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1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Bold S mall" -1 10 "Times" 1 1 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Page Num ber" -1 33 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Maple Co mment" -1 21 "Courier" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Capt ion Reference" -1 207 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 } {CSTYLE "2D Inert Output" -1 208 "Times" 1 12 144 144 144 1 2 2 2 2 1 2 0 0 0 1 }{CSTYLE "2D Comment" -1 18 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "Times" 1 12 0 128 128 1 2 2 1 2 2 2 0 0 0 1 }{CSTYLE "Prompt" -1 1 "Courier" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Text" -1 209 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Bold" -1 39 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 } {CSTYLE "Caption Text" -1 210 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{PSTYLE "" -1 215 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }} {SECT 0 {PARA 215 "" 0 "" {TEXT 209 64 "MAPLE WORKSHEET #6: 3d Plots , Spacecurves and Parametric Plots." }}{SECT 1 {PARA 3 "" 0 "" {TEXT 211 39 "Load the plots and plottools packages:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "with(plots): with(plottools):" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 211 105 "Ordinary 3d plots of functions with op tions. The color specifications are given by hue, RGB and by zhue:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "plot3d(\n" }{MPLTEXT 1 0 27 " x^2-y^2, x=-2..2, y=-2..2,\n" }{MPLTEXT 1 0 75 "grid=[40,40], scaling= constrained, style=patchnogrid, orientation=[66,32],\n" }{MPLTEXT 1 0 15 "color=x^2+y^2,\n" }{MPLTEXT 1 0 21 "orientation=[66,32],\n" } {MPLTEXT 1 0 45 "titlefont=[TIMES,ROMAN,18],title=`Saddle I`);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "plot3d(\n" }{MPLTEXT 1 0 27 " x^2-y^2, x=-2..2, y=-2..2,\n" }{MPLTEXT 1 0 54 "grid=[60,60], scaling= constrained, style=patchnogrid,\n" }{MPLTEXT 1 0 40 "color=[ sin(x*y), cos(x*y), tan(x*y) ],\n" }{MPLTEXT 1 0 20 "orientation=[90,0],\n" } {MPLTEXT 1 0 47 "titlefont=[TIMES,ROMAN,18], title=`Saddle II`);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "plot3d(\n" }{MPLTEXT 1 0 25 " x^2-y^2,x=-2..2,y=-2..2,\n" }{MPLTEXT 1 0 68 "grid=[40,40], scaling=co nstrained, style=patchnogrid, shading=zhue,\n" }{MPLTEXT 1 0 21 "orien tation=[66,32],\n" }{MPLTEXT 1 0 47 "titlefont=[TIMES,ROMAN,18],title= `Saddle III`);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "plot3d(\n" }{MPLTEXT 1 0 33 "y^2-x^3+3*x-2, x=-2..2, y=-2..2,\n" }{MPLTEXT 1 0 48 "grid=[50,50], style=patchcontour, shading=zhue,\n" }{MPLTEXT 1 0 23 "orientation=[-109,22],\n" }{MPLTEXT 1 0 53 "titlefont=[TIMES,ROMAN ,18], title=`Elliptic Curves`);" }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 212 152 "Another way to create the last figure is to use contourplot. Noti ce that we defined z as a function and we do not need to display the x and y variables." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "z:=(x,y )->y^2-x^3+3*x-2:\n" }{MPLTEXT 1 0 13 "contourplot(\n" }{MPLTEXT 1 0 17 "z, -2..2, -2..2,\n" }{MPLTEXT 1 0 89 "grid=[30,30], contours=[-1,- 1/2,-1/4,0,1/4,1/2,1], filled=false, thickness=1, axes=none);" }}}} {SECT 0 {PARA 0 "" 0 "" {TEXT 212 72 "Here is an example of the real p art of the complex function colored by \n" }{TEXT 212 20 "the imaginar y part: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "plot3d(\n" } {MPLTEXT 1 0 35 "exp(x)*cos(y), x=-1..2, y=0..4*Pi,\n" }{MPLTEXT 1 0 33 "grid=[30,50], style=patchnogrid,\n" }{MPLTEXT 1 0 22 "color=exp(x) *sin(y) );" }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 211 12 "Spacecurves:" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "s:=\{\}:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "for n from 1 by 1 to 4 do \n" }{MPLTEXT 1 0 14 "s:=s union \{ \n" }{MPLTEXT 1 0 51 "[cos(t), cos(n*Pi/12)*sin(t) , sin(n*Pi/12)*sin(t)]\n" }{MPLTEXT 1 0 1 "\}" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "f:= \+ transform((x,y,z) -> [x/(1-z),y/(1-z),0]):" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 38 "a:=spacecurve(s, t=0..2*Pi,color=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "display3d(a, \n" }{MPLTEXT 1 0 34 " scaling=constrained, thickness=2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "b:=f(spacecurve(s, t=0..2*Pi, \n" }{MPLTEXT 1 0 28 "c olor=blue, numpoints=100)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "display3d(\{a,b\},\n" }{MPLTEXT 1 0 34 "scaling=constrained, thick ness=2);" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 211 32 "Combining plots an d spacecurves:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "f:=(x,y)-> cos(x^2+y^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$I\"xG6\"I\"yGF%F%6$ I)operatorGF%I&arrowGF%F%-I$cosG6$%*protectedGI(_syslibGF%6#,&*$9$\"\" #\"\"\"*$9%F3F4F%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "gr aph := plot3d(\n" }{MPLTEXT 1 0 42 "f-0.05, -Pi^2/8..Pi^2/8, -Pi^2/8.. Pi^2/8,\n" }{MPLTEXT 1 0 55 "grid=[50,50],style=patchnogrid,axes=none, shading=zhue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "x0:=Pi^2/ 22:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "y0:=Pi^2/22:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "z0:=f(x0,y0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I$cosG6$%*protectedGI(_syslibG6\"6#,$*$I#PiGF%\"\" %#\"\"\"\"$U#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "a:=D[1](f) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$I\"xG6\"I\"yGF%F%6$I)operatorGF %I&arrowGF%F%,$*&-I$sinG6$%*protectedGI(_syslibGF%6#,&*$9$\"\"#\"\"\"* $9%F5F6F6F4F6!\"#F%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 " b:=D[2](f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$I\"xG6\"I\"yGF%F%6$I) operatorGF%I&arrowGF%F%,$*&-I$sinG6$%*protectedGI(_syslibGF%6#,&*$9$\" \"#\"\"\"*$9%F5F6F6F8F6!\"#F%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "s:=a(x0,y0);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*&-I$ sinG6$%*protectedGI(_syslibG6\"6#,$*$I#PiGF'\"\"%#\"\"\"\"$U#F0F-\"\"# #!\"\"\"#6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "t:=b(x0,y0);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*&-I$sinG6$%*protectedGI(_syslibG6 \"6#,$*$I#PiGF'\"\"%#\"\"\"\"$U#F0F-\"\"##!\"\"\"#6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "tangentplane:=plot3d(\n" }{MPLTEXT 1 0 58 "s*(x-x0)+t*(y-y0)+z0,x=-Pi^2/8..Pi^2/8,y=-Pi^2/8..Pi^2/8,\n" } {MPLTEXT 1 0 40 "grid=[30,30],style=wireframe,axes=none):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "xcurve:=spacecurve([x,y0,f(x,y0),x= -Pi^2/8..Pi^2/8],color=red,thickness=4):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "ycurve:=spacecurve([x0,y,f(x0,y),y=-Pi^2/8..Pi^2/8],c olor=blue,thickness=4):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 " xline:=spacecurve([x,y0,z0+s*(x-x0),x=-Pi^2/8..Pi^2/8],color=red,thick ness=4):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "yline:=spacecur ve([x0,y,z0+t*(y-y0),y=-Pi^2/8..Pi^2/8],color=blue,thickness=4):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "display(\{ graph, tangentpla ne, xcurve, ycurve, xline, yline \},\n" }{MPLTEXT 1 0 55 "scaling=cons trained, orientation=[122,43], shading=XY,\n" }{MPLTEXT 1 0 57 "titlef ont=[TIMES,ROMAN,18], title=`Partial Derivatives`);" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 211 17 "Parametric plots:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "plot3d(\n" }{MPLTEXT 1 0 68 "[ cos(u)*cos(v), si n(u)*cos(v), sin(v) ], u=0..2*Pi, v=-Pi/4..Pi/4,\n" }{MPLTEXT 1 0 75 " grid=[40,30], style=patchnogrid, orientation=[45,38], scaling=constrai ned,\n" }{MPLTEXT 1 0 23 "shading=zhue, color=u,\n" }{MPLTEXT 1 0 54 " titlefont=[TIMES,ROMAN,18], title=`Spherical Belt I`);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "plot3d(\n" }{MPLTEXT 1 0 68 "[ cos(u )*cos(v), sin(u)*cos(v), sin(v) ], u=0..2*Pi, v=-Pi/4..Pi/4,\n" } {MPLTEXT 1 0 75 "grid=[40,30], style=patchnogrid, orientation=[45,38], scaling=constrained,\n" }{MPLTEXT 1 0 23 "shading=zhue, color=v,\n" } {MPLTEXT 1 0 55 "titlefont=[TIMES,ROMAN,18], title=`Spherical Belt II` );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "display3d(\{\n" } {MPLTEXT 1 0 5 "seq(\n" }{MPLTEXT 1 0 50 "plot3d( [ cos(u)*cos(v), sin (u)*cos(v), sin(v) ],\n" }{MPLTEXT 1 0 43 "u=2*k*Pi/10..(2*k+1)*Pi/10, v=-Pi/2..Pi/2,\n" }{MPLTEXT 1 0 43 "grid=[10,30], style=patchnogrid, \+ color=u),\n" }{MPLTEXT 1 0 10 "k=0..9)\},\n" }{MPLTEXT 1 0 56 "orienta tion=[46,47], scaling=constrained, shading=zhue,\n" }{MPLTEXT 1 0 60 " titlefont=[TIMES,ROMAN,18], title=`Meridians of Longitude`);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "display3d(\{\n" }{MPLTEXT 1 0 5 "seq(\n" }{MPLTEXT 1 0 50 "plot3d( [ cos(u)*cos(v), sin(u)*cos(v), sin(v) ],\n" }{MPLTEXT 1 0 39 "u=0..2*Pi, v=2*k*Pi/20..(2*k+1)*Pi/20, \n" }{MPLTEXT 1 0 43 "grid=[40,10], style=patchnogrid, color=u),\n" } {MPLTEXT 1 0 11 "k=-5..4)\},\n" }{MPLTEXT 1 0 56 "orientation=[46,47], scaling=constrained, shading=zhue,\n" }{MPLTEXT 1 0 59 "titlefont=[TI MES,ROMAN,18], title=`Parallels of Latitude`);" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 211 5 "Tori:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 " tor:=(R,r,c1,c2,c3,a,b,c,d)->plot3d(\n" }{MPLTEXT 1 0 72 "[ cos(u)*(R+ r*cos(v)), sin(u)*(R+r*cos(v)), r*sin(v) ], u=a..b, v=c..d,\n" } {MPLTEXT 1 0 69 "style=patch, shading=zhue, grid=[60,30], color=COLOR( RGB,c1,c2,c3) ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "display (\{\n" }{MPLTEXT 1 0 37 "tor(3,1.5,1,0,1,0,2*Pi,Pi/2,3*Pi/2),\n" } {MPLTEXT 1 0 28 "tor(3,1,1,1,0,0,Pi,0,2*Pi),\n" }{MPLTEXT 1 0 34 "tor( 3,0.5,0,1,1,0,2*Pi,0,2*Pi) \},\n" }{MPLTEXT 1 0 43 "scaling=constraine d, orientation=[-87,73],\n" }{MPLTEXT 1 0 51 "titlefont=[TIMES,ROMAN,1 8], title=`Clifford Tori`);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "plot3d([u*co s(v),u*sin(v),v/2], u=0..3, v=-Pi..4*Pi,\n" }{MPLTEXT 1 0 74 "scaling= constrained,grid=[10,100],style=patch,orientation=[9,46],color=u,\n" } {MPLTEXT 1 0 53 "titlefont=[TIMES,ROMAN,18],title=`Spiral Staircase`); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "plot3d(\n" }{MPLTEXT 1 0 68 "[ 5*cos(u)*(2+v*cos(u/2)), 5*sin(u)*(2+v*cos(u/2)), 5*v*sin(u/2) ],\n" }{MPLTEXT 1 0 19 "u=0..2*Pi,v=-1..1,\n" }{MPLTEXT 1 0 42 "grid= [50,20], style=patchnogrid, color=u,\n" }{MPLTEXT 1 0 42 "scaling=cons trained, orientation=[39,63],\n" }{MPLTEXT 1 0 49 "titlefont=[TIMES,RO MAN,18], title=`Mobius Band`);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "plot3d(\n" }{MPLTEXT 1 0 79 "[ r*cos(theta), r*sin(theta), r^(1 /2)*cos(theta/2) ], r=0..0.5, theta=0..4*Pi,\n" }{MPLTEXT 1 0 41 "grid =[10,100], style=patch, color=theta,\n" }{MPLTEXT 1 0 57 "titlefont=[T IMES,ROMAN,18], title=`Complex Square Root`);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "plot3d(\n" }{MPLTEXT 1 0 79 "[ r*cos(theta), r*si n(theta), r^(1/3)*cos(theta/3) ], r=0..0.4, theta=0..6*Pi,\n" } {MPLTEXT 1 0 60 "grid=[10,140], style=patch, orientation=[46,62], colo r=1-r,\n" }{MPLTEXT 1 0 56 "titlefont=[TIMES,ROMAN,18], title=`Complex Cubic Root`);" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 211 45 "We now creat e two models of the Klein Bottle:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "plot([sin(theta),sin(2*theta),theta=0..2*Pi],\n" }{MPLTEXT 1 0 34 "scaling=constrained, thickness=2,\n" }{MPLTEXT 1 0 52 "titlefont =[TIMES,ROMAN,18], title=`The Lemniscate`);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "kb1 := (2+cos(r/2)*sin(theta)-sin(r/2)*sin(2*theta ))*cos(r):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "kb2 := (2+cos (r/2)*sin(theta)-sin(r/2)*sin(2*theta))*sin(r):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "kb3 := sin(r/2)*sin(theta)+cos(r/2)*sin(2*thet a):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "plot3d(\n" }{MPLTEXT 1 0 43 "[kb1, kb2, kb3], r=0..2*Pi, theta=0..2*Pi,\n" }{MPLTEXT 1 0 57 "grid=[40,40], color=r, style=patch, scaling=constrained,\n" } {MPLTEXT 1 0 52 "titlefont=[TIMES,ROMAN,18], title=`Klein Bottle I`);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "with(plots):with(plottools):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "pl1:=plot3d(\n" }{MPLTEXT 1 0 52 "[ 2*sin(t), 2*cos(t), u ], t=Pi/2..5*Pi/2, u=-4..0,\n" }{MPLTEXT 1 0 48 "grid=[25,10], style=patch, scaling=constrained):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "pl2 := plot3d(\n" }{MPLTEXT 1 0 79 "[ (4+2*cos(Pi*u/14 +4*Pi/14))*sin(t), (4+2*cos(Pi*u/14 +4*Pi/14))*cos (t), u ],\n" }{MPLTEXT 1 0 26 "t=Pi/2..5*Pi/2, u=-4..10,\n" }{MPLTEXT 1 0 48 "grid=[25,30], style=patch, scaling=constrained):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "pl3 := plot3d(\n" }{MPLTEXT 1 0 81 "[ cos(t)*(5+2*cos(u))+5, 2*sin(u), sin(t)*(5+2*cos(u)) ], t=Pi/2..Pi, u=Pi..3*Pi,\n" }{MPLTEXT 1 0 48 "grid=[18,25], style=patch, scaling=co nstrained):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "pl4 := plot3 d(\n" }{MPLTEXT 1 0 61 "[ cos(t)*(5+2*cos(u))+5, 2*sin(u), sin(t)*(5+2 *cos(u))+10 ],\n" }{MPLTEXT 1 0 25 "t=-Pi/2..Pi, u=Pi..3*Pi,\n" } {MPLTEXT 1 0 48 "grid=[50,25], style=patch, scaling=constrained):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "pl5 := plot3d(\n" }{MPLTEXT 1 0 81 "[ cos(t)*(4+2*cos(u)), sin(t)*(4+2*cos(u)), 2*sin(u)-4], t=Pi. .3*Pi, u=Pi..2*Pi,\n" }{MPLTEXT 1 0 48 "grid=[25,20], style=patch, sca ling=constrained):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "displ ay3d(\n" }{MPLTEXT 1 0 25 "\{ pl1,pl2,pl3,pl4,pl5 \},\n" }{MPLTEXT 1 0 40 "orientation=[-87,-128],style=wireframe,\n" }{MPLTEXT 1 0 52 "tit lefont=[TIMES,ROMAN,18],title=`Klein Bottle II`);" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 211 46 "Combining 3d parametric plots and spacecurves:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "st1 := plot3d(\n" } {MPLTEXT 1 0 64 "[ cos(v)*cos(u), cos(v)*sin(u), sin(v) ], u=0..2*Pi, \+ v=-Pi..Pi,\n" }{MPLTEXT 1 0 43 "grid=[15,30], style=wireframe, color=b lue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "st2 := plot3d(\n" }{MPLTEXT 1 0 45 "[ v*cos(u), v*sin(u), 0], u=0..2*Pi, v=0..4,\n" } {MPLTEXT 1 0 43 "grid=[30,15], style=wireframe, color=cyan):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "st3 := spacecurve(\n" } {MPLTEXT 1 0 45 "[0, (1-t)/sqrt(2), t+(1-t)/sqrt(2)], t=0..1,\n" } {MPLTEXT 1 0 27 "thickness=5, color=yellow):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "st4 := spacecurve(\n" }{MPLTEXT 1 0 52 "[0, t/sqr t(2)+(1-t)/(sqrt(2)-1), t/sqrt(2)],t=0..1,\n" }{MPLTEXT 1 0 24 "thickn ess=5, color=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "displ ay3d(\n" }{MPLTEXT 1 0 24 "\{ st1, st2, st3, st4 \},\n" }{MPLTEXT 1 0 43 "scaling=constrained, orientation=[34,70], \n" }{MPLTEXT 1 0 61 "ti tlefont=[TIMES,ROMAN,18],title=`Stereographic Projection`);" }}}} {SECT 1 {PARA 3 "" 0 "" {TEXT 211 10 "Tubeplots:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "F := (x,y) ->sin(x):\n" }{MPLTEXT 1 0 11 "tube plot(\{\n" }{MPLTEXT 1 0 56 "[cos(t), sin(t), 0], [0, sin(t)-1, cos(t) ]\}, t=0..2*Pi,\n" }{MPLTEXT 1 0 12 "radius=1/4,\n" }{MPLTEXT 1 0 29 " numpoints=50, tubepoints=30,\n" }{MPLTEXT 1 0 43 "color=F, style=patch , orientation=[26,76],\n" }{MPLTEXT 1 0 49 "titlefont=[TIMES,ROMAN,18] , title=`Linked Tori`);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 " k := 3.0^(1/2): N := 15:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "tubeplot(\{\n" }{MPLTEXT 1 0 49 "[ 1+k*cos(t), k*sin(t), 0.3*sin(3*t) , t=0..2*Pi,\n" }{MPLTEXT 1 0 51 "radius=.3, numpoints=trunc(6.4*N), t ubepoints=N ],\n" }{MPLTEXT 1 0 56 "[ -1/2+k*cos(t), k/2+k*sin(t), 0.3 *sin(3*t), t=0..2*Pi,\n" }{MPLTEXT 1 0 51 "radius=.3, numpoints=trunc( 6.4*N), tubepoints=N ],\n" }{MPLTEXT 1 0 57 "[ -1/2+k*cos(t), -k/2+k*s in(t), 0.3*sin(3*t), t=0..2*Pi,\n" }{MPLTEXT 1 0 52 "radius=.3, numpoi nts=trunc(6.4*N), tubepoints=N] \},\n" }{MPLTEXT 1 0 59 "scaling=const rained, orientation=[63,37], color=[1,.2,.6],\n" }{MPLTEXT 1 0 53 "tit lefont=[TIMES,ROMAN,18], title=`Borromian Rings`);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "tubeplot(\n" }{MPLTEXT 1 0 38 "[ -10*cos(t) -2*cos(5*t)+15*sin(2*t), \n" }{MPLTEXT 1 0 36 "-15*cos(2*t)+10*sin(t)- 2*sin(5*t), \n" }{MPLTEXT 1 0 15 "10*cos(3*t) ],\n" }{MPLTEXT 1 0 12 " t= 0..2*Pi,\n" }{MPLTEXT 1 0 30 "numpoints=100, tubepoints=20,\n" } {MPLTEXT 1 0 43 "radius=3, style=patchnogrid, shading=zhue);" }}}} {SECT 1 {PARA 3 "" 0 "" {TEXT 211 144 "Links and knots. Recall the thr ee golden rectangles that were used to create the icosahedron. We ins cribe in each golden rectangle an ellipse.\n" }{TEXT 211 80 "Around ea ch ellipse we draw a tube using tubeplot and obtain three linked tori. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "with(plots):with(plottools):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "g:=(1+sqrt(5))/2:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "goldrec1 := polygonplot3d(\n " }{MPLTEXT 1 0 44 "[ [0,g,1], [0,-g,1], [0,-g,-1], [0,g,-1] ],\n" } {MPLTEXT 1 0 37 "style=line, color=gold, thickness=5):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "goldrec2 := polygonplot3d(\n" } {MPLTEXT 1 0 44 "[ [g,1,0], [-g,1,0], [-g,-1,0], [g,-1,0] ],\n" } {MPLTEXT 1 0 35 "style=line,color=gold,thickness=5):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "goldrec3 := polygonplot3d(\n" }{MPLTEXT 1 0 44 "[ [1,0,g], [-1,0,g], [-1,0,-g], [1,0,-g] ],\n" }{MPLTEXT 1 0 38 "style=line, color=gold, thickness=5):" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 11 "display3d(\n" }{MPLTEXT 1 0 33 "\{ goldrec1, goldre c2, goldrec3\},\n" }{MPLTEXT 1 0 42 "scaling=constrained, orientation= [49,60]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "ell1 := spacec urve([g*cos(t), sin(t), 0], t=0..2*Pi,\n" }{MPLTEXT 1 0 24 "thickness= 3, color=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "ell2 := s pacecurve([0, g*cos(t), sin(t)], t=0..2*Pi,\n" }{MPLTEXT 1 0 31 "thick ness=3, color=aquamarine):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "ell3 := spacecurve([cos(t), 0, g*sin(t)], t=0..2*Pi,\n" }{MPLTEXT 1 0 25 "thickness=3, color=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "display3d(\n" }{MPLTEXT 1 0 52 "\{ goldrec1, goldrec2 , goldrec3, ell1, ell2, ell3 \},\n" }{MPLTEXT 1 0 42 "scaling=constrai ned, orientation=[49,60]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "display3d(\n" }{MPLTEXT 1 0 22 "\{ ell1, ell2, ell3 \},\n" } {MPLTEXT 1 0 42 "scaling=constrained, orientation=[49,60]);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "tubell1 := tubeplot(\n" } {MPLTEXT 1 0 34 "[g*cos(t), sin(t), 0], t=0..2*Pi,\n" }{MPLTEXT 1 0 56 "radius=0.2, style=patchnogrid, tubepoints=16,color=red):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "tubell2 := tubeplot(\n" } {MPLTEXT 1 0 34 "[0, g*cos(t), sin(t)], t=0..2*Pi,\n" }{MPLTEXT 1 0 61 "radius=0.2,style=patchnogrid,tubepoints=16,color=aquamarine):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "tubell3 := tubeplot(\n" } {MPLTEXT 1 0 34 "[cos(t), 0, g*sin(t)], t=0..2*Pi,\n" }{MPLTEXT 1 0 57 "radius=0.2, style=patchnogrid, tubepoints=16,color=blue):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "display3d(\n" }{MPLTEXT 1 0 31 "\{ tubell1, tubell2, tubell3 \},\n" }{MPLTEXT 1 0 41 "scaling=cons trained,orientation=[49,60],\n" }{MPLTEXT 1 0 23 "lightmodel = `light2 `,\n" }{MPLTEXT 1 0 56 "titlefont=[TIMES,ROMAN,18],title=`Three Linked Tori I`);" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 211 26 "We draw curves o n a torus." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "torus0 := plot 3d(\n" }{MPLTEXT 1 0 70 "[cos(u)*(3+cos(v)), sin(u)*(3+cos(v)), sin(v) ], u=0..2*Pi, v=0..2*Pi,\n" }{MPLTEXT 1 0 44 "style=patchnogrid, grid= [50,80], shading=Z):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "wir etorus := plot3d(\n" }{MPLTEXT 1 0 70 "[cos(u)*(3+cos(v)), sin(u)*(3+c os(v)), sin(v)], u=0..2*Pi, v=0..2*Pi,\n" }{MPLTEXT 1 0 42 "style=wire frame, grid=[50,80], shading=Z):" }}}{PARA 0 "" 0 "" {TEXT 212 34 "The curves will be raised by 0.05." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "s:=1.05:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "win1 := \+ spacecurve(\n" }{MPLTEXT 1 0 65 "[cos(t)*(3+s*cos(t)), sin(t)*(3+s*cos (t)), s*sin(t)], t=0..2*Pi,\n" }{MPLTEXT 1 0 40 "numpoints=500, thickn ess=3, color=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "disp lay3d(\{ torus0, win1 \}, scaling=constrained, orientation=[25,66]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "win2 := spacecurve(\n" } {MPLTEXT 1 0 79 "[cos(t-2*Pi/3)*(3+s*cos(t)), sin(t-2*Pi/3)*(3+s*cos(t )), s*sin(t)], t=0..2*Pi,\n" }{MPLTEXT 1 0 40 "numpoints=500, thicknes s=3, color=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "win3 : = spacecurve(\n" }{MPLTEXT 1 0 79 "[cos(t-4*Pi/3)*(3+s*cos(t)), sin(t- 4*Pi/3)*(3+s*cos(t)), s*sin(t)], t=0..2*Pi,\n" }{MPLTEXT 1 0 40 "numpo ints=500, thickness=3, color=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "display3d(\n" }{MPLTEXT 1 0 31 "\{ torus0, win1, win2 , win3 \}, \n" }{MPLTEXT 1 0 66 "lightmodel = `light1`, scaling=constr ained, orientation=[113,46]);" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 211 49 "For transparency we use wireframe with fine grid." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "display3d(\n" }{MPLTEXT 1 0 33 "\{ wireto rus, win1, win2, win3 \},\n" }{MPLTEXT 1 0 41 "scaling=constrained,ori entation=[63,57]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "tub1 \+ := tubeplot(\n" }{MPLTEXT 1 0 65 "[cos(t)*(3+s*cos(t)), sin(t)*(3+s*co s(t)), s*sin(t)], t=0..2*Pi,\n" }{MPLTEXT 1 0 42 "radius=0.3, style=pa tchnogrid, color=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "t ub2 := tubeplot(\n" }{MPLTEXT 1 0 78 "[cos(t-2*Pi/3)*(3+s*cos(t)), sin (t-2*Pi/3)*(3+s*cos(t)), s*sin(t),t=0..2*Pi],\n" }{MPLTEXT 1 0 49 "rad ius=0.3, style=patchnogrid, color=aquamarine):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "tub3 := tubeplot(\n" }{MPLTEXT 1 0 78 "[cos(t-4 *Pi/3)*(3+s*cos(t)), sin(t-4*Pi/3)*(3+s*cos(t)), s*sin(t),t=0..2*Pi], \n" }{MPLTEXT 1 0 43 "radius=0.3, style=patchnogrid, color=blue):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "display3d(\n" }{MPLTEXT 1 0 22 "\{ tub1, tub2, tub3 \},\n" }{MPLTEXT 1 0 77 "lightmodel = `light1` , scaling=constrained, orientation=[113,46], shading=Z,\n" }{MPLTEXT 1 0 58 "titlefont=[TIMES,ROMAN,18], title=`Three Linked Tori II`);" }} }}{SECT 1 {PARA 3 "" 0 "" {TEXT 211 28 "We now consider torus knots." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "win:=(R,r,w1,w2)->spacecur ve(\n" }{MPLTEXT 1 0 80 "[cos(w2*t)*(R+r*cos(w1*t)), sin(w2*t)*(R+r*co s(w1*t)), r*sin(w1*t)], t=0..2*Pi,\n" }{MPLTEXT 1 0 40 "numpoints=500, thickness=3, color=navy):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "tub:=(R,r,w1,w2)->tubeplot(\n" }{MPLTEXT 1 0 70 "[cos(w2*t)*(R+r*c os(w1*t)), sin(w2*t)*(R+r*cos(w1*t)), r*sin(w1*t)], \n" }{MPLTEXT 1 0 24 "t=0..2*Pi, radius=r/2, \n" }{MPLTEXT 1 0 53 "style=patchnogrid, nu mpoints=40*w1, tubepoints=8*w2,\n" }{MPLTEXT 1 0 54 "scaling=constrain ed, orientation=[127,64], shading=Z):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "display3d(\n" }{MPLTEXT 1 0 29 "\{ wiretorus, win(3,1 ,3,2) \},\n" }{MPLTEXT 1 0 42 "scaling=constrained,orientation=[127,64 ]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "display3d(tub(3,1,3, 2), titlefont=[TIMES,ROMAN,18], title=`Torus Knot I`);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "display3d(tub(3,1,5,2), titlefont=[ TIMES,ROMAN,18], title=`Torus Knot II`);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "display3d(tub(3,1,10,1), titlefont=[TIMES,ROMAN,18], \+ title=`Torus Knot III`);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "smalltorus := tubeplot(\n" }{MPLTEXT 1 0 77 "[cos(t)*(3+(1/5)*cos(t)) , sin(t)*(3+(1/5)*cos(t)), (1/5)*sin(t)], t=0..2*Pi,\n" }{MPLTEXT 1 0 57 "radius=1/2, style=patchnogrid, grid=[50,20], color=cyan):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "display3d(\{tub(3,1,10,1),sm alltorus\},\n" }{MPLTEXT 1 0 54 "scaling=constrained, orientation=[127 ,52], shading=Z,\n" }{MPLTEXT 1 0 51 "titlefont=[TIMES,ROMAN,18], titl e=`Torus Knot IV`);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "spac ecurve([t*cos(t),t*sin(t),0.2*t],t=0..6*Pi,\n" }{MPLTEXT 1 0 26 "numpo ints=100,color=navy);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "tu beplot([t*cos(t),t*sin(t),0.1*t],t=0..6*Pi,radius=0.7*exp(0.2*t),\n" } {MPLTEXT 1 0 29 "tubepoints=40,numpoints=100);" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 211 21 "Programming in Maple:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "with(plots):with(plottools):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "torus0:=proc(R,r,c1,c2,c3,a,b,c,d) \n" }{MPLTEXT 1 0 63 "plot3d([ cos(u)*(R+r*cos(v)), sin(u)*(R+r*cos(v)), r*sin(v) ] ,\n" }{MPLTEXT 1 0 16 "u=a..b, v=c..d,\n" }{MPLTEXT 1 0 73 "style=patc h,shading=zhue, grid=[60,30], color=COLOR(RGB,c1,c2,c3) ); end:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "torus0(3, 1.5, 0, 1, 1, 0, 2 *Pi, Pi/2, 3*Pi/2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "prin t(torus0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6+I\"RG6\"I\"rGF%I#c1GF% I#c2GF%I#c3GF%I\"aGF%I\"bGF%I\"cGF%I\"dGF%F%F%F%-I'plot3dG6$%*protecte dGI(_syslibGF%6)7%*&-I$cosGF06#I\"uGF%\"\"\",&9$F:*&9%F:-F76#I\"vGF%F: F:F:*&-I$sinGF0F8F:F;F:*&F>F:-FDF@F:/F9;9)9*/FA;9+9,/I&styleGF%I&patch GF%/I(shadingGF%I%zhueGF%/I%gridGF%7$\"#g\"#I/I&colorGF%-I&COLORGF06&I $RGBGF09&9'9(F%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "Cheb yshev:=proc( n ) \n" }{MPLTEXT 1 0 14 "local p, k; \n" }{MPLTEXT 1 0 22 "p[0] := 1; p[1] := x;\n" }{MPLTEXT 1 0 33 "if n<=1 then RETURN(eva l(p)) fi;\n" }{MPLTEXT 1 0 21 "for k from 2 to n do\n" }{MPLTEXT 1 0 36 "p[k] := expand( 2*x*p[k-1]-p[k-2] )\n" }{MPLTEXT 1 0 4 "od;\n" } {MPLTEXT 1 0 16 "RETURN(eval(p))\n" }{MPLTEXT 1 0 13 "end: " } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "Chebyshev(5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"I&falseG%*protectedGE\\[l'\"\"!\"\"\"F(I\"xGF #\"\"#,&*$F)F*F*!\"\"F(\"\"$,&*$F)F.\"\"%F)!\"$\"\"&,(*$F)F3\"#;F0!#?F )F3F1,(*$F)F1\"\")F,!\")F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "a:=Chebyshev(5):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "p lot(\{ seq(a[i],i=0..5) \}, x=-1..1, thickness=3, tickmarks=[0,0]);" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "curves := proc(n)\n" } {MPLTEXT 1 0 12 "local s, k;\n" }{MPLTEXT 1 0 10 "s[0]:=\{\};\n" } {MPLTEXT 1 0 34 "if n<=0 then RETURN(eval(s) ) fi;\n" }{MPLTEXT 1 0 23 "for k from 1 to n do \n" }{MPLTEXT 1 0 22 "s[k]:=s[k-1] union \{ \+ \n" }{MPLTEXT 1 0 52 "[cos(t), cos(k*Pi/12)*sin(t), sin(k*Pi/12)*sin(t )],\n" }{MPLTEXT 1 0 85 "[ cos(t)/(1-sin(t)*sin(k*Pi/12)), (cos(k*Pi/1 2)*sin(t))/(1-sin(t)*sin(k*Pi/12)), 0] \n" }{MPLTEXT 1 0 3 "\};\n" } {MPLTEXT 1 0 4 "od;\n" }{MPLTEXT 1 0 16 "RETURN(eval(s))\n" }{MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "curves := pr oc(n)\n" }{MPLTEXT 1 0 12 "local s, k;\n" }{MPLTEXT 1 0 7 "s:=\{\};\n" }{MPLTEXT 1 0 34 "if n<=0 then RETURN(eval(s) ) fi;\n" }{MPLTEXT 1 0 23 "for k from 1 to n do \n" }{MPLTEXT 1 0 14 "s:=s union \{ \n" } {MPLTEXT 1 0 52 "[cos(t), cos(k*Pi/12)*sin(t), sin(k*Pi/12)*sin(t)],\n " }{MPLTEXT 1 0 85 "[ cos(t)/(1-sin(t)*sin(k*Pi/12)), (cos(k*Pi/12)*si n(t))/(1-sin(t)*sin(k*Pi/12)), 0] \n" }{MPLTEXT 1 0 3 "\};\n" } {MPLTEXT 1 0 4 "od;\n" }{MPLTEXT 1 0 16 "RETURN(eval(s))\n" }{MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "display3d(\n " }{MPLTEXT 1 0 75 "spacecurve(curves(5), t=0..2*Pi, thickness=2, colo r=blue, numpoints=200), \n" }{MPLTEXT 1 0 41 "scaling=constrained, ori entation=[1,70]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "f := p roc(x) if x<0 then -1 else 1 fi; end; " }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#I\"xG6\"F%F%F%@%29$\"\"!!\"\"\"\"\"F%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "plot(f,-1..1, \n" }{MPLTEXT 1 0 41 "thickness= 2, color=red, tickmarks=[0,0]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "g1 := proc(u,v,t) if u " 0 "" {MPLTEXT 1 0 44 "animate3d([g1,g2,g3],0..2*Pi,0..10,0..2*Pi, \n" }{MPLTEXT 1 0 46 "style=patchnogrid, grid=[40,50],shading=zhue,\n" }{MPLTEXT 1 0 62 "titlefont=[TIMES,ROMAN,18], title=`Rolling out the \+ Cylinder`);" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 211 9 "Seashells" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "g:=(1+sqrt(5))/2:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "F:=(x,y)->sin(x):with(plots) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "tubeplot([g^(-t)*cos(t ),g^(-t)*sin(t),1.001^(3*t)],\n" }{MPLTEXT 1 0 39 "t=0..4*Pi,radius=(. 75)^t,numpoints=50,\n" }{MPLTEXT 1 0 56 "tubepoints=30,color=F,style=p atch,orientation=[-51,31],\n" }{MPLTEXT 1 0 21 "scaling=constrained);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "tubeplot([(1.17^(-t))*co s(t),(1.17^(-t))*sin(t),\n" }{MPLTEXT 1 0 21 "((t+3)/(t+2.04))^t],\n" }{MPLTEXT 1 0 35 "t=0..8*Pi,radius=1.5^(-(.5+.4*t)),\n" }{MPLTEXT 1 0 27 "numpoints=60,tubepoints=40," }{MPLTEXT 1 0 34 "style=patch,orienta tion=[-51,31],\n" }{MPLTEXT 1 0 21 "scaling=constrained);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "tubeplot([(-k/3)*cos(k),(-k/3)*sin( k),.0062*k],\n" }{MPLTEXT 1 0 30 "k=0..8*Pi,radius=.125*(k/4)+1," } {MPLTEXT 1 0 27 "numpoints=75,tubepoints=40," }{MPLTEXT 1 0 34 "style= patch,orientation=[-51,31],\n" }{MPLTEXT 1 0 21 "scaling=constrained); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 15 10 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }