\"\"!Fahm-%*THICKNESSG6#\"\" #-F$6%7$7$$FahmFahmFjhm7$$\"3++++++++]F*$\"3'fQWy.a-m)F*-F]hm6&F_hmFah mFahmFahm-%*LINESTYLEGFdhm-F$6%7$7$$F`hmFahmFjhmF[imF`imFbim-%*AXESTIC KSG6$\"\"$F\\jm-%+AXESLABELSG6%%\"xG%\"yG-%%FONTG6#%(DEFAULTG-%(SCALIN GG6#%,CONSTRAINEDG-%%VIEWG6$FejmFejm" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 96 "In order \+ to complete the hexagon, we need to obtain further points spaced out a round the circle." }}{PARA 0 "" 0 "" {TEXT -1 11 "The option " }{TEXT 0 19 "scaling=constrained" }{TEXT -1 75 " causes the same scale to be \+ used on both the horizontal and vertical axes." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 243 "p0 := [1,0] :\np1 := [cos(Pi/3),sin(Pi/3)]:\np2 := [cos(2*Pi/3),sin(2*Pi/3)]:\np3 \+ := [cos(Pi),sin(Pi)]:\np4 := [cos(4*Pi/3),sin(4*Pi/3)]:\np5 := [cos(5* Pi/3),sin(5*Pi/3)]:\nplot([p0,p1,p2,p3,p4,p5,p0],scaling=constrained,t ickmarks=[3,3],thickness=2);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6(-%'CURVESG6$7)7$$\"\"\"\"\"!$F*F*7$$\"3++++++++]!#= $\"3'fQWy.a-m)F/7$$!3++++++++]F/F07$$!\"\"F*F+7$F3$!3'fQWy.a-m)F/7$F-F 9F'-%'COLOURG6&%$RGBG$\"#5F7F+F+-%*THICKNESSG6#\"\"#-%*AXESTICKSG6$\" \"$FI-%+AXESLABELSG6$Q!6\"FM-%(SCALINGG6#%,CONSTRAINEDG-%%VIEWG6$%(DEF AULTGFV" 1 2 0 1 10 2 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curv e 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 " Another way to do this is to use the Maple command " }{TEXT 0 3 "seq" }{TEXT -1 49 " to form the sequence of points for the vertices." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 120 "vertices := [seq([cos(2*Pi*i/6),sin(2*Pi*i/6)],i=0..6)];\nplot(ve rtices,scaling=constrained,tickmarks=[3,3],thickness=2);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%)verticesG7)7$\"\"\"\"\"!7$#F'\"\"#,$*&F+!\"\" \"\"$F*F'7$#F.F+F,7$F.F(7$F1,$*&F+F.F/F*F.7$F*F4F&" }}{PARA 13 "" 1 " " {GLPLOT2D 400 300 300 {PLOTDATA 2 "6(-%'CURVESG6$7)7$$\"\"\"\"\"!$F* F*7$$\"3++++++++]!#=$\"3'fQWy.a-m)F/7$$!3++++++++]F/F07$$!\"\"F*F+7$F3 $!3'fQWy.a-m)F/7$F-F9F'-%'COLOURG6&%$RGBG$\"#5F7F+F+-%*THICKNESSG6#\" \"#-%*AXESTICKSG6$\"\"$FI-%+AXESLABELSG6$Q!6\"FM-%(SCALINGG6#%,CONSTRA INEDG-%%VIEWG6$%(DEFAULTGFV" 1 2 0 1 10 2 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 130 "We can also draw a filled in polygon. We can make do with one less vertex, as it is not necessary to close the polygon when using " }{TEXT 0 11 "polygonplot" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "vertices := [seq([cos(2*Pi*i/6),sin(2*Pi*i/6)],i = 0..5)];\nplots [polygonplot](vertices,scaling=constrained,tickmarks=[3,3],color=pink, axes=none);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)verticesG7(7$\"\"\" \"\"!7$#F'\"\"#,$*&F+!\"\"\"\"$F*F'7$#F.F+F,7$F.F(7$F1,$*&F+F.F/F*F.7$ F*F4" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%)POLYG ONSG6#7(7$$\"\"\"\"\"!$F*F*7$$\"+++++]!#5$\"+SSDg')F/7$$!+++++]F/F07$$ !\"\"F*F+7$F3$!+SSDg')F/7$F-F9-%'COLOURG6&%$RGBGF)$\"*w6%Hv!\"*$\"+9Vy gzF/-%*AXESTICKSG6$\"\"$FH-%(SCALINGG6#%,CONSTRAINEDG-%*AXESSTYLEG6#%% NONEG" 1 2 0 1 10 0 2 6 1 1 1 1.000000 45.000000 45.000000 0 0 "Curve \+ 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 122 "T he advantage of this second method is that, with a minor change, we ca n draw a regular polygon with any number of sides. " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 154 "vertices \+ := [seq([cos(2*Pi*i/13),sin(2*Pi*i/13)],i = 0..12)];\nplots[polygonplo t](vertices,scaling=constrained,tickmarks=[3,3],color=aquamarine,axes= none);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%)verticesG7/7$\"\"\"\"\"!7 $-%$cosG6#,$*(\"\"#F'\"#8!\"\"%#PiGF'F'-%$sinGF,7$-F+6#,$*(\"\"%F'F0F1 F2F'F'-F4F77$-F+6#,$*(\"\"'F'F0F1F2F'F'-F4F>7$,$-F+6#,$*(\"\"&F'F0F1F2 F'F'F1-F4FF7$,$-F+6#,$*(\"\"$F'F0F1F2F'F'F1-F4FN7$,$-F+6#,$*&F0F1F2F'F 'F1-F4FV7$FT,$FYF17$FL,$FRF17$FD,$FJF17$F=,$FBF17$F6,$F;F17$F*,$F3F1" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%)POLYGONSG6# 7/7$$\"\"\"\"\"!$F*F*7$$\"+d-ca))!#5$\"+> " 0 "" {MPLTEXT 1 0 57 "ngon := n -> [seq([cos(2*Pi*i/n),sin(2*Pi*i/n)],i=1..n)];" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%ngonGf*6#%\"nG6\"6$%)operatorG%&arr owGF(7#-%$seqG6$7$-%$cosG6#,$**\"\"#\"\"\"%#PiGF7%\"iGF79$!\"\"F7-%$si nGF3/F9;F7F:F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 104 "To get a list of the vertices of a regular polygon wit h a given number of sides, all we have to type is:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "ngon(4);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7&7$\"\"!\"\"\"7$!\"\"F%7$F%F(7$F&F%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "We can now draw a regular octagon with the command:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "plots[polygo nplot](ngon(8),scaling=constrained,tickmarks=[3,3],color=khaki,axes=no ne);" }}{PARA 13 "" 1 "" {GLPLOT2D 338 286 286 {PLOTDATA 2 "6'-%)POLYG ONSG6#7*7$$\"+5y1rq!#5F(7$$\"\"!F-$\"\"\"F-7$$!+5y1rqF*F(7$$!\"\"F-F,7 $F1F17$F,F47$F(F17$F.F,-%'COLOURG6&%$RGBG$\")THNi!\")F>$\")-\\DPF@-%*A XESTICKSG6$\"\"$FF-%(SCALINGG6#%,CONSTRAINEDG-%*AXESSTYLEG6#%%NONEG" 1 2 0 1 10 0 2 6 1 1 1 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 37 "Rotating polygon animat ion - 1 colour" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "In order to construct a rotation animation we first see h ow to draw a single rotated polygon." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "rotngon := (n,h) -> [seq([ cos(2*Pi*i/n+h),sin(2*Pi*i/n+h)],i=1..n)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(rotngonGf*6$%\"nG%\"hG6\"6$%)operatorG%&arrowGF)7#-% $seqG6$7$-%$cosG6#,&**\"\"#\"\"\"%#PiGF8%\"iGF89$!\"\"F89%F8-%$sinGF4/ F:;F8F;F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 14 "\nThe function " } {TEXT 0 7 "rotngon" }{TEXT -1 182 " can be used to obtain the vertices of a regular polygon with n sides which has been rotated through an a ngle of h radians from the basic position with one vertex at the point (1,0)." }}{PARA 0 "" 0 "" {TEXT -1 64 "For example, we can rotate the octagon considered above through " }{XPPEDIT 18 0 "10^o;" "6#)\"#5%\" oG" }{TEXT -1 11 " by taking " }{XPPEDIT 18 0 "h = Pi/18;" "6#/%\"hG*& %#PiG\"\"\"\"#=!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "plots[polygonplot](rotngo n(8,Pi/18),scaling=constrained,\n tickmarks=[3,3],color=aquam arine,axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 127 "In order to obtain an an imation, we need to form a sequence of rotated polygons to form the in dividual frames of the animation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 65 "We have already seen how to form a sequen ce by using the command " }{TEXT 0 3 "seq" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 22 "Another way to form a " }{TEXT 259 8 "sequence" } {TEXT -1 47 " is to use a programming construction called a " }{TEXT 259 8 "for loop" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "seqn := NULL:\nfor i from 1 \+ to 10 do\n seqn := seqn,i^2;\nend do:\nseqn;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6,\"\"\"\"\"%\"\"*\"#;\"#D\"#O\"#\\\"#k\"#\")\"$+\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 154 "To const ruct the individual frames, first take the angle subtended by a single side of the polygon at the centre, and divide this by the number of f rames." }}{PARA 0 "" 0 "" {TEXT -1 11 "This gives " }{XPPEDIT 18 0 "h \+ = 2*Pi/(n*`.`*numframes);" "6#/%\"hG*(\"\"#\"\"\"%#PiGF'*(%\"nGF'%\".G F'%*numframesGF'!\"\"" }{TEXT -1 9 ", where \"" }{TEXT 260 9 "numframe s" }{TEXT -1 68 "\", as its name suggests, is a variable to hold the n umber of frames." }}{PARA 0 "" 0 "" {TEXT -1 55 "Then h is the rotatio n angle between successive frames." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 68 "Now form a sequence consisting of the suc cessively rotated polygons." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 303 "n := 10:\nnumframes := 20:\nh := ( 2*Pi)/(n*numframes);\nseqn := NULL:\nfor j from 0 to numframes-1 do\n \+ frame := plots[polygonplot](rotngon(n,h*j),color=tan);\n seqn := s eqn,frame;\nend do:\nplots[display]([seqn],insequence=true,scaling=con strained,\n title=\"animation to rotate polygon\",axes=none); " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 111 "To \+ play the animation, click on the graphic and use the controls in the c ontext bar, or use the animation menu." }}{PARA 0 "" 0 "" {TEXT -1 59 "You can make the polygon rotate continuously by selecting \"" }{TEXT 263 1 "C" }{TEXT -1 33 "ontinuous\" in the animation menu." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 54 "A polygon subdivided into diffe rent coloured triangles" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 175 "The following command is used to describe a triangl e with one vertex at the origin and two vertices on the unit circle sp aced so that the corresponding angle at the centre is " }{XPPEDIT 18 0 "2*Pi/n;" "6#*(\"\"#\"\"\"%#PiGF%%\"nG!\"\"" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "trg := (j,n) -> [[0,0],[cos(2*Pi*(j-1)/n),sin(2*Pi*(j-1)/n)],\n \+ [cos(2*Pi*j/n),sin(2*Pi*j/n)]];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$trgGf*6$%\"jG%\"nG6\"6$%)operatorG%&arrowGF)7%7$\"\" !F/7$-%$cosG6#,$**\"\"#\"\"\"%#PiGF7,&9$F7F7!\"\"F79%F;F7-%$sinGF37$-F 26#,$**F6F7F8F7F:F7FFAF)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 120 "As an illustration we draw the 4th a nd 8th triangles which form part of the interior of a regular polygon \+ with 13 sides." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 252 "triangle1 := plots[polygonplot](trg(4,13),color =brown):\ntriangle2 := plots[polygonplot](trg(8,13),color=pink):\ncrcl e := plot([cos(t),sin(t),t=0..2*Pi]):\nplots[display]([crcle,triangle1 ,triangle2],scaling=constrained,\n tickmarks=[3,3],axes=none); " }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6*-%'CURVESG6$ 7S7$$\"\"\"\"\"!$F*F*7$$\"3w\"4hRPij!**!#=$\"3Ikwb#=y_O\"F/7$$\"3E8J#) )4-Qn*F/$\"3%[#\\ff*)GLDF/7$$\"3-N5')yke[#*F/$\"3gLj&[K5J!QF/7$$\"3?go z=42`')F/$\"3j9NXR4U7]F/7$$\"3Sb](G._U!zF/$\"3t_H-qceDhF/7$$\"3_\\$R'o Td#3(F/$\"3!GO#*3qU&fqF/7$$\"3?H$>jubk6'F/$\"3!\\@@&\\!>8\"zF/7$$\"3VG uIc:x5]F/$\"3-$>p(Qh-a')F/7$$\"37C3nW'R,#QF/$\"3aK+16bcT#*F/7$$\"3$oY@ iF'QDDF/$\"3s&Q\"[M\"oen*F/7$$\"3OB^hAo8X8F/$\"33)fVPO<\"4**F/7$$!3+qB /u(p5(f!#@$\"2%HhJ<#)******!#<7$$!33)\\T#fB[i8F/$\"3ap%>wGZn!**F/7$$!3 )e:d.:#=YEF/$\"3PA>=\\D`V'*F/7$$!3%*yV1J*3Jx$F/$\"3IzF#e`m3E*F/7$$!3V( \\AOF:B/&F/$\"3vU'yI2&oN')F/7$$!3[igNw%*[RgF/$\"3!G;)RQ+BqzF/7$$!3/XUw YjJ*3(F/$\"3AfvOg?x_qF/7$$!3&=e_b?/U!zF/$\"3u3HvXQF/7$$!3!p!RS4Nij'* F/$\"3o$p9m#Q%=d#F/7$$!3#p(pT>+.2**F/$\"31V?00]Ug8F/7$$!3/gKG4>&*****F /$\"3vCA\\O%485$!#?7$$!3__Is=!Q%3**F/$!3q#4*>c=8]8F/7$$!3)>b`sYlSn*F/$ !3UNbFGIGKDF/7$$!37_J.8AEj#*F/$!3q4&=j`Bsw$F/7$$!3%\\F)4Kh=u')F/$!3kEN X')3zv\\F/7$$!3EB*z7xng%zF/$!37W(H!pUCrgF/7$$!3XD84Pfc5rF/$!3Y?#o?\"yM JqF/7$$!3!Q%y6Ike[gF/$!3)4j2yeGL'zF/7$$!3g@UD=%)o!*\\F/$!3I_Mh6Mil')F/ 7$$!3o+1s\"[\"ysPF/$!3#\\IN,%***4E*F/7$$!30yCH\"el'3EF/$!3=V>(fp[Pl*F/ 7$$!3g67Cvnc\"H\"F/$!3;$RoI*>C;**F/7$$!3EzyOHMQdIFas$!3W1O#>E`*****F/7 $$\"3GIAj`Ks(G\"F/$!3lYZ3X=u;**F/7$$\"3EKRqX8/bDF/$!3U$[o%H'z!o'*F/7$$ \"3@%)yb&Q)zNQF/$!36.2<#>x]B*F/7$$\"3O7w4w0I.]F/$!3wHgb5wMe')F/7$$\"3@ \\#)[*R]$4hF/$!3/a*[=H2o\"zF/7$$\"33/wY'Q+$*4(F/$!3)4d)eg?sUqF/7$$\"3[ f=s$Ry,!zF/$!3D1$H)zD(p_v\\F/7$$\"3!G$ e@`4+D#*F/$!3i&>2ndo*fQF/7$$\"31mGAp))oa'*F/$!3%)f]nRQ=0EF/7$$\"3@zb'f `bp!**F/$!3/up91t'4O\"F/7$F($\"36YKhSr8/#)!#F-%'COLOURG6&%$RGBG$\"#5! \"\"F+F+-%)POLYGONSG6$7%7$F+F+7$$\"+/oO07!#5$\"+T()3F**Fh[l7$$!+o)[ga$ Fh[l$\"+GC;]$*Fh[l-Fjz6&F\\[l$\")#)eqk!\")$\"))eqk\"Fd\\lFe\\l-Fa[l6$7 %Fd[l7$$!+u\"=%4(*Fh[l$!+Vm:$R#Fh[l7$$!+![2^[(Fh[l$!+%eE7j'Fh[l-Fjz6&F \\[lF)$\"*w6%Hv!\"*$\"+9VygzFh[l-%*AXESTICKSG6$\"\"$F^^l-%+AXESLABELSG 6$Q!6\"Fb^l-%(SCALINGG6#%,CONSTRAINEDG-%*AXESSTYLEG6#%%NONEG-%%VIEWG6$ %(DEFAULTGF__l" 1 2 0 1 10 0 2 9 1 1 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 125 "Now we setup sequences of \"odd\" and \" even\" triangles, so that we can provide alternating colours as we go round the polygon." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "oddt := n -> [seq(trg((2*i-1),n),i=1..trun c((n+1)/2))];\nevent := n -> [seq(trg((2*i),n),i=1..trunc(n/2))];" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%oddtGf*6#%\"nG6\"6$%)operatorG%&arr owGF(7#-%$seqG6$-%$trgG6$,&*&\"\"#\"\"\"%\"iGF6F6F6!\"\"9$/F7;F6-%&tru ncG6#,&*&#F6F5F6F9F6F6FAF6F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% &eventGf*6#%\"nG6\"6$%)operatorG%&arrowGF(7#-%$seqG6$-%$trgG6$,$*&\"\" #\"\"\"%\"iGF6F69$/F7;F6-%&truncG6#,$*&#F6F5F6F8F6F6F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "We plot the res ults for a regular 10 sided polygon or " }{TEXT 259 7 "decagon" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 101 "Of course, you will on ly get alternating colours all the way round the polygon if the polygo n has an " }{TEXT 259 20 "even number of sides" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 164 "first := plots[polygonplot](oddt(10),color=brown):\nsecond := plo ts[polygonplot](event(10),color=pink):\nplots[display]([first,second], scaling=constrained,axes=none);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%)POLYGONSG6(7%7$$\"\"!F)F(7$$\"\"\"F)F(7$$\"+V*p, 4)!#5$\"+CD&y(eF07%F'7$$\"+Q*p,4$F0$\"+l^c5&*F07$$!+Q*p,4$F0F77%F'7$$! +V*p,4)F0F17$$!\"\"F)F(7%F'7$F>$!+CD&y(eF07$F:$!+l^c5&*F07%F'7$F5FH7$F .FE-%'COLOURG6&%$RGBG$\")#)eqk!\")$\"))eqk\"FSFT-F$6(7%F'F-F47%F'F9F=7 %F'F@FD7%F'FGFK7%F'FLF*-FN6&FPF,$\"*w6%Hv!\"*$\"+9VygzF0-%(SCALINGG6#% ,CONSTRAINEDG-%*AXESSTYLEG6#%%NONEG" 1 2 0 1 10 0 2 9 1 1 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 134 ": For a po lygon with a large number of sides it is probably better to remove the boundary lines for the triangles by using the option " }{TEXT 0 17 "s tyle=patchnogrid" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 200 "first := plots[polygonplot] (oddt(30),color=brown,style=patchnogrid):\nsecond := plots[polygonplot ](event(30),color=pink,style=patchnogrid):\nplots[display]([first,seco nd],scaling=constrained,axes=none);" }}{PARA 13 "" 1 "" {GLPLOT2D 403 368 368 {PLOTDATA 2 "6&-%)POLYGONSG637%7$$\"\"!F)F(7$$\"\"\"F)F(7$$\"+ 2gZ\"y*!#5$\"+4p6z?F07%F'7$$\"+xXXN\"*F0$\"+IkOnSF07$$\"+V*p,4)F0$\"+C D&y(eF07%F'7$$\"+igI\"p'F0$\"+c#[9V(F07$$\"+++++]F0$\"+SSDg')F07%F'7$$ \"+Q*p,4$F0$\"+l^c5&*F07$$\"+HYGX5F0$\"+a*=_%**F07%F'7$$!+HYGX5F0FR7$$ !+Q*p,4$F0FM7%F'7$$!+++++]F0FG7$$!+igI\"p'F0FB7%F'7$$!+V*p,4)F0F<7$$!+ xXXN\"*F0F77%F'7$$!+2gZ\"y*F0F17$$!\"\"F)F(7%F'7$Feo$!+4p6z?F07$Fao$!+ IkOnSF07%F'7$F^o$!+CD&y(eF07$Fjn$!+c#[9V(F07%F'7$Fgn$!+SSDg')F07$FY$!+ l^c5&*F07%F'7$FV$!+a*=_%**F07$FPFaq7%F'7$FKF]q7$FEFjp7%F'7$F@Ffp7$F:Fc p7%F'7$F5F_p7$F.F\\p-%'COLOURG6&%$RGBG$\")#)eqk!\")$\"))eqk\"FcrFdr-%& STYLEG6#%,PATCHNOGRIDG-F$637%F'F-F47%F'F9F?7%F'FDFJ7%F'FOFU7%F'FXFfn7% F'FinF]o7%F'F`oFdo7%F'FgoF[p7%F'F^pFbp7%F'FepFip7%F'F\\qF`q7%F'FcqFeq7 %F'FfqFhq7%F'FiqF[r7%F'F\\rF*-F^r6&F`rF,$\"*w6%Hv!\"*$\"+9VygzF0Ffr-%( SCALINGG6#%,CONSTRAINEDG-%*AXESSTYLEG6#%%NONEG" 1 2 0 1 10 0 2 9 1 1 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 38 "Rotating polygon animation - 2 co lours" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 " We can construct an animation which rotates the two-colour polygon as \+ follows." }}{PARA 0 "" 0 "" {TEXT -1 99 "First define functions to con struct the vertices of the rotated triangular sections of the polygon. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 243 "trgh := (j,n,h) -> [[0,0],[cos(2*Pi*(j-1)/n+h),sin(2*Pi*(j-1) /n+h)],\n [cos(2*Pi*j/n+h),sin(2*Pi*j/n+h)]];\noddth := (n,h) -> [seq(trgh((2*i-1),n,h),i=1..trunc((n+1)/2))];\neventh := (n, h) -> [seq(trgh((2*i),n,h),i=1..trunc(n/2))];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%trghGf*6%%\"jG%\"nG%\"hG6\"6$%)operatorG%&arrowGF*7% 7$\"\"!F07$-%$cosG6#,&**\"\"#\"\"\"%#PiGF8,&9$F8F8!\"\"F89%FF8-F@FCF*F*F*" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%&oddthGf*6$%\"nG%\"hG6\"6$%)operatorG%&arrowGF)7#-% $seqG6$-%%trghG6%,&*&\"\"#\"\"\"%\"iGF7F7F7!\"\"9$9%/F8;F7-%&truncG6#, &*&#F7F6F7F:F7F7FCF7F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'event hGf*6$%\"nG%\"hG6\"6$%)operatorG%&arrowGF)7#-%$seqG6$-%%trghG6%,$*&\" \"#\"\"\"%\"iGF7F79$9%/F8;F7-%&truncG6#,$*&#F7F6F7F9F7F7F)F)F)" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "The frame s for the animation can be constructed using the following commands." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 418 "n := 10:\nnumframes := 30:\nh := 4*Pi/(n*numframes):\nanimatn : = NULL:\nfor j from 0 to numframes-1 do\n first := plots[polygonplot ](oddth(10,h*j),color=brown);\n second := plots[polygonplot](eventh( 10,h*j),color=pink);\n frame := plots[display](\{first,second\});\n \+ animatn := animatn,frame;\nend do:\nplots[display]([animatn],inseque nce=true,scaling=constrained,\n title=\"animation to rotate p olygon\",axes=none);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%(ANIMATEG6@7$-%)POLYGONSG6(7%7$$\"\"!F-F,7$$\"+V*p,4) !#5$\"+CD&y(eF17$$\"+Q*p,4$F1$\"+l^c5&*F17%F+7$$!+Q*p,4$F1F77$$!+V*p,4 )F1F27%F+7$$!\"\"F-F,7$F>$!+CD&y(eF17%F+7$F;$!+l^c5&*F17$F5FI7%F+7$F/F E7$$\"\"\"F-F,-%'COLOURG6&%$RGBGFP$\"*w6%Hv!\"*$\"+9VygzF1-F(6(7%F+FNF .7%F+F4F:7%F+F=FA7%F+FDFH7%F+FKFM-FR6&FT$\")#)eqk!\")$\"))eqk\"F_oF`o7 $-F(6(7%F+7$$\"+,$G7***F1$\"+s`c(=%!#67$$\"+uX$p$yF1$\"+-yZ6iF17%F+7$$ \"+,#)>*o#F1$\"+pciJ'*F17$$!+n/s&[$F1$\"+(*)>GP*F17%F+7$$!+1C@H$)F1$\" +%\\:R`&F17$$!+,$G7***F1$!+s`c(=%F[p7%F+7$$!+uX$p$yF1$!+-yZ6iF17$$!+,# )>*o#F1$!+pciJ'*F17%F+7$$\"+n/s&[$F1$!+(*)>GP*F17$$\"+1C@H$)F1$!+%\\:R `&F1F[o-F(6(7%F+F\\pFbp7%F+FgpF]q7%F+FbqFhq7%F+F]rFcr7%F+FhrFfoFQ7$-F( 6(7%F+7$$\"+b0&*pvF1$\"+Tg?MlF17$$\"+(p3NG#F1$\"+I!*yN(*F17%F+7$$!+oe: vQF1$\"+::j=#*F17$$!+-Ek`&)F1$\"+%4q-=&F17%F+7$$!+#fG\\'**F1$!+OVyn$)F [p7$$!+b0&*pvF1$!+Tg?MlF17%F+7$$!+(p3NG#F1$!+I!*yN(*F17$$\"+oe:vQF1$!+ ::j=#*F17%F+7$$\"+-Ek`&)F1$!+%4q-=&F17$$\"+#fG\\'**F1$\"+OVyn$)F[pFQ-F (6(7%F+FivFhs7%F+F]tFct7%F+FhtF^u7%F+FcuFiu7%F+F^vFdvF[o7$-F(6(7%F+7$$ \"+tio*G(F1$\"+g5ZXoF17$$\"+UJ\"Q(=F1$\"+3D(G#)*F17%F+7$$!+=HzdUF1$\"+ B0F[!*F17$$!++o1j()F1$\"+Tn`<[F17%F+7$$!+8q9@**F1$!+OBL`7F17$$!+tio*G( F1$!+g5ZXoF17%F+7$$!+UJ\"Q(=F1$!+3D(G#)*F17$$\"+=HzdUF1$!+B0F[!*F17%F+ 7$$\"++o1j()F1$!+Tn`<[F17$$\"+8q9@**F1$\"+OBL`7F1FQ-F(6(7%F+FjzFiw7%F+ F^xFdx7%F+FixF_y7%F+FdyFjy7%F+F_zFezF[o7$-F(6(7%F+7$$\"+r.'*f)*F1$\"+n uon;F17$$\"+0Mj'*pF1$\"+'zEZ9(F17%F+7$$\"+#GI3Y\"F1$\"+ILs#*)*F17$$!+` .'Hj%F1$\"+\"zN?'))F17%F+7$$!+-w6d*)F1$\"+$z^jW%F17$$!+r.'*f)*F1$!+nuo n;F17%F+7$$!+0Mj'*pF1$!+'zEZ9(F17$$!+#GI3Y\"F1$!+ILs#*)*F17%F+7$$\"+`. 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}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 39 "_________________________________ ______" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q2 " }}{PARA 0 "" 0 "" {TEXT -1 65 " Const ruct an animation which rotates a regular 12 sided polygon." }}{PARA 0 "" 0 "" {TEXT -1 39 "_______________________________________" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 39 "__ _____________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q3 " }} {PARA 0 "" 0 "" {TEXT -1 214 "Draw a 12-sided polygon subdivided into1 2 triangles, with one vertex at the centre, and coloured with three di fferent colours sequentially around the polygon. Then construct an ani mation which rotates this polygon." }}{PARA 0 "" 0 "" {TEXT -1 39 "___ ____________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 39 "_______________________________________ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 24 "Code for drawing picture" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 791 "cs60 := evalf(cos(Pi/3)):\n sn60 := evalf(sin(Pi/3)):\nC := [0,0]:\np0 := [1,0]:\np1 := [cs60,sn60 ]:\np2 := [cs60,0]:\npts := plot([[C,p0,p1]$4],symbol=[circle$2,diamon d,cross],symbolsize=[15,10$3],\n color=[black,magenta$3],style=point) :\nlines := plot([[C,p1],[p1,p2],[p0,p1]],linestyle=2,color=black):\nc rcle := plot([cos(t),sin(t),t=0..2*Pi],thickness=2):\nt1 := plots[tex tplot]([[0.95,0.93,`(cos( ),sin( ))`],\n [0.25,-0.07,`1/2`] ,[0.75,-0.07,`1/2`],[0.23,0.53,`1`],\n [0.77,0.53,`1`],[0.4,0.36,`/ \+ 3`],[0.4,0.34,`__`],[0.4,0.25,`2`]],font=[HELVETICA,9]):\nt2 := plots [textplot]([0.35,0.34,`v`],font=[HELVETICA,7]):\nt3 := plots[textplot] ([1.04,.93,`p/3 p/3`],font=[SYMBOL,9]):\nplots[display]([crcl e,pts,lines,t1,t2,t3],scaling=constrained,\n labels=[``,``],tickmar ks=[2,2]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "4 0 0 " 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }