{VERSION 3 0 "IBM INTEL NT" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "TXT CMD" -1 257 "MS Sans Serif" 0 0 128 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "Bookmark" 20 258 "" 0 0 0 128 0 1 1 0 0 0 0 1 0 0 0 } {CSTYLE "word" -1 259 "" 0 0 128 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "boo kmark" -1 260 "" 0 0 0 128 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 " " 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 0 0 128 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Text \+ Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 255 255 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 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 "R3 \+ Font 0" -1 256 1 {CSTYLE "" -1 -1 "Helvetica" 1 12 0 0 0 0 2 1 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 2" -1 257 1 {CSTYLE "" -1 -1 "Courier" 1 14 0 0 0 0 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 3" -1 258 1 {CSTYLE "" -1 -1 "Helvetica" 1 12 0 0 0 0 1 1 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 4" -1 259 1 {CSTYLE "" -1 -1 "Lucida" 1 14 0 0 0 0 1 1 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 \+ Font 5" -1 260 1 {CSTYLE "" -1 -1 "Lucida" 1 14 0 0 0 0 1 1 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 6" -1 261 1 {CSTYLE "" -1 -1 "Lucida" 1 14 0 0 0 0 1 1 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 7" -1 262 1 {CSTYLE "" -1 -1 "Ch arter" 1 18 0 0 0 0 1 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 8" -1 263 1 {CSTYLE "" -1 -1 "Charter" 1 18 0 0 0 0 1 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 9" -1 264 1 {CSTYLE "" -1 -1 "Charter" 1 18 0 0 0 0 1 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 10" -1 265 1 {CSTYLE "" -1 -1 "Charter" 1 18 0 0 0 0 1 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 11" -1 266 1 {CSTYLE "" -1 -1 "Charter" 1 18 0 0 0 0 1 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 12" -1 267 1 {CSTYLE "" -1 -1 "Lucida" 1 14 0 0 0 0 1 1 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R 3 Font 13" -1 268 1 {CSTYLE "" -1 -1 "Lucida" 1 14 0 0 0 0 1 1 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 14" -1 269 1 {CSTYLE "" -1 -1 "Lucida" 1 14 0 0 0 0 1 1 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 15" -1 270 1 {CSTYLE "" -1 -1 "Lucida" 1 14 0 0 0 0 1 1 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 16" -1 271 1 {CSTYLE "" -1 -1 "Courier" 1 14 0 0 0 0 1 1 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R 3 Font 17" -1 272 1 {CSTYLE "" -1 -1 "Courier" 1 14 0 0 0 0 1 1 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 18" -1 273 1 {CSTYLE "" -1 -1 "Courier" 1 14 0 0 0 0 1 1 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 19" -1 274 1 {CSTYLE "" -1 -1 "Courier" 1 14 0 0 0 0 1 1 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 20" -1 275 1 {CSTYLE "" -1 -1 "Courier" 1 14 0 0 0 0 1 1 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R 3 Font 21" -1 276 1 {CSTYLE "" -1 -1 "Lucidatypewriter" 1 14 0 0 0 0 2 1 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 2 2" -1 277 1 {CSTYLE "" -1 -1 "Lucidatypewriter" 1 14 0 0 0 0 2 1 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 23" -1 278 1 {CSTYLE "" -1 -1 "Lucidatypewriter" 1 14 0 0 0 0 2 1 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 24" -1 279 1 {CSTYLE "" -1 -1 "Lucidatypewriter" 1 14 0 0 0 0 2 1 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 25" -1 280 1 {CSTYLE " " -1 -1 "Lucidatypewriter" 1 14 0 0 0 0 2 1 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 26" -1 281 1 {CSTYLE "" -1 -1 "Lucidatypewriter" 1 14 0 0 0 0 2 1 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 27" -1 282 1 {CSTYLE "" -1 -1 "Luci datypewriter" 1 14 0 0 0 0 2 1 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 28" -1 283 1 {CSTYLE "" -1 -1 "Lucidatypewr iter" 1 14 0 0 0 0 2 1 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 3 284 1 {CSTYLE "" -1 -1 "" 0 1 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 "newpage" -1 285 1 {CSTYLE " " -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 1 0 -1 0 }{PSTYLE "vfill" -1 286 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Definition" -1 287 1 {CSTYLE "" -1 -1 "" 0 0 0 64 128 1 0 0 0 0 0 0 0 0 0 }0 0 0 -1 4 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Theorem" -1 288 1 {CSTYLE "" -1 -1 "" 0 0 219 36 36 1 0 0 0 0 0 0 0 0 0 }0 0 0 -1 4 4 0 0 0 0 0 0 -1 0 } {PSTYLE "Problem" -1 289 1 {CSTYLE "" -1 -1 "" 0 0 0 0 255 1 0 0 0 0 0 0 0 0 0 }0 0 0 -1 4 4 3 4 0 0 0 0 -1 0 }{PSTYLE "dblnorm" -1 290 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "code" 2 291 1 {CSTYLE "" -1 -1 "Comic Sans MS" 0 0 128 0 128 1 0 1 0 0 0 0 3 0 0 }0 0 0 -1 -1 -1 3 12 0 0 0 0 -1 0 } {PSTYLE "asis" 0 292 1 {CSTYLE "" -1 -1 "Arial Narrow" 1 12 128 64 0 1 0 0 0 0 0 1 3 0 0 }0 0 0 -1 -1 -1 3 6 0 0 0 0 0 0 }{PSTYLE "subprobl em" 0 293 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "diagram" -1 294 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "dblnorm.mws" -1 295 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Item" 0 296 1 {CSTYLE "" -1 -1 "Lucida Sans" 1 12 0 0 0 1 0 0 0 0 0 0 0 0 0 }0 0 0 -1 6 -1 3 0 0 0 0 0 -1 0 }} {SECT 0 {SECT 0 {PARA 284 "" 0 "" {TEXT -1 16 "Rolling polygons" }} {EXCHG {PARA 267 "" 0 "motions" {TEXT -1 1 " " }}{PARA 267 "" 0 "" {TEXT -1 43 "1. Reflections, Rotations, and Translation\n" }}{PARA 0 " " 0 "" {TEXT -1 272 "A rigid motion is defined to be a transformation T of points where if p and q are points then the distance from p to q is the same as the distance from T(p) to T(q). It is a famous theor em of Euclidean geometry that each rigid motion is a composition of \+ reflections. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 190 "We want to get a formula for reflecting a point about a line. Here is one way. First define a word which takes two point s in the plane and returns the equation for the line through them. " } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 276 " lineqn := proc(A,B)\n l ocal x,y;\n (y-A[2])*(B[1]-A[1])-(x-A[1])*(B[2]-A[2]) end:\n distol := proc(X,A,B)\n local eq,a,b;\n eq := lineqn(A,B);\n \+ a := coeff(eq,x);\n b := coeff(eq,y);\n subs(\{y = X[2], x = X[1]\},eq)/\n sqrt(a^2+b^2)\n end:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "lineqn([3,2],[1,-2]);" }}{PARA 2 " " 1 "" {TEXT -1 47 " -2 y - 8 + 4 x" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 82 "Next, get a word which returns t he reflection of a point pt about the line lin. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 230 "reflect := proc(pt,lin)\n local \+ linperp,cen;\n linperp := coeff(lin,y)*(x-pt[1])-coeff(l in,x)*(y - pt[2]);\n cen := solve(\{linperp,lin\},\{y,x\});\n \+ subs(cen,[2*x-pt[1],2*y-pt[2]]);\n end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "reflect([-4,0],y-x);" }}{PARA 2 "" 1 "" {TEXT -1 44 " [0, -4]" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 444 "Reflect is not a very efficient w ord, because it actually carries out the algorithm we normally use whe n we are doing the problem by hand: Write the equation for the line \+ through pt which is perpendicular to lin, the solve the two equations \+ simutaneously for the point of intersection, then calculate the reflec tion. But we can use reflect to generate the formula for calculating \+ the reflection of a general point [h,k] about a general line " } {XPPEDIT 18 0 "a*x +b*y +c =0" "6#/,(*&%\"aG\"\"\"%\"xGF'F'*&%\"bGF'% \"yGF'F'%\"cGF'\"\"!" }{TEXT -1 50 ". This would be a more efficient way to compute." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "formula := reflect([h,k],a*x+b*y+c);" }}{PARA 2 "" 1 "" {TEXT -1 334 " \+ 2 2\n \+ -b h + b a k + c a -b c - b h a + a k\n fo rmula := [-2 ------------------- - h, 2 ------------------- - k]\n \+ 2 2 2 2\n \+ b + a b + a" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "We can make this into a function. " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "ref := unapply(formula,h,k,a ,b,c);" }}{PARA 2 "" 1 "" {TEXT -1 313 " ref := (h, k, a, b, c) ->\n \n 2 2\n \+ -b h + b a k + c a -b c - b h a + a k\n [-2 -------- ----------- - h, 2 ------------------- - k]\n 2 2 2 2\n b + a \+ b + a" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "Now, make a more ef ficient version of reflect" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "effref := proc(pt,eqn)\nref(pt[1],pt[2],coeff(eqn,x,1),\ncoeff(eqn ,y,1),subs(\{x=0,y=0\},eqn)) \nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "effref( [-4,0],x-y);" }}{PARA 2 "" 1 "" {TEXT -1 44 " [0, -4]" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 28 "Now, lets make a hexagon." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "fig := [[-4,0],[0, 2],[1,4],[0,5],[-15,19],[-15,1 5] ];\neq1 := x-y;" }}{PARA 2 "" 1 "" {TEXT -1 71 " fig := [[- 4, 0], [0, 2], [1, 4], [0, 5], [-15, 19], [-15, 15]]" }}{PARA 2 "" 1 " " {TEXT -1 46 " eq1 := x - y" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "rfig := [seq(effref(fig[i],e q1),i=1..nops(fig))];" }}{PARA 2 "" 1 "" {TEXT -1 72 " rfig := [[0, -4], [2, 0], [4, 1], [5, 0], [19, -15], [15, -15]]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 " rlin := plot(x,x=-10..10):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 133 "pfig := display([rlin,polygonplot(fig,color=yellow,style=patch) ]):\nrpfig := display([rlin,polygonplot(rfig,color=blue,style=patch)]) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "display([rlin,pfig,rpf ig],scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6*-%'CURVESG6$7S7$$!#5\"\"!F(7$$!1nmm;p0k&*!#:F,7$$!1LL$3 s%HaF.FK7$$!1******\\$*4)*\\F.FN7$$!1++ +]_&\\c%F.FQ7$$!1+++]1aZTF.FT7$$!1mm;/#)[oPF.FW7$$!1LLL$=exJ$F.FZ7$$!1 LLLL2$f$HF.Fgn7$$!1++]PYx\"\\#F.Fjn7$$!1MLLL7i)4#F.F]o7$$!1****\\P'psm \"F.F`o7$$!1****\\74_c7F.Fco7$$!1:LL$3x%z#)!#;Ffo7$$!1ILL3s$QM%FhoFjo7 $$!1^omm;zr)*!#=F]p7$$\"1WLLezw5VFhoFap7$$\"1.++v$Q#\\\")FhoFdp7$$\"1N LLe\"*[H7F.Fgp7$$\"1++++dxd;F.Fjp7$$\"1,++D0xw?F.F]q7$$\"1,+]i&p@[#F.F `q7$$\"1+++vgHKHF.Fcq7$$\"1lmmmZvOLF.Ffq7$$\"1,++]2goPF.Fiq7$$\"1KL$eR <*fTF.F\\r7$$\"1-++])Hxe%F.F_r7$$\"1mm;H!o-*\\F.Fbr7$$\"1,+]7k.6aF.Fer 7$$\"1mmm;WTAeF.Fhr7$$\"1****\\i!*3`iF.F[s7$$\"1NLLL*zym'F.F^s7$$\"1OL L3N1#4(F.Fas7$$\"1pm;HYt7vF.Fds7$$\"1-+++xG**yF.Fgs7$$\"1qmmT6KU$)F.Fj s7$$\"1OLLLbdQ()F.F]t7$$\"1++]i`1h\"*F.F`t7$$\"1-+]P?Wl&*F.Fct7$$\"#5F *Fft-%'COLOURG6&%$RGBG$Fgt!\"\"F*F*F#-%)POLYGONSG6%7(7$$!\"%F*F*7$F*$ \"\"#F*7$$\"\"\"F*$\"\"%F*7$F*$\"\"&F*7$$F.F*$\"#>F*7$Fav$\"#:F*-Fit6& F[u$\"*++++\"!\")FivF*-%&STYLEG6#%&PATCHGF#-F_u6%7(7$F*Fcu7$FfuF*7$F[v Fiu7$F^vF*7$FbvFav7$FevFav-Fit6&F[uF*F*FivF\\w-%(SCALINGG6#%,CONSTRAIN EDG-%+AXESLABELSG6$Q\"x6\"%!G-%%VIEWG6$;F(Fft%(DEFAULTG" 1 2 0 1 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 188 "Now, lets do rotations. A rotation of pt through an a ngle of theta radians about the origin is the product of two reflectio ns, the first about the x-axis and the second about the line " } {XPPEDIT 18 0 "y = theta/2*x" "6#/%\"yG*(%&thetaG\"\"\"\"\"#!\"\"%\"xG F'" }{TEXT -1 85 ". To rotate about a point cen, first subtract cen \+ then rotate then add cen back on." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "rot := proc(pt,theta,cen)\n cen+effref(effref(pt-cen ,y),y-tan(theta/2)*x) end;" }}{PARA 2 "" 1 "" {TEXT -1 93 " rot := pr oc(pt, theta, cen)\n cen + effref(effref(pt - cen, y), y - tan(1/2* theta)*x)\nend" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "evalf(rot ([1,0],Pi/2,[0,0]));" }}{PARA 2 "" 1 "" {TEXT -1 44 " \+ [0, 1.]" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 92 "No w here is a word to roll a polygon s steps along the line thru its first two vertices." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " } {MPLTEXT 1 0 647 "rotit := proc(gon1,clr,s )\n local gon,ang,movie,h ill,t,i,j,k,l,tmpgon;\n gon := gon1:\n hill := plot([op(e xpand(t*gon[1]+(1-t)*gon[2])),t=-4..4]);\n movie := plots[display] ([hill,polygonplot(gon,color=clr)]);\n for k from 1 to s do\n \+ ang := evalf(Pi-linalg[angle](gon[1]-gon[2], gon[3]-gon[2] ));\n f or i from 1 to 3 do \n tmpgon :=evalf([seq(rot(gon[j],-i/3*ang, gon[2]),\nj = 1 .. nops(gon))]);\n movie := movie,display([hil l,polygonplot(tmpgon,\ncolor = clr)]);\n od;\n gon := [seq(tmp gon[l],l=2..nops(tmpgon)),tmpgon[1]]:\n od;\n plots[display]([ movie],scaling=constrained,insequence=true);\n end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "printlevel:=1;" }}{PARA 2 "" 1 "" {TEXT -1 48 " printlevel := 1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "rotit([[0,0],[1,.5],[1,2]],red,4); " }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6$-%(ANIMATEG6 /7&-%'CURVESG6$7S7$$\"\"&\"\"!$\"1+++++++D!#:7$$\"1nmmmFiD[F1$\"1MLL$Q 6GT#F17$$\"1LLLo!)*Qn%F1$\"1nm;M!\\pL#F17$$\"1mmmwxE.XF1$\"1LLL))Qj^AF 17$$\"1mmmOk]JVF1$\"1LLL=Kvl@F17$$\"1MLL[9cgTF1$\"1nm;C2G!3#F17$$\"1mm mhN2-SF1$\"1LL$3yO5+#F17$$\"1+++N&oz$QF1$\"1++]nU)*=>F17$$\"1nmm\")3Do OF1$\"1LL$3WDT$=F17$$\"1+++:v2*\\$F1$\"1++]d(Q&\\1DLF1$ \"1nmmc4`i;F17$$\"1nmmw))yrJF1$\"1LLLQW*ee\"F17$$\"1+++S(R#**HF1$\"1++ +q)>'*\\\"F17$$\"1++++@)f#GF1$\"1+++]5*HT\"F17$$\"1+++gi,fEF1$\"1+++I \"3&H8F17$$\"1nmm\"G&R2DF1$\"1LL$3k(p`7F17$$\"1LLLtK5FBF1$\"1nmmO;bj6F 17$$\"1MLL$HsV<#F1$\"1nmmYh=(3\"F17$$\"1+++b)4n*>F1$\"1,++v#\\N)**!#;7 $$\"1MLL$\\[%R=F1$\"1pmmmCC(>*Ffq7$$\"1+++by!pm\"F1$\"1*****\\FRXL)Ffq 7$$\"1+++l$3E]\"F1$\"1+++D=/8vFfq7$$\"1LLL$3z6L\"F1$\"1mmm;a*el'Ffq7$$ \"1LLL)[`P<\"F1$\"1nmmTuwoeFfq7$$\"1nmm;([R+\"F1$\"1MLL$eV(>]Ffq7$$\"1 tmm;Gpv#)Ffq$\"1PLL3k%y8%Ffq7$$\"1/++]YISnFfq$\"1-++DB:qLFfq7$$\"1rmmm L/#3&Ffq$\"1NLL$o@5a#Ffq7$$\"1.+++s*)oLFfq$\"1,+++'[Wo\"Ffq7$$\"13+++z \"Hp\"Ffq$\"1U+++&*ek%)!#<7$$\"1H1++]qM$Ffq$!1JLLL&4Nn\"Ffq7$$!1)***** **HSu]Ffq$!1*******\\,s`#Ffq7$$!1ILL$ep'RmFfq$!1lmm\"zM)>LFfq7$$!1**** ***R>4N)Ffq$!1*******pfa<%Ffq7$$!1emm;@2h**Ffq$!1HLLeg`!)\\Ffq7$$!1+++ lXTk6F1$!1)****\\#G2AeFfq7$$!1mmmmd'*G8F1$!1KLLL)G[k'Ffq7$$!1+++DcB,:F 1$!1)****\\7yh](Ffq7$$!1MLLt>:n;F1$!1ommm)fdL)Ffq7$$!1LLL.a#o$=F1$!1mm m;q7%=*Ffq7$$!1nmm^Q40?F1$!1LL$e#pa-5F17$$!1+++!3:(f@F1$!1+++Sv&)z5F17 $$!1nmmc%GpL#F1$!1LLLGUYo6F17$$!1LLL8-V&\\#F1$!1nmm1^rZ7F17$$!1+++XhUk EF1$!1++]sI@K8F17$$!1+++:oNI6!\"*$\"+E&QR%R!#5F[\\l7$$ \"+RP4T:Ff]l$\"+?f+**=Ff]lFb\\lFg\\lF[]l7&F'-Fg[l6$7%7$$!)M/>wFf]l$\"+ %*\\3I!)Fi]lF[\\l7$$\"+gWK4?Ff]l$\"+@Mi4;Ff]lFb\\lFg\\lF[]l7&F'-Fg[l6$ 7%7$$\"*3Gd0\"Ff]l$\"+%R?3<\"Ff]lF[\\l7$$\"+(yS;M#Ff]l$\"+$R?3<\"Ff]lF b\\lFg\\lF[]l7&F'-Fg[l6$7%7$$\"*+!ov(*Ff]l$\"+XQw%z\"Ff]lFf_l7$$\"*q$R )Q*Ff]l$\"+co77HFf]lFb\\lFg\\lF[]l7&F'-Fg[l6$7%7$$\"+gVxr>Ff]l$\"+i\\] CEFf]lFf_l7$$\"+(f3w\"GFf]l$\"+/TkbLFf]lFb\\lFg\\lF[]l7&F'-Fg[l6$7%7$$ \"+*yS;C$Ff]l$\"+#R?3P#Ff]lFf_l7$$\"+!zS;M%Ff]l$\"+!R?3<#Ff]lFb\\lFg\\ lF[]l7&F'-Fg[l6$7%7$$\"+(*e7cAFf]l$\"+<`VxHFf]lF`bl7$$\"+4$>%\\PFf]l$ \"+H37>JFf]lFb\\lFg\\lF[]l7&F'-Fg[l6$7%7$$\"+te-NNFf]l$\"+&GNjD%Ff]lF` bl7$$\"+#H\\Yk%Ff]l$\"+C3,ZKFf]lFb\\lFg\\lF[]l7&F'-Fg[l6$7%7$$\"+(zS;M &Ff]l$\"+(Q?3<%Ff]lF`bl7$$\"+$zS;M&Ff]l$\"+)Q?3n#Ff]lFb\\lFg\\lF[]l7&F '-Fg[l6$7%7$$\"+)f0'GUFf]l$\"+WU@lDFf]lFjdl7$$\"+OXt#)eFf]l$\"+1j#)pSF f]lFb\\lFg\\lF[]l7&F'-Fg[l6$7%7$$\"+e.XlUFf]l$\"+#*)GQ(HFf]lFjdl7$$\"+ c_'4N'Ff]l$\"+/QW!y$Ff]lFb\\lFg\\lF[]l7&F'-Fg[l6$7%7$$\"+-O@ZWFf]l$\"+ 'yS;M$Ff]lFjdl7$$\"+#e\"G$o'Ff]l$\"+v2kTLFf]lFb\\lFg\\lF[]l-%(SCALINGG 6#%,CONSTRAINEDG" 1 2 0 1 0 2 9 1 4 1 1.000000 44.000000 46.000000 0 } }}}}{EXCHG {PARA 289 "" 0 "" {TEXT 261 9 "Problem: " }{TEXT -1 32 " \+ Roll a polyhedron up a hill." }}{PARA 289 "" 0 "" {TEXT 262 8 "Proble m:" }{TEXT -1 68 " Redo this completely using words from the plotto ols package. " }}}}{MARK "1 1 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 }