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1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE " Heading 3" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 10 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 }} {SECT 0 {SECT 0 {PARA 3 "" 0 "" {TEXT 256 31 "Egy \351rdekes geometria i alakzat:" }{TEXT -1 10 " az Oloid\n" }{TEXT 314 13 "K\351sz\355tette : " }{TEXT 316 17 "dr Szilassi Lajos" }{TEXT 317 1 "\n" }{TEXT 318 32 "HANS DIRB\326CK \351s HELMUTH STACHEL:" }{TEXT 313 30 " The Develo pment of the Oloid " }{TEXT 315 65 "(Journal for Geometry and Graphics Volume 1 (1977) No2. 105-118) " }{TEXT 319 37 " c. cikk felhaszn\341l \341s\341val. \nE-mail: " }{TEXT 320 26 "szilassi@jgytf.u-szeged.hu" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "restart: with(plots):with( plottools):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changeco ords has been redefined\n" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 9 "A fe ladat" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 193 "Adott a s\355kban k\351t ugyanakkora - egys\351gnyi - sugar\372 k\366r, melyek s\355kjai mer \365legesek egym\341sra, \351s k\366z\351ppontjaik t\341vols\341ga eg ys\351gnyi. \315gy mindk\351t k\366rvonal illeszkedik a m\341sik k\366 r k\366z\351ppontj\341ra. " }{TEXT 257 5 "Oloid" }{TEXT -1 23 "nak nev ezz\374k a k\351t k\366r " }{TEXT 258 14 "konvex burk\341t:" }{TEXT -1 80 " azt a legkisebb konvex t\351rgeometriai alakzatot, amely mindk \351t k\366rt tartalmazza." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "Feladatunk az, hogy ennek az alakzatnak a tulajdo ns\341gait megismerj\374k. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 116 "Az oloidot meghat\341roz\363 k\366r\366ket \+ \372gy helyezz\374k koordin\341tarendszerbe, hogy az alakzat legyen sz immetrikus az orig\363ra.\nA" }{TEXT 284 3 " ka" }{TEXT -1 7 " k\366r \+ az" }{TEXT 285 3 " xy" }{TEXT -1 11 " s\355kban, a " }{TEXT 287 2 "kb " }{TEXT -1 4 " az " }{TEXT 286 2 "xz" }{TEXT -1 24 " s\355kban helyez kedj\351k el\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {SECT 0 {PARA 5 "" 0 "" {TEXT -1 7 "rajz1: " }{TEXT 288 25 "bemutatjuk a k\351t k\366rlapot" }{TEXT 332 0 "" }{TEXT 333 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 334 3 "n: " }{TEXT -1 22 "a beoszt\341s finoms\341ga; \n" }{TEXT 341 1 "a" }{TEXT -1 4 " \351s " }{TEXT 335 1 "b" }{TEXT -1 26 " a k\351t k\366rvonal pontjai, " }{TEXT 336 2 "CA" }{TEXT -1 4 " \+ \351s " }{TEXT 337 2 "CB" }{TEXT -1 14 " a k\351t k\366rlap " }{TEXT 338 2 "RV" }{TEXT 339 0 "" }{TEXT 340 0 "" }{TEXT -1 20 " a k\366rvona lak rajza." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 263 "n:=30: \na :=[seq( [ 1/2+cos(i*2*Pi/n),sin(i*2*Pi/n),0],i=1..n)]:\nb:=[seq( [-1/2 +cos(i*2*Pi/n),0,sin(i*2*Pi/n)],i=1..n)]:\nCA:=polygonplot3d(a, color= yellow): \nCB:=polygonplot3d(b,color=cyan): \nRV:=polygonplot3d([a ,b],linestyle=16,style=wireframe,color=blue):\n" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 40 "display3d(CA,CB,RV,scaling=constrained);" }} {PARA 13 "" 1 "" {GLPLOT3D 400 300 300 {PLOTDATA 3 "6&-%)POLYGONSG6$7@ 7%$\"+,w9y9!\"*$\"+4p6z?!#5$\"\"!F/7%$\"+eaa89F*$\"+IkOnSF-F.7%$\"+%*p ,48F*$\"+CD&y(eF-F.7%$\"+118p6F*$\"+c#[9V(F-F.7%$\"\"\"F/$\"+SSDg')F-F .7%$\"+Q*p,4)F-$\"+l^c5&*F-F.7%$\"+HYGXgF-$\"+a*=_%**F-F.7%$\"+r`raRF- FLF.7%$\"+i+$)4>F-FGF.7%F.FBF.7%$!+igI\"p\"F-F=F.7%$!+V*p,4$F-F8F.7%$! +xXXNTF-F3F.7%$!+2gZ\"y%F-F+F.7%$!+++++]F-F.F.7%Fin$!+4p6z?F-F.7%Ffn$! +IkOnSF-F.7%FY$!+CD&y(eF-F.7%FV$!+c#[9V(F-F.7%F.$!+SSDg')F-F.7%FR$!+l^ c5&*F-F.7%FO$!+a*=_%**F-F.7%FJFapF.7%FEF^pF.7%F@F[pF.7%F;FhoF.7%F6FeoF .7%F1FboF.7%F(F_oF.7%$\"+++++:F*F.F.-%'COLOURG6&%$RGBG$\"*++++\"!\")Fa qF.-F$6$7@7%$\"+2gZ\"y%F-F.F+7%$\"+xXXNTF-F.F37%$\"+V*p,4$F-F.F87%$\"+ igI\"p\"F-F.F=7%F.F.FB7%$!+i+$)4>F-F.FG7%$!+r`raRF-F.FL7%$!+HYGXgF-F.F L7%$!+Q*p,4)F-F.FG7%$!\"\"F/F.FB7%$!+118p6F*F.F=7%$!+%*p,48F*F.F87%$!+ eaa89F*F.F37%$!+,w9y9F*F.F+7%$!+++++:F*F.F.7%F]tF.F_o7%FjsF.Fbo7%FgsF. Feo7%FdsF.Fho7%FasF.F[p7%F^sF.F^p7%F[sF.Fap7%FhrF.Fap7%FerF.F^p7%F.F.F [p7%FarF.Fho7%F^rF.Feo7%F[rF.Fbo7%FhqF.F_o7%$\"+++++]F-F.F.-F^q6&F`qF. FaqFaq-F$6'F&Ffq-F^q6&F`qF.F.Faq-%*LINESTYLEG6#\"#;-%&STYLEG6#%%LINEG- %(SCALINGG6#%,CONSTRAINEDG" 1 2 0 1 10 0 2 1 1 1 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }}}}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 16 "Az oloid alkot\363i" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "Ha egy " }{TEXT 289 2 "G " }{TEXT -1 133 "geometriai alakzathoz \+ \351rint\365s\355kokat illeszt\374nk az \366sszes lehets\351ges m\363d on, majd vessz\374k az ezek, mint hat\341sr\355kok \341ltal meghat\341 rozott " }{TEXT 261 1 "G" }{TEXT -1 60 "-t tartalmaz\363 f\351lterek k \366z\366s r\351sz\351t, akkor ezzel megadtuk a " }{TEXT 290 2 "G " } {TEXT -1 8 "alakzat " }{TEXT 291 13 "konvex burk\341t" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 33 "Eset\374nkben - mivel az alakzat az " }{TEXT 263 2 " x" }{TEXT -1 51 " tengely k\366r\374li der\351ksz\366 g\373 forgat\341sra, valamint az " }{TEXT 264 2 "yz" }{TEXT -1 83 " s \355kra szimmetrikus, elegend\365 azokat az \351rint\365s\355kokat viz sg\341lni, melyek metszik az " }{TEXT 265 1 "x" }{TEXT -1 79 " tengely pozit\355v fel\351t. (Ezekhez hozz\341 kell m\351g venn\374nk azt az \+ egyet, amely az " }{TEXT 266 1 "x" }{TEXT -1 34 " tengellyel p\341rhuz amos.) \nLegyen " }{TEXT 259 11 "P:= [m,0,0]" }{TEXT -1 34 " az x ten gely tetsz\365leges pontja, " }{TEXT 260 1 "M" }{TEXT -1 8 "-b\365l az " }{TEXT 269 1 "a" }{TEXT -1 46 " k\366rh\366z h\372zott (egyik) \351 rint\365 \351rint\351si pontja " }{TEXT 267 12 "A:=[xb,yb,0]" }{TEXT -1 4 ", a " }{TEXT 268 1 "b" }{TEXT -1 22 " k\366rh\366z h\372zott \+ \351rint\365 " }{TEXT 270 13 "B:=[xb,0,zb]." }}{PARA 0 "" 0 "" {TEXT -1 16 "Fusson v\351gig az " }{TEXT 262 1 "A" }{TEXT -1 29 " pont az a \+ k\366r negyedk\366r\351n: " }{TEXT 271 11 "t=[0..Pi/2)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 342 15 "ax, ay, bx, by " }{TEXT -1 43 "a k\366rvonalakat le\355r\363 param\351teres f\374ggv\351nyek" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 " ax:= t -> 1/2+cos(t): \n ay:= sin:\n bx:= t-> \+ 1/2*(cos(t)-1)/(cos(t)+1):\n bz:= t-> sqrt((2*cos(t)+1))/(cos(t)+1):\n " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "Az" }{TEXT 343 2 " m" }{TEXT -1 33 " param\351ter \351rt\351ke is kifejezhet\365 " }{TEXT 272 1 "t " }{TEXT -1 24 "-vel, kihaszn\341lva, hogy " }{TEXT 292 11 "OM=1/cos(t )" }{TEXT -1 7 ", ahol " }{TEXT 293 13 "O:=[1/2,0,0] " }{TEXT -1 3 "az " }{TEXT 294 1 "a" }{TEXT -1 17 " k\366r k\366z\351ppontja:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "mx:= t -> 1/2+1/cos(t):" }}} {SECT 0 {PARA 5 "" 0 "" {TEXT -1 5 "Rajz2" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "Ez a rajz csak akkor \341ll\355that\363 el\365, ha az R1 \+ rajzot el\365z\365leg el\365\341ll\355tottuk!" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 7 "p:=1.1;" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pG$\"#6!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 477 "tx:= line([mx(p)+0.5,0,0], [-2,0,0],color=blue):\nty:= line ([0,-1.5,0], [0,1.5,0] ,color=blue):\ntz:= line([0,0,-1.5], [0,0,1.5] \+ ,color=blue):\nMA:= line([mx(p),0,0], [ax(p),ay(p),0] ,color=red):\nOA := line([1/2,0,0], [ax(p),ay(p),0] ,color=black):\n\nMp:=textplot3d([m x(p),0,0,`M`],color=black):\nAp:=textplot3d([ax(p),ay(p),0,`A`],color= black):\n\nXp:=textplot3d([mx(p)+0.6,0,0,`X`],color=black):\nYp:=textp lot3d([0,1.5,0,`Y`],color=black):\nZp:=textplot3d([0,0,1.5,`Z`],color= black):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "display3d(CA,CB, RV,\n tx,ty,tz,\n Mp,Ap,\n MA,OA,\n Xp,Yp,Zp,\nscaling = constrained);" }}{PARA 13 "" 1 "" {GLPLOT3D 641 421 421 {PLOTDATA 3 "60-%)POLYGONSG6$7@7%$\"+,w9y9!\"*$\"+4p6z?!#5$\"\"!F/7%$\"+eaa89F*$\" +IkOnSF-F.7%$\"+%*p,48F*$\"+CD&y(eF-F.7%$\"+118p6F*$\"+c#[9V(F-F.7%$\" \"\"F/$\"+SSDg')F-F.7%$\"+Q*p,4)F-$\"+l^c5&*F-F.7%$\"+HYGXgF-$\"+a*=_% **F-F.7%$\"+r`raRF-FLF.7%$\"+i+$)4>F-FGF.7%F.FBF.7%$!+igI\"p\"F-F=F.7% $!+V*p,4$F-F8F.7%$!+xXXNTF-F3F.7%$!+2gZ\"y%F-F+F.7%$!+++++]F-F.F.7%Fin $!+4p6z?F-F.7%Ffn$!+IkOnSF-F.7%FY$!+CD&y(eF-F.7%FV$!+c#[9V(F-F.7%F.$!+ SSDg')F-F.7%FR$!+l^c5&*F-F.7%FO$!+a*=_%**F-F.7%FJFapF.7%FEF^pF.7%F@F[p F.7%F;FhoF.7%F6FeoF.7%F1FboF.7%F(F_oF.7%$\"+++++:F*F.F.-%'COLOURG6&%$R GBG$\"*++++\"!\")FaqF.-F$6$7@7%$\"+2gZ\"y%F-F.F+7%$\"+xXXNTF-F.F37%$\" +V*p,4$F-F.F87%$\"+igI\"p\"F-F.F=7%F.F.FB7%$!+i+$)4>F-F.FG7%$!+r`raRF- F.FL7%$!+HYGXgF-F.FL7%$!+Q*p,4)F-F.FG7%$!\"\"F/F.FB7%$!+118p6F*F.F=7%$ !+%*p,48F*F.F87%$!+eaa89F*F.F37%$!+,w9y9F*F.F+7%$!+++++:F*F.F.7%F]tF.F _o7%FjsF.Fbo7%FgsF.Feo7%FdsF.Fho7%FasF.F[p7%F^sF.F^p7%F[sF.Fap7%FhrF.F ap7%FerF.F^p7%F.F.F[p7%FarF.Fho7%F^rF.Feo7%F[rF.Fbo7%FhqF.F_o7%$\"++++ +]F-F.F.-F^q6&F`qF.FaqFaq-F$6'F&Ffq-F^q6&F`qF.F.Faq-%*LINESTYLEG6#\"#; -%&STYLEG6#%%LINEG-%'CURVESG6$7$7%$\"+*Q/Y?$F*F.F.7%$!\"#F/F.F.Fgu-Fbv 6$7$7%F.$!#:FbsF.7%F.$\"#:FbsF.Fgu-Fbv6$7$7%F.F.F_w7%F.F.FbwFgu-%%TEXT G6%7%$\"+*Q/Yq#F*F.F.%\"MG-F^q6&F`qF/F/F/-Fjw6%7%$\"+97'f`*F-$\"+,O27* )F-F.%\"AGF`x-Fbv6$7$F\\xFdx-F^q6&F`qFaqF.F.-Fbv6$7$F`uFdxF`x-Fjw6%7%$ \"+*Q/YI$F*F.F.%\"XGF`x-Fjw6%Faw%\"YGF`x-Fjw6%Fhw%\"ZGF`x-%(SCALINGG6# %,CONSTRAINEDG" 1 2 0 1 10 0 2 1 1 1 1 1.000000 46.000000 44.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7 " "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" }}}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 108 "K\374l\366nb\366z\365 (Pi/2-n\351 l kisebb) t \351rt\351kre kipr\363b\341lhatjuk, mik\351nt v\341ltozik \+ a rajz a param\351ter megv\341laszt\341s\341t\363l.:" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "\315rjuk fel az " }{TEXT 295 1 "M" } {TEXT -1 31 " pont koordin\341t\341j\341ban szerepl\365 " }{TEXT 296 2 "mx" }{TEXT -1 18 " f\374ggv\351nyek\351nt az " }{TEXT 297 1 "M" } {TEXT -1 8 "-b\365l a " }{TEXT 298 1 "b" }{TEXT -1 50 " k\366rh\366z \+ h\372zott (egyik) \351rint\365 \351rint\351si pontj\341nak, " }{TEXT 299 1 "B" }{TEXT -1 20 "-nek a koordin\341t\341it:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "bx:= t->-1/2+1/(mx(t)+1/2);\nbz:= t-> sqrt (1-(bx(t)+1/2)^2);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %#bxGR6#%\"tG6\"6$%)operatorG%&arrowGF(,&#!\"\"\"\"#\"\"\"*&F0F0,&-%#m xG6#9$F0#F0F/F0F.F0F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#bzGR6# %\"tG6\"6$%)operatorG%&arrowGF(-%%sqrtG6#,&\"\"\"F0*$),&-%#bxG6#9$F0#F 0\"\"#F0F9F0!\"\"F(F(F(" }}}{EXCHG {PARA 256 "" 1 "" {TEXT -1 112 "A f \374ggv\351nyeket ki\355ratva (\351s egyszer\373s\355tve) l\341that \363, hogy mindk\351t f\374ggv\351ny egyetlen f\374ggetlen v\341ltoz \363t tartalmaz." }{TEXT 344 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "bx(t);simplify(bx(t));" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&#!\"\"\"\"#\"\"\"*&F'F',&F'F'*&F'F'-%$cosG6#%\"t GF%F'F%F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,&-%$cosG6#%\"tG\"\" \"F*!\"\"F*,&F&F*F*F*F+#F*\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "bz(t);simplify(bz(t));" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$-%%sqrtG6#,&\"\"\"F(*&F(F(*$),&F(F(*&F(F(-%$cosG6#%\" tG!\"\"F(\"\"#F(F2F2F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$-%%sqrtG6# *&,&-%$cosG6#%\"tG\"\"#\"\"\"F.F.*$),&F)F.F.F.F-F.!\"\"F." }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 7 "Rajz 3\n" }{TEXT 274 8 "Az oloid" }{TEXT 277 2 " t" }{TEXT 278 38 " param\351ter\351hez tartoz\363 \351rint\365 s\355kja az " }{TEXT 275 3 "ABM" }{TEXT 276 5 " s\355k:" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "Ez a rajz csak akkor \341ll\355that\363 e l\365, ha az R1 \351s R2 rajzokat is el\365\341ll\355tottuk!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 460 "p:=1.1:\nMA:= line([mx(p),0 ,0], [ax(p),ay(p),0] ,color=red):\nOA:= line([1/2,0,0], [ax(p),ay(p),0 ] ,color=black):\nXp:=textplot3d([mx(p)+0.6,0,0,`X`],color=black):\nMp :=textplot3d([mx(p),0,0,`M`],color=black):\nAp:=textplot3d([ax(p),ay(p ),0,`A`],color=black):\nMB:= line([mx(p), 0,0],[bx(p),0,bz(p)] ,color= red):\nAB:= line([ax(p),ay(p),0],[bx(p),0,bz(p)] ,color=red):\nMAB:= p olygonplot3d([[mx(p), 0,0],[ax(p),ay(p),0],[bx(p),0,bz(p)]],color=cyan ,style=patchnogrid):\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 " display3d(CA,CB,RV,\n tx,ty,tz,\n Mp,Ap,\n MA,MB,AB,MAB,\n Xp,Yp,Zp,\nscaling= constrained);" }}{PARA 13 "" 1 "" {GLPLOT3D 574 410 410 {PLOTDATA 3 "62-%)POLYGONSG6$7@7%$\"+,w9y9!\"*$\"+4p6z?!#5 $\"\"!F/7%$\"+eaa89F*$\"+IkOnSF-F.7%$\"+%*p,48F*$\"+CD&y(eF-F.7%$\"+11 8p6F*$\"+c#[9V(F-F.7%$\"\"\"F/$\"+SSDg')F-F.7%$\"+Q*p,4)F-$\"+l^c5&*F- F.7%$\"+HYGXgF-$\"+a*=_%**F-F.7%$\"+r`raRF-FLF.7%$\"+i+$)4>F-FGF.7%F.F BF.7%$!+igI\"p\"F-F=F.7%$!+V*p,4$F-F8F.7%$!+xXXNTF-F3F.7%$!+2gZ\"y%F-F +F.7%$!+++++]F-F.F.7%Fin$!+4p6z?F-F.7%Ffn$!+IkOnSF-F.7%FY$!+CD&y(eF-F. 7%FV$!+c#[9V(F-F.7%F.$!+SSDg')F-F.7%FR$!+l^c5&*F-F.7%FO$!+a*=_%**F-F.7 %FJFapF.7%FEF^pF.7%F@F[pF.7%F;FhoF.7%F6FeoF.7%F1FboF.7%F(F_oF.7%$\"+++ ++:F*F.F.-%'COLOURG6&%$RGBG$\"*++++\"!\")FaqF.-F$6$7@7%$\"+2gZ\"y%F-F. F+7%$\"+xXXNTF-F.F37%$\"+V*p,4$F-F.F87%$\"+igI\"p\"F-F.F=7%F.F.FB7%$!+ i+$)4>F-F.FG7%$!+r`raRF-F.FL7%$!+HYGXgF-F.FL7%$!+Q*p,4)F-F.FG7%$!\"\"F /F.FB7%$!+118p6F*F.F=7%$!+%*p,48F*F.F87%$!+eaa89F*F.F37%$!+,w9y9F*F.F+ 7%$!+++++:F*F.F.7%F]tF.F_o7%FjsF.Fbo7%FgsF.Feo7%FdsF.Fho7%FasF.F[p7%F^ sF.F^p7%F[sF.Fap7%FhrF.Fap7%FerF.F^p7%F.F.F[p7%FarF.Fho7%F^rF.Feo7%F[r F.Fbo7%FhqF.F_o7%$\"+++++]F-F.F.-F^q6&F`qF.FaqFaq-F$6'F&Ffq-F^q6&F`qF. F.Faq-%*LINESTYLEG6#\"#;-%&STYLEG6#%%LINEG-%'CURVESG6$7$7%$\"+*Q/Y?$F* F.F.7%$!\"#F/F.F.Fgu-Fbv6$7$7%F.$!#:FbsF.7%F.$\"#:FbsF.Fgu-Fbv6$7$7%F. F.F_w7%F.F.FbwFgu-%%TEXTG6%7%$\"+*Q/Yq#F*F.F.%\"MG-F^q6&F`qF/F/F/-Fjw6 %7%$\"+97'f`*F-$\"+,O27*)F-F.%\"AGF`x-Fbv6$7$F\\xFdx-F^q6&F`qFaqF.F.-F bv6$7$F\\x7%$!+8+\\z=F-F.$\"+ENl+&*F-F]y-Fbv6$7$FdxFbyF]y-F$6%7%F\\xFd xFbyFcu-F^v6#%,PATCHNOGRIDG-Fjw6%7%$\"+*Q/YI$F*F.F.%\"XGF`x-Fjw6%Faw% \"YGF`x-Fjw6%Fhw%\"ZGF`x-%(SCALINGG6#%,CONSTRAINEDG" 1 2 0 1 10 0 2 1 1 1 1 1.000000 46.000000 44.000000 0 0 "Curve 1" "Curve 2" "Curve 3" " Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 "I" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 17 "Az alkot\363k hossza " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 166 "AZ oloid MAB \351rint\365s \355kja a fel\374letet az AB szakasz ment\351n \351rinti. \nNevezz \374k az \351rint\365s\355k \351s az oloid k\366z\366s r\351sz\351t az oloid alkot\363inak!\nVizsg\341ljuk meg, mik\351nt f\374gg a " } {TEXT 392 7 "d(t)=AB" }{TEXT -1 18 " alkot\363 hossza a " }{TEXT 393 1 "t" }{TEXT -1 14 " param\351tert\365l!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "d:= t->sqrt((bx(t)-ax(t))^2+(0-ay(t))^2+(bz(t)-0)^2); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"dGR6#%\"tG6\"6$%)operatorG%&ar rowGF(-%%sqrtG6#,(*$),&-%#bxG6#9$\"\"\"-%#axGF5!\"\"\"\"#F7F7*$)-%#ayG F5F;F7F7*$)-%#bzGF5F;F7F7F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "d(t);simplify(d(t));\nd(0);\nlimit(d(t),t=Pi/2);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#*$-%%sqrtG6#,**$),(!\"\"\"\"\"*&F,F,,&F,F,*&F,F, -%$cosG6#%\"tGF+F,F+F,F0F+\"\"#F,F,*$)-%$sinGF2F4F,F,F,F,*&F,F,*$)F.F4 F,F+F+F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$-%%sqrtG6#\"\"$\"\"\"" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#*$-%%sqrtG6#\"\"$\"\"\"" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#*$-%%sqrtG6#\"\"$\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "A fentiek szerint " }{TEXT 279 37 "az oloid b\341rmely alkot\363ja ugyanakkora" }{TEXT -1 4 ", a " }{TEXT 280 2 "t " }{TEXT -1 59 "param\351ter megv\341laszt\341s\341t\363l f\374ggetlen\374l.\nE z a meg\341llap\355t\341s " }{TEXT 281 6 "t = 0 " }{TEXT -1 5 " \351s " }{TEXT 282 10 " t = Pi/2 " }{TEXT -1 31 " esetre is \351rv\351nyes , amikor az " }{TEXT 300 3 "ABM" }{TEXT -1 24 " h\341romsz\366g nem l \351tezik. " }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 22 "Az oloid megjelen\355t\351se" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 143 "Az oloidr\363l k\351sz\355tett rajzokhoz \372jra defini\341ljuk az eddig haszn\341lt v\341ltoz\363kat, \355gy \+ ez a programr\351szlet az el\365z\365ekt\365l f\374ggetlen\374l is fut tathat\363." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "restart: wit h(plots):with(plottools):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been redefined\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "Ugyanazokat a v\341ltoz\363kat \351s f\374ggv\351nyeket f ogjuk haszn\341lni:\n" }{TEXT 273 1 "a" }{TEXT -1 4 " \351s " }{TEXT 325 1 "b" }{TEXT -1 26 " a k\351t k\366rvonal pontjai, " }{TEXT 326 2 "CA" }{TEXT -1 4 " \351s " }{TEXT 327 2 "CB" }{TEXT -1 14 " a k\351t k\366rlap " }{TEXT 328 2 "RV" }{TEXT 329 0 "" }{TEXT 330 0 "" }{TEXT -1 20 " a k\366rvonalak rajza." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 260 "n:=30:\na:=[seq( [ 1/2+cos(i*2*Pi/n),sin(i*2*Pi/n),0],i=1..n) ]:\nb:=[seq( [-1/2+cos(i*2*Pi/n),0,sin(i*2*Pi/n)],i=1..n)]:\nCA:=polyg onplot3d(a, color=yellow): \nCB:=polygonplot3d(b,color=cyan): \nRV :=polygonplot3d([a,b],linestyle=16,style=wireframe,color=blue):\n" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 331 15 "ax, ay, bx, by " } {TEXT -1 80 "a k\366rvonalakat le\355r\363 param\351teres f\374ggv\351 nyek (az egyszer\373s\355tett alakot vett\374k \341t)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 " ax:= t -> 1/2+cos(t); \n ay:= sin :ay(t);\n bx:= t-> 1/2*(cos(t)-1)/(cos(t)+1);\n bz:= t-> sqrt((2*cos(t )+1))/(cos(t)+1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#axGR6#%\"tG6\" 6$%)operatorG%&arrowGF(,&#\"\"\"\"\"#F.-%$cosG6#9$F.F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$sinG6#%\"tG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#bxGR6#%\"tG6\"6$%)operatorG%&arrowGF(,$*&,&-%$cosG6#9$\"\"\"F 3!\"\"F3,&F/F3F3F3F4#F3\"\"#F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%#bzGR6#%\"tG6\"6$%)operatorG%&arrowGF(*&-%%sqrtG6#,&-%$cosG6#9$\"\"# \"\"\"F6F6,&F1F6F6F6!\"\"F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 324 1 "n" }{TEXT -1 93 " jelentse tov\341bbra is a beoszt \341s finoms\341g\341t. Ennek most p\341rosnak kell lennie.\nAhhoz, ho gy a " }{TEXT 346 1 "t" }{TEXT -1 132 " param\351tert tov\341bbra is \+ hat\341rozatlan v\341ltoz\363k\351nt kezelhess\374k, c\351lszer\373 az onnal megfosztanunk a ciklusban kapott konkr\351t \351rt\351k\351t\365 l. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 180 "n:=12: \nfor i f rom 0 to n do\n t:=i*Pi/n-Pi/2:\n A[i]:=[ax(t),ay(t), 0]:\n B[i]:=[ bx(t), 0, bz(t)]:\n L[i]:=line(A[i],B[i]\n , color=COLOR(HUE,(4*n+i) /8/n)\n):\n od:\nunassign('t');" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 345 11 "Az alkot\363k:" }{TEXT -1 168 " (csak negyedr\351sz\374ket \341ll \355tottuk el\365 k\374l\366n-k\374l\366n, a t\366bbit ebb\365l kapjuk geometriai transzform\341ci\363kkal. 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A \"vektoriszorzat\" helyett haszn\341lhatn\341nk a " }{TEXT 364 6 "linalg" }{TEXT -1 19 " elj\341r\341sai k\366z\374l a " }{TEXT 365 9 "crossprod" }{TEXT -1 319 " f\374ggv\351nyt is, de ennek a bekap csol\341sa tov\341bb terheln\351 - t\366bbnyire felesleges dolgokkal \+ - a programunkat.\nA - m\341shol is haszn\341lhat\363 - egys\351gvekto rt el\365\341ll\355t\363 elj\341r\341sn\341l \374gyeln\374nk kellett a rra, hogy ne \341lljon le a programunk 0-val val\363 oszt\341s miatt, \+ ha a bemen\365 adata a nullvektor. (B\341r ez itt nem fog el\365fordul ni.)\nAz " }{TEXT 366 1 "E" }{TEXT -1 113 " elj\341r\341s feladata, ho gy egy AB szakaszra illessze a tervezett t\351glalapot. \nEnnek az elj \341r\341snak a bemen\365 adata az " }{TEXT 370 1 "A" }{TEXT -1 4 " \+ \351s " }{TEXT 371 1 "B" }{TEXT -1 25 " pont, tov\341bb\341 egy olyan \+ " }{TEXT 378 1 "V" }{TEXT -1 22 "vektor, amely vagy az " }{TEXT 372 2 "CA" }{TEXT -1 8 " vagy a " }{TEXT 373 2 "CB" }{TEXT -1 17 " k\366rlap ot \351rinti " }{TEXT 374 1 "A" }{TEXT -1 11 "-ban, ill. " }{TEXT 375 1 "B" }{TEXT -1 79 "-ben, \355gy biztosan benne van a keresett s\355kb an, csak (t\366bbnyire) nem mer\365leges " }{TEXT 376 2 "AB" }{TEXT -1 79 "-re. Az elj\341r\341s l\351nyege, hogy el\365sz\366r el\365\341 ll\355tja erre az \351rint\365re, valamint az " }{TEXT 377 2 "AB" } {TEXT -1 13 "-re mer\365leges" }{TEXT 379 2 " N" }{TEXT -1 23 " vektor t, majd erre \351s " }{TEXT 380 2 "AB" }{TEXT -1 61 "-re ism\351t mer \365legest \341ll\355t,(melyet jel\366lhet\374nk ugyanazzal az " } {TEXT 381 1 "N" }{TEXT -1 40 " -nel) \355gy megkapjuk az \351rint\365 \+ t\351glalap " }{TEXT 382 2 "AB" }{TEXT -1 196 "-re mer\365leges oldal \341t.\n Negyedik bemen\365 adat az \351rint\365s\355k sz\355ne, amely jelen esetben \351pp a fel\374let adott \351rint\351si pontj\341hoz ( pontosabban szakasz\341hoz) tartoz\363 sz\355n ellent\351tes sz\355ne, amellyel mag\341t " }{TEXT 383 2 "AB" }{TEXT -1 149 "-t is rajzoltuk. Akinek szimpatikusabb lenne, ha az \351rintkez\365 fel\374letdarab \+ \351s az \351rint\365s\355k sz\355ne legyen azonos, , \355rja \341t ez t a bemen\365 adatot rendre " }{TEXT 367 21 "i/8/n , (i+2*n)/8/n " } {TEXT -1 10 "stb.-re. \n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 429 "vektoriszorzat:= (a,b) -> [a[2]*b[3]-a[3]*b[2],\n \+ a[3]*b[1]-a[1]*b[3],\n a[1]*b[2]- a[2]*b[1]]:\nEgysegnyi:= proc (a)\nlocal d;\nd:=sqrt(a[1]^2+a[2]^2+a[3 ]^2):\nif is(d>0) then [a[1]/d,a[2]/d,a[3]/d] \n else a; fi;\n end proc:\n\nE:=proc(A,B,V,s)\nlocal N;\nN:=vektoriszorzat(V,B-A);\nN: =Egysegnyi(vektoriszorzat(N,B-A)):\npolygonplot3d([A+N,A-N,B-N,B+N],co lor=COLOR(HUE,s));\nend proc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 365 "e:=Pi/2/n:\nfor i from 0 to 2*n-1 do\n t:=e*i:\nER[i ]:=E(A [i ],B[i ],[cos(t), sin(t),0],(4*n+i)/8/n):\nER[i+2*n]:=E(B[i+2* n],A[i+2*n],[cos(t),0, sin(t)],(6*n+i)/8/n):\nER[i+4*n]:=E(A[i+4*n],B[ i+4*n],[cos(t),-sin(t),0],(i)/8/n):\nER[i+6*n]:=E(B[i+6*n],A[i+6*n],[c 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Pi/2] " }{TEXT -1 114 "r\351szt fogjuk felh aszn\341lni, mivel ebb\365l a r\351szb\365l az eg\351sz fel\374let \+ \366ssze\341ll\355that\363 egybev\341g\363s\341gi transzform\341ci\363 kkal." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 325 "kax:= t->2*sqrt(3 )/9*(sqrt(2*(1+2*cos(t))*(1-cos(t)))+\narccos(sqrt(2)*cos(t)/sqrt(1+co s(t))));\nkay:= t ->sqrt(3)/9*(4*(1-cos(t))+log(2/(1+cos(t))));\nkbx:= t-> 2*sqrt(3)/9*(-sqrt(2*(1+2*cos(t))*(1-cos(t)))/(1+cos(t))+\narcco s(sqrt(2)*cos(t)/sqrt(1+cos(t))));\nkby:= t -> sqrt(3)/9*((11+7*cos(t) )/(1+cos(t))+log(2/(1+cos(t))));\n" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$kaxGR6#%\"tG6\"6$%)operatorG%&arrowGF(,$*&-%%sqrtG6# \"\"$\"\"\",&-F/6#,$*&,&F2F2*&\"\"#F2-%$cosG6#9$F2F2F2,&F2F2F;!\"\"F2F :F2-%'arccosG6#*&*&-F/6#F:F2F;F2F2-F/6#,&F2F2F;F2F@F2F2#F:\"\"*F(F(F( " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$kayGR6#%\"tG6\"6$%)operatorG%&a rrowGF(,$*&-%%sqrtG6#\"\"$\"\"\",(\"\"%F2*&F4F2-%$cosG6#9$F2!\"\"-%$lo 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1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7 " "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Cur ve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 18" "Curve 19" "Curve 2 0" "Curve 21" "Curve 22" "Curve 23" }}}}{SECT 0 {PARA 257 "" 0 "" {TEXT 306 27 "Egy ellen\365rz\351si lehet\365s\351g:\n" }{TEXT 307 37 "Vizsg\341ljuk meg, hogy igaz-e, hogy az " }{TEXT 308 2 "AB" }{TEXT 309 22 " szakasz hossza - a " }{TEXT 310 1 "t" }{TEXT 311 83 " param \351tert\365l f\374ggetlen\374l - a s\355kbafejt\351s ut\341n is ugya nakkora-e, m\351gpedig sqrt(3)!" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "d:=sqrt((kbx(t)-kax(t))^2+(kby(t)-kay(t))^2);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(d);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6# >%\"dG*$-%%sqrtG6#,&*$),&*&-F'6#\"\"$\"\"\",&*&*&-F'6#\"\"#F1-F'6#*&,& F1F1*&F7F1-%$cosG6#%\"tGF1F1F1,&F1F1F=!\"\"F1F1F1,&F1F1F=F1FBFB-%'arcc osG6#*&*&F5F1F=F1F1*$-F'6#FCF1FBF1F1#F7\"\"**&#F7FMF1*&F.F1,&*&F5F1F8F 1F1FDF1F1F1FBF7F1F1*$),&*&F.F1,&*&,&\"#6F1*&\"\"(F1F=F1F1F1FCFBF1-%#ln G6#,$*&F1F1FCFBF7F1F1#F1FM*&#F1FMF1*&F.F1,(\"\"%F1*&FaoF1F=F1FBFgnF1F1 F1FBF7F1F1F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$-%%sqrtG6#\"\"$\"\" \"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Az eredm\351ny megnyugtat \363." }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 249 "Az eg\351sz fel\374let s\355kbafejt\351s\351hez sz\374ks\351g\374nk lesz m\351g a b k\366r k \351p\351t el\365\341ll\355t\363 fb vonal t=2*Pi/3 \351rt\351khez tart oz\363 pontj\341ra, ugyanis ebbe a pontba kell eltolni, \351s \"lefel \351 ford\355tani\" az fa vonalat, hogy az pontosan le\355rja az fb vo nal kritikus r\351sz\351t . " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "ex:=kbx(2*Pi/3);ey:=kby(2*Pi/3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#exG,$*&-%%sqrtG6#\"\"$\"\"\"%#PiGF+#\"\"#\"\"*" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#eyG,$*&-%%sqrtG6#\"\"$\"\"\",&\"#:F+-%#lnG6# \"\"%F+F+#F+\"\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 95 "Ennek a felh aszn\341l\341s\341val \366sszesen nyolc egybev\341g\363 r\351szb\365l \+ fel\351p\355thet\365 a fel\374let kifejtett k\351pe:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 466 "ka[0]:=plot([-kax(t),kay(t),t=0..Pi/2],c olor=black):\nka[1]:=plot([ kax(t),kay(t),t=0..Pi/2],color=black):\nka [2]:=plot([-kax(t)+ex,-kay(t)+ey,t=0..Pi/2],color=black):\nka[3]:=plot ([ kax(t)+ex,-kay(t)+ey,t=0..Pi/2],color=black):\nka[4]:=plot([-kax(t) +2*ex,kay(t),t=0..Pi/2],color=black):\nka[5]:=plot([ kax(t)+2*ex,kay(t ),t=0..Pi/2],color=black):\nka[6]:=plot([-kax(t)+3*ex,-kay(t)+ey,t=0.. 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54" "Curve 55" "Curve 56" "Curve 57" "Curve 58" "Curve 59 " "Curve 60" "Curve 61" "Curve 62" "Curve 63" "Curve 64" "Curve 65" "C urve 66" "Curve 67" "Curve 68" "Curve 69" "Curve 70" "Curve 71" "Curve 72" "Curve 73" "Curve 74" "Curve 75" "Curve 76" "Curve 77" "Curve 78 " "Curve 79" "Curve 80" "Curve 81" "Curve 82" "Curve 83" "Curve 84" "C urve 85" "Curve 86" "Curve 87" "Curve 88" "Curve 89" "Curve 90" "Curve 91" "Curve 92" "Curve 93" "Curve 94" "Curve 95" "Curve 96" "Curve 97 " "Curve 98" "Curve 99" "Curve 100" "Curve 101" "Curve 102" "Curve 103 " "Curve 104" "Curve 105" "Curve 106" "Curve 107" "Curve 108" "Curve 1 09" "Curve 110" "Curve 111" "Curve 112" "Curve 113" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 246 "Annak, aki az itt el\365\341ll\355tott rajz al apj\341n szeretn\351 elk\351sz\355teni az alakzat modellj\351t (valami lyen kem\351nyebb pap\355rra nyomtatva az oloid h\341l\363zat\341t), s z\374ks\351ge lehet arra, hogy egyszerre egy-egy negyedr\351sznyit, fa gy fel\351nyit nyomtasson kia h\341l\363zatb\363l:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 696 "f1:= display( ka[0],kb[0],\n \+ kb[1],ka[1],\n seq(L[i],i=0..2*n),\n seq(P[i ],i=0..2*n-1),\ninsequence=false,axes=none,style=patchnogrid):\nf2:= d isplay( ka[2],kb[2],\n kb[3],ka[3],\n seq(L[ i],i=2*n..4*n),\n seq(P[i],i=2*n..4*n-1),\ninsequence=fal se,axes=none,style=patchnogrid):\nf3:= display( ka[4],kb[4],\n \+ kb[5],ka[5],\n seq(L[i],i=4*n..6*n),\n \+ seq(P[i],i=4*n..6*n-1),\ninsequence=false,axes=none,style=patchnogrid ):\nf4:= display( ka[6],kb[6],\n kb[7],ka[7],\n \+ seq(L[i],i=6*n..8*n),\n seq(P[i],i=6*n..8*n-1),\ninse quence=false,axes=none,style=patchnogrid):\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "f1 ;f2;f3;f4;" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6C-% 'CURVESG6$7S7$$!\"!\"\"!$F*F*7$$!33hx0tsQBM!#>$\"3kis.f9!e2&!#@7$$!3-S Y&*zdr*R'F/$\"3')G#z7'exuFA$\"3mO!f6Fe!e;F/7$$!3qWRIxx$oE#FA$\"3g&RQ6%p3YAF/7$$!3))yW22lh 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}{TEXT -1 219 ", ha egy s\355kon g\366rd\355tv e olyan mozg\341st v\351gez, amely sor\341n a t\366megk\366z\351ppont j\341nak egy s\355kt\363l m\351rt t\341vols\341ga nem v\341ltozik. Ez \351rt gurul pl. a g\366mb minden ir\341nyba, a henger egyenes vonal \+ ment\351n, a k\372p egy k\366r\355v ment\351n, stb." }}{PARA 0 "" 0 " " {TEXT -1 507 "Ha az oloid modellj\351t egy s\355klap\372 asztalon pr \363b\341ljuk g\366rd\355teni, egy saj\341tos kacs\341z\363 mozg\341sn ak lehet\374nk tan\372i.\n(\311ppen a h\341l\363zat\341t k\366vetve mo zog, amely - mint l\341ttuk - periodikus.)\n M\351gis csak akkor mondh atjuk, hogy \"gurul\" az oloid, ha bel\341tjuk, hogy b\341rmely \351ri nt\365s\355kja ugyanolyan t\341vols\341gra van a t\366megk\366z\351ppo ntj\341t\363l, amely jelen esetben a k\351t k\366r k\366z\351ppontj \341t \366sszek\366t\365 szakasz felez\365pontja.(A mi koordin\341tare ndszer\374nkben az orig\363.)\nA k\351rd\351s eld\366nt\351se c\351lj \341b\363l meghat\341roztuk az orig\363nak a - tetsz\365leges - " } {TEXT 348 2 "AB" }{TEXT -1 34 " alkot\363ra es\365 mer\365leges vet \374let\351t:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 91 "T\351rj\374nk vi ssza az elej\351n bevezetett f\374ggv\351nyekhez, melyek az oloid alko t\363it hat\341rozt\341k meg:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 " ax:= t -> 1/2+cos(t): \n ay:= sin:\n bx:= t-> 1/2*(cos(t)-1)/( cos(t)+1):\n bz:= t-> sqrt((2*cos(t)+1))/(cos(t)+1):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "ax(t);ay(t);\nbx(t);bz(t);" }{TEXT -1 0 " " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&#\"\"\"\"\"#F%-%$cosG6#%\"tGF%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$sinG6#%\"tG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,&-%$cosG6#%\"tG\"\"\"F*!\"\"F*,&F&F*F*F*F+#F*\"\"# " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&*$-%%sqrtG6#,&-%$cosG6#%\"tG\"\" #\"\"\"F.F.F.,&F)F.F.F.!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "L egyen egy " }{TEXT 349 13 "P=[px,py,pz] " }{TEXT -1 4 " az " }{TEXT 352 2 "AB" }{TEXT -1 16 " szakasznak egy " }{TEXT 350 2 " p" }{TEXT -1 37 " param\351tert\365l f\374gg\365 fut\363pontja, ahol " }{TEXT 351 7 " 0 " 0 "" {MPLTEXT 1 0 83 "px:= t-> \+ ax(t)+p*(bx(t)-ax(t)):\npy:= t-> ay(t)+p*(0-ay(t)):\npz:= t-> 0+p*(bz( t)-0):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "Ezek a f\374ggv\351nyek m\351g tartalmazz\341k a " }{TEXT 353 1 "p" }{TEXT -1 13 " param\351t ert.." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "Fel\355rjuk az " }{TEXT 354 2 "AB" }{TEXT -1 4 " \351s " }{TEXT 355 2 "PO" }{TEXT -1 35 " vekt orok skal\341rszorzat\341t, melynek " }{TEXT 356 1 "0" }{TEXT -1 33 "- nak kell lennie akkor, ha azt a " }{TEXT 357 1 "p" }{TEXT -1 26 " \351 rt\351ket keress\374k, melyre " }{TEXT 358 2 "OP" }{TEXT -1 11 " mer \365leges " }{TEXT 359 2 "AB" }{TEXT -1 4 "-re." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 57 "e:=(bx(t)-ax(t))*px(t)+(0-ay(t))*py(t)+(bz(t)- 0)*pz(t)=0;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"eG/,(*&,(*&,&-%$cos G6#%\"tG\"\"\"F/!\"\"F/,&F+F/F/F/F0#F/\"\"##F/F3F0F+F0F/,(F2F/F+F/*&% \"pGF/F(F/F/F/F/*&-%$sinGF-F/,&F9F/*&F7F/F9F/F0F/F0*&*&,&F+F3F/F/F/F7F /F/*$)F1F3F/F0F/\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "p: =solve(e,p);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pG, $*&,.*$)-%$cosG6#%\"tG\"\"$\"\"\"\"\"#*&F.F/)F*F0F/F/*(F0F/)-%$sinGF,F 0F/F*F/F/*&F.F/F*F/F/*&F0F/F4F/F/F/F/F/,.F(F/*$F2F/F/*&F0F/F*F/F/*&F4F /F*F/F/*$F4F/F/F0F/!\"\"#F/F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(p);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#, $*&,(*$)-%$cosG6#%\"tG\"\"#\"\"\"F-\"\"$F-*&\"\"&F-F(F-F-F-,&F(F-F-F-! \"\"#F-\"\"'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "Most m\341r a " } {TEXT 360 8 "px,py,pz" }{TEXT -1 22 " f\374ggv\351nyek kiz\341r\363lag " }{TEXT 361 1 "t" }{TEXT -1 12 " f\374ggv\351nyei:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "xp:=simplify(px(t));\nyp:=simplify(py(t)) ;\nzp:=simplify(pz(t));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xpG,$* &*&-%$cosG6#%\"tG\"\"\",(F(!\"'\"\"%!\"\"*$)F(\"\"$F,F,F,F,,(*$)F(\"\" #F,F,*&F7F,F(F,F,F,F,F0#F0\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%# ypG,$*&*&-%$sinG6#%\"tG\"\"\",(*$)-%$cosGF*\"\"#F,F,F0!\"\"\"\"$F3F,F, ,&F0F,F,F,F3#F3\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 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