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{TEXT 239 49 "Warning, the name c hangecoords has been redefined" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 207 34 "#sphere: radius=2r, center=(0,0,0)" }{MPLTEXT 1 0 2 "\n" } {MPLTEXT 1 0 8 "r:=10:\n" }{MPLTEXT 1 0 6 "sphere" }{MPLTEXT 1 0 53 ": =[2*r*cos(v)*cos(u),2*r*cos(v)*sin(u),2*r*sin(v)]:\n" }{MPLTEXT 1 0 8 "sphere_s" }{MPLTEXT 1 0 68 ":=plot3d(sphere,u=0..2*Pi,v=-Pi/2..Pi/2,g rid=[36,36],color=magenta):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "display(sphere_s,axes=framed,style=hidden,orientation=[30,45],scal ing=CONSTRAINED);" }}{PARA 13 "" 1 "" {TEXT 240 0 "" }{TEXT 240 0 "" } }}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 208 28 "#vertical cylinder: ra dius=r" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 4 "vcyl" }{MPLTEXT 1 0 30 " :=[r*(cos(u)+1),r*sin(u),v]:\n" }{MPLTEXT 1 0 6 "vcyl_s" }{MPLTEXT 1 0 58 ":=plot3d(vcyl,u=0..2*Pi,v=-30..30,grid=[36,2],color=blue):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "display(vcyl_s,axes=framed,s tyle=hidden,orientation=[30,45],scaling=CONSTRAINED);" }}{PARA 13 "" 1 "" {TEXT 240 0 "" }{TEXT 240 0 "" }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 209 60 "#the line of intersection of the sphere and the cyl inder: \n" }{MPLTEXT 1 210 48 "#this is a Viviani curve, which has the equation" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 94 "viviani:=spacecurve( [2*r*cos(t)*cos(t),2*r*cos(t)*sin(t),2*r*sin(t)],t=0..2*Pi,color=green ):\n" }{MPLTEXT 1 0 97 "display([sphere_s,vcyl_s,viviani],axes=NONE,st yle=patch,orientation=[30,45],scaling=CONSTRAINED);" }}{PARA 13 "" 1 " " {TEXT 240 0 "" }{TEXT 240 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 136 "display([sphere_s,vcyl_s,viviani],axes=NONE,style=PATCHNOGRID ,orientation=[30,45],scaling=CONSTRAINED,lightmodel='light4',shading=Z HUE);" }}{PARA 13 "" 1 "" {TEXT 240 0 "" }{TEXT 240 0 "" }}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 211 30 "#horizontal cylinder: radius=r" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 7 "hcyl:=[" }{MPLTEXT 1 0 27 "r*(c os(u)+1),v,r*sin(u)]:\n" }{MPLTEXT 1 0 6 "hcyl_s" }{MPLTEXT 1 0 60 ":= plot3d(hcyl,u=0..2*Pi,v=-30..30,grid=[36,2],color=grey):\n" }{MPLTEXT 1 0 81 "display(hcyl_s,axes=framed,style=hidden,orientation=[30,45],sc aling=CONSTRAINED);" }}{PARA 13 "" 1 "" {TEXT 240 0 "" }{TEXT 240 0 "" }}}{EXCHG {PARA 210 "> " 0 "" {MPLTEXT 1 212 46 "#the line of interse ction of the two cylinders" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 103 "di splay([vcyl_s,hcyl_s],orientation=[30,45],style=PATCHNOGRID,lightmodel ='light2',scaling=CONSTRAINED);" }}{PARA 13 "" 1 "" {TEXT 240 0 "" } {TEXT 240 0 "" }}}{EXCHG {PARA 211 "> " 0 "" {MPLTEXT 1 213 41 "#we de fine a larger horizontal cylinder\n" }{MPLTEXT 1 214 54 "#the radius o f the horizontal cylinder is m*r, 1 " 0 "" {MPLTEXT 1 217 65 "#torus: radius of the directrix=a, radius of the meridian=b, a>b " }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 420 "a:=12: b:=8:\ndirectrix:=spac ecurve([a*cos(u),a*sin(u),0],u=0..2*Pi,color=blue):\nmeridian:=spacecu rve([a+b*cos(v),0,b*sin(v)],v=0..2*Pi,color=red):\ncenterm:=spacecurve ([a+0.3*cos(v),0,0.3*sin(v)],v=0..2*Pi,color=red):\ncenterd:=spacecurv e([0.2*cos(u),0.2*sin(u),0],u=0..2*Pi,color=blue):\ntorus:=[(a+b*cos(v ))*cos(u),(a+b*cos(v))*sin(u),b*sin(v)]:\ntorus_s:=plot3d(torus,u=0..2 *Pi,v=0..2*Pi,grid=[36,36],color=pink):\n" }{MPLTEXT 1 0 232 "display( [directrix,meridian,centerm,centerd],axes=framed,style=hidden,orientat ion=[30,45],scaling=CONSTRAINED);\ndisplay([torus_s,directrix,meridian ,centerm,centerd],axes=framed,style=hidden,orientation=[30,45],scaling =CONSTRAINED);" }}{PARA 13 "" 1 "" {TEXT 240 0 "" }{TEXT 240 0 "" }} {PARA 13 "" 1 "" {TEXT 240 0 "" }{TEXT 240 0 "" }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 218 64 "#the line of intersection of the torus and \+ the vertical cylinder" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 104 "display ([torus_s,vcyl_s],orientation=[30,45],style=PATCHNOGRID,lightmodel='li ght2',scaling=CONSTRAINED);" }}{PARA 13 "" 1 "" {TEXT 240 0 "" }{TEXT 240 0 "" }}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 219 15 "#a larger to rus" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 17 "an:=15: bn:=10:\n" } {MPLTEXT 1 0 64 "torn:=[(an+bn*cos(v))*cos(u),(an+bn*cos(v))*sin(u),bn *sin(v)]:\n" }{MPLTEXT 1 0 68 "torn_s:=plot3d(torn,u=0..2*Pi,v=0..2*Pi ,grid=[36,36],color=brown):\n" }{MPLTEXT 1 0 81 "display(torn_s,axes=f ramed,style=hidden,orientation=[30,45],scaling=CONSTRAINED);" }}{PARA 13 "" 1 "" {TEXT 240 0 "" }{TEXT 240 0 "" }}}{EXCHG {PARA 215 "> " 0 " " {MPLTEXT 1 220 45 "#the larger torus and the horizontal cylinder" } {MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 221 45 "#they have 2 common horizontal tangent planes" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 105 "display([torn _s,hcyl_s],orientation=[37,24],style=PATCHNOGRID,lightmodel='light2',s caling=CONSTRAINED);\n" }}{PARA 13 "" 1 "" {TEXT 240 0 "" }{TEXT 240 0 "" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 222 52 "#the larger toru s and the larger horizontal cylinder" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 104 "display([torn_s,mcyl_s],orientation=[105,60],style=PATCHNOGRI D,lightmodel='light2',scaling=CONSTRAINED);" }}{PARA 13 "" 1 "" {TEXT 240 0 "" }{TEXT 240 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 223 47 "#the original torus and the horizontal cylinder" }{MPLTEXT 1 0 107 "\ndisplay([torus_s,hcyl_s],orientation=[119,72],style=PATCHNOGRID ,lightmodel='light3',scaling=CONSTRAINED);" }}{PARA 13 "" 1 "" {TEXT 240 0 "" }{TEXT 240 0 "" }}}{EXCHG {PARA 217 "> " 0 "" {MPLTEXT 1 224 38 "#a plane is defined by a paralelogram " }{MPLTEXT 1 225 2 "\n" } {MPLTEXT 1 226 74 "#the plane passes through the origin and is given b y the vectors v1 and v2" }{MPLTEXT 1 0 1 " " }{MPLTEXT 1 0 2 "\n" } {MPLTEXT 1 0 45 "v1:=[sqrt(an*an-bn*bn),0,bn]: v2:=[0,bn,0]:\n" } {MPLTEXT 1 0 5 "plane" }{MPLTEXT 1 0 34 ":=evalm(s*v1+t*v2):\nA:=[0,0, 0]:\n" }{MPLTEXT 1 0 247 "plane_s:=plot3d(plane,s=-2..2,t=-3..3,grid=[ 2,2],color=khaki):\nlia:=spacecurve(evalm(A+t*v1),t=0..1,color=blue): \nlib:=spacecurve(evalm(A+t*v2),t=0..1,color=green):\ndisplay([plane_s ,lia,lib],axes=framed,orientation=[108,56],scaling=CONSTRAINED);" }} {PARA 13 "" 1 "" {TEXT 240 0 "" }{TEXT 240 0 "" }}}{EXCHG {PARA 218 "> " 0 "" {MPLTEXT 1 227 90 "#this plane is a special tangent plane of t he torus\n#it intersects the torus in 2 circles" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 117 "display([torn_s,plane_s],axes=framed,style=PATCHNOG RID,orientation=[108,56],lightmodel='light3',scaling=CONSTRAINED);" }} {PARA 13 "" 1 "" {TEXT 240 0 "" }{TEXT 240 0 "" }}}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 228 74 "#a paralel translation of the plane is given by the vector d*vec, -10 " 0 "" {MPLTEXT 1 0 72 "pl anetr_s:=plot3d(planetr,s=-2-d..2-d,t=-3..3,grid=[2,2],color=khaki):\n " }{MPLTEXT 1 0 119 "display([torn_s,planetr_s],axes=framed,style=PATC HNOGRID,orientation=[108,56],lightmodel='light3',scaling=CONSTRAINED); " }}{PARA 13 "" 1 "" {TEXT 240 0 "" }{TEXT 240 0 "" }}}{EXCHG {PARA 220 "> " 0 "" {MPLTEXT 1 229 36 "#the vertical cylinder and the plane" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 117 "display([plane_s,vcyl_s],axes =framed,style=PATCHNOGRID,orientation=[120,50],lightmodel='light2',sca ling=CONSTRAINED);" }}{PARA 13 "" 1 "" {TEXT 240 0 "" }{TEXT 240 0 "" }}}{EXCHG {PARA 221 "> " 0 "" {MPLTEXT 1 230 28 "#the torus with varyi ng data" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 143 "animate3d([(q*an+bn*c os(v))*cos(u),(q*an+bn*cos(v))*sin(u),bn*sin(v)],u=0..2*Pi,v=0..2*Pi,q =bn/an..5*bn/an,grid=[24,24],scaling=CONSTRAINED);\n" }{MPLTEXT 1 231 50 "#click onto the picture and call animate then play" }}{PARA 13 "" 1 "" {TEXT 240 0 "" }{TEXT 240 0 "" }}{PARA 222 "> " 0 "" {MPLTEXT 1 241 0 "" }}}{EXCHG {PARA 223 "> " 0 "" {MPLTEXT 1 232 48 "#animation w ith the changing torus and the plane" }{MPLTEXT 1 0 229 "\nplane_b:=pl ot3d(plane,s=-6..6,t=-6..6,grid=[2,2],color=yellow):\ndisplay([animate 3d([(q*an+bn*cos(v))*cos(u),(q*an+bn*cos(v))*sin(u),bn*sin(v)],u=0..2* Pi,v=0..2*Pi,q=bn/an..5*bn/an,grid=[24,24]),plane_b],scaling=CONSTRAIN ED);\n" }}}{EXCHG {PARA 224 "> " 0 "" {MPLTEXT 1 233 46 "#animation wi th the moving plane and the torus" }{MPLTEXT 1 234 95 "\ndisplay([anim ate3d(evalm(plane+f*vek),s=-12..12,t=-3..3,f=-10..10,grid=[2,2],color= yellow),\n" }{MPLTEXT 1 235 9 " torn_s]," }{MPLTEXT 1 0 31 "scaling=CO NSTRANED, axes=none);" }}{PARA 13 "" 1 "" {TEXT 240 0 "" }}}{EXCHG {PARA 13 "" 1 "" {TEXT 240 0 "" }{TEXT 240 0 "" }}}{EXCHG {PARA 225 "> " 0 "" {MPLTEXT 1 236 55 "#Generate pictures by choosing other pairs \+ of surfaces!" }{MPLTEXT 1 237 2 "\n" }{MPLTEXT 1 238 38 "#Have a lot o f fun!\n#M. Szilvasi-Nagy" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {SECT 0 {PARA 3 "" 0 "" {TEXT 242 0 "" }}}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }