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-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 }} {SECT 0 {SECT 1 {PARA 3 "" 0 "" {TEXT -1 33 "Az arkhim\351d\351szi koc ka el\365\341ll\355t\341sa" }{TEXT 300 13 "\nK\351sz\355tette: " } {TEXT 301 18 "dr. Szilassi Lajos" }{TEXT 302 62 "\n S ZTE JGYTF Matematika Tansz\351k\nE-mail: " }{TEXT 303 26 "szilass i@jgytf.u-szeged.hu" }}{EXCHG {PARA 0 "" 0 "" {TEXT 258 136 " Az arkhi m\351d\351szi kocka egy olyan f\351lig szab\341lyos poli\351der, melyn ek minden cs\372cs\341ra egy n\351gyzet, \351s n\351gy szab\341lyos h \341romsz\366g illeszkedik." }{TEXT -1 91 " \n6 n\351gyzetlapja \351s \+ 32 h\341romsz\366glapja van. A n\351gyzetlapok egy kocka lapjaira ill eszkednek." }}{PARA 257 "" 0 "" {GLPLOT3D 316 278 278 {PLOTDATA 3 "6$- %)POLYGONSG6H7&7%$\"+G,*oV&!\"*$\"+@u(f&HF*$\"#5\"\"!7%$!+@u(f&HF*F(F- 7%$!+G,*oV&F*F1F-7%F+F4F-7&7%F4$!#5F/F17%F1F9F(7%F(F9F+7%F+F9F47&7%F1F 4F97%F(F1F97%F+F(F97%F4F+F97&7%F9F1F47%F9F(F17%F9F+F(7%F9F4F+7&7%F-F(F +7%F-F1F(7%F-F4F17%F-F+F47&7%F+F-F(7%F(F-F17%F1F-F47%F4F-F+7%F'FIFN7%F QF0FF7%F=F@FK7%F?FDF87%F6FJF<7%FGF;F37%FLFOFA7%FPFBFE7%F'F0FN7%FNFQF07 %FFFGF37%F0F3FF7%F3F6F;7%F;F " 0 "" {MPLTEXT 1 0 37 "restart :\nwith(plots):with(plottools):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "K\351zenfekv\365 jel\366l\351s, hogy a gyakran haszn\341lt pontoka t " }{TEXT 316 7 "A, B, C" }{TEXT -1 4 " \351s " }{TEXT 317 1 "D" } {TEXT -1 15 " jel\366lje. De a " }{TEXT 318 1 "D" }{TEXT -1 166 " t \366bb c\351lra lefoglalt bet\373 a MAPLE-ben, (pl. f\374ggv\351ny der iv\341l\341s\341ra). Mivel mi most ezt a szerep\351t nem fogjuk kihasz n\341lni, oldjuk fel, ett\365l kezdve b\341rmit jel\366lhet\374nk " } {TEXT 319 1 "D" }{TEXT -1 54 "-vel: (Egy \351rt\351kad\341s el\365sz \366r hib\341t jelent ut\363bb nem.) " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "D:=a;unprotect(D);D:=a;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Adjuk meg a felhaszn\341lt pontok koordin\341t\341it:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 906 "kocka:=polygonplot3d([\n \+ [[ 10, 10, 10],[-10, 10, 10]],\n [[-10, 10, 10],[-10,-10, 10]],\n [[-10,-10, 10],[ 10,-10, 10]],\n \+ [[ 10,-10, 10],[ 10, 10, 10]],\n [[ 10, 10 ,-10],[-10, 10,-10]],\n [[-10, 10,-10],[-10,-10,-10]],\n [[-10,-10,-10],[ 10,-10,-10]],\n [[ 10,-1 0,-10],[ 10, 10,-10]],\n [[ 10, 10, 10],[ 10, 10,-10]], \n [[ 10,-10, 10],[ 10,-10,-10]],\n [[-10, -10, 10],[-10,-10,-10]],\n [[-10, 10, 10],[-10, 10,-10]] ]):\nL:=(x,y) ->\n[ [[ x, y, 10],[ y, -x, 10],[ -x, -y, 10],[ -y, \+ x, 10]],\n [[ y, 10, x],[ x, 10, -y],[ -y, 10, -x],[ -x, 10, y]] ,\n [[ 10, x, y],[ 10, y, -x],[ 10, -x, -y],[ 10, -y, x]]]:\nJel: =(x,y) ->[[ x, y, 10,`A`],\n [ 10, x, y,`B`],\n \+ [ y, 10, x,`C`],\n [ -y, x, 10,`D`]]:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "x:=6:y:=2:" }{TEXT -1 2 "( \+ " }{TEXT 309 1 "x" }{TEXT -1 4 " \351s " }{TEXT 310 2 "y " }{TEXT -1 92 "hely\351re m\341s-m\341s sz\341mot \355rva figyelj\374k meg, hogy \+ miben v\341ltozik a konstrukci\363, \351s miben nem.)" }{MPLTEXT 1 0 198 "\nlapok :=(x,y) -> polygonplot3d(L(x,y)):\nfelirat:=(x,y) -> tex tplot3d(Jel(x,y),color=BLACK):\ndisplay3d(kocka,lapok(x,y),felirat(x,y ),scaling= constrained,title=`H\341rom n\351gyzet`,orientation=[30,80] );" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "Megfigyelhet j\374k, hogy b\341rmely" }{TEXT 311 7 " [x, y]" }{TEXT -1 15 " \351rt \351kp\341rra az " }{TEXT 312 3 "ABC" }{TEXT -1 103 " h\341romsz\366g \+ szab\341lyos lesz, ahol\n A[ x, y, 10], B[ 10, x, y], C[ y, 10 , x] . Ez igazolhat\363 is: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "x;y; unassign('x','y');x;y;\n" }{TEXT -1 14 "Megfosztottuk " } {TEXT 314 1 "x" }{TEXT -1 7 "-et \351s " }{TEXT 315 1 "y" }{TEXT -1 105 "-t a konkr\351t \351rt\351k\351t\365l, amely a rajz elk\351sz\355 t\351s\351hez kellett, ezut\341n \341ltal\341nosan v\351gezhet\365k a \+ sz\341m\355t\341sok. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "\315rjuk fel a kapott pontok t\341vols\341gait:\n" }{MPLTEXT 1 0 234 "seq(prin t(\nJel(x,y)[i,4],`=[`,Jel(x,y)[i,1],Jel(x,y)[i,2],Jel(x,y)[i,3],`]`), i=1..4);\nAB:=(10-x)^2+(x-y)^2+(y-10)^2;\nBC:=(y-10)^2+(10-x)^2+(x-y)^ 2;\nCA:=(x-y)^2+(y-10)^2+(10-x)^2;\nDA:=(-y-x)^2+(x-y)^2+0;\nDC:=(-y-y )^2+(x-10)^2+(x-10)^2;" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " " }{TEXT -1 3 "Az " }{TEXT 313 4 "ABC " }{TEXT -1 96 "h\341romsz\366g szab\341lyos, hiszen - amellett, hogy rendezett form \341ban l\341tszik - a MAPLE is tan\372s\355tja: " }{MPLTEXT 1 0 21 " \nis(AB=BC);is(BC=CA);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "x:=7;y:=2;" }{TEXT -1 14 "Kib\365v\355tj\374k az " } {TEXT 323 1 "L" }{TEXT -1 13 " m\341trixot az " }{TEXT 324 3 "ABC" } {TEXT -1 14 " h\341romsz\366ggel:" }{MPLTEXT 1 0 245 "\nL:=(x,y) ->\n[ [[ x, y, 10],[ y, -x, 10],[ -x, -y, 10],[ -y, x, 10]],\n [[ y, \+ 10, x],[ x, 10, -y],[ -y, 10, -x],[ -x, 10, y]],\n [[ 10, x, y], [ 10, y, -x],[ 10, -x, -y],[ 10, -y, x]],\n [[ x, y, 10],[ 10, x , y],[ y, 10, x]] ]:\n" }{TEXT -1 3 "Az " }{TEXT 320 1 "L" }{TEXT -1 45 " m\341trixot megv\341ltoztattuk (b\365v\355tett\374k), de a " } {TEXT 321 9 "lapok(xy)" }{TEXT -1 60 " f\374gv\351nyt nem kell \372jra megadnunk, mivel mind\355g az aktu\341lis " }{TEXT 322 1 "L" }{TEXT -1 19 " t\366mb\366t haszn\341lja." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "display3d(kocka,lapok(x,y),felirat(x,y),scaling= con strained,title=`N\351gyzetek \351s egy h\341romsz\366g`,orientation=[3 0,80]);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 256 "" 1 "" {TEXT -1 19 "V\341lasszuk most meg " }{TEXT 263 2 "x-" }{TEXT -1 6 "et \351s " }{TEXT 264 1 "y" }{TEXT -1 16 "-t \+ \372gy, hogy az " }{TEXT 266 3 "ABC" }{TEXT -1 40 " h\341romsz\366g ol dala egyezzen meg a n\351gyzet" }{TEXT 267 4 " AD " }{TEXT -1 37 "olda l\341val!\nIsm\351t meg kell fosztanunk " }{TEXT 268 1 "x" }{TEXT -1 6 "-et \351s" }{TEXT 269 2 " y" }{TEXT -1 27 "-t a konkr\351t \351rt \351k\351t\365l, ez " }{TEXT 270 2 "CA" }{TEXT -1 40 "-t is hat\341roz atlan kifejez\351ss\351 alak\355tja." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "CA;DC;" }{TEXT -1 33 "(most m\351g van konkr\351t sz \341m\351rt\351ke)" }{MPLTEXT 1 0 30 "\nunassign('x','y');\nx;y;CA;DC; " }{TEXT -1 33 "Most m\341r egyik sem jelent sz\341mot." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "m:=solve(CA=DC,y);\n" }{TEXT -1 278 "El kell d\366nten\374nk, hogy a kapott m\341sodfok\372 egyenlet g y\366kei k\366z\374l melyik a sz\341munkra megfelel\365. A k\351t gy \366k\366t tetsz\351s szerinti sorrendben adja meg a MAPLE, nek\374nk \+ ki kell v\341lasztanunk azt, amelyben a gy\366k\366s kifejez\351s pozi t\355v, ugyanis a feladat geometriai tartalm\341b\363l ad\363dik, hogy " }{TEXT 271 3 "y>0" }{TEXT -1 26 "-nak teljes\374lnie kell, ha " } {TEXT 272 7 "0 \+ " 0 "" {MPLTEXT 1 0 35 "y:=-1/2*x-5+1/2*sqrt(x^2+60*x+100);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 291 "L:=(x,y) ->\n[ [ [ x, y, 10],[ y, -x, 10],[ -x, -y, 10],[ -y, x, 10]],\n [[ y, 10 , x],[ x, 10, -y],[ -y, 10, -x],[ -x, 10, y]],\n [[ 10, x, y],[ \+ 10, y, -x],[ 10, -x, -y],[ 10, -y, x]],\n [[ x, y, 10],[ 10, x, \+ y],[ y, 10, x]],\n [[ x, y, 10],[ y, 10, x],[ -y, x, 10]] ]: \n" }{TEXT -1 34 "Tov\341bb b\365v\355tett\374k az alakzatot az " } {TEXT 273 4 "ACD " }{TEXT -1 73 "h\341romsz\366ggel. A polygonplot elj \341r\341s csak konkr\351t x \351s y \351rt\351kkel m\373k\366dik." }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 " \332jabb rajzokat k\351sz\355thet\374nk, most m\341r csak x-et v\341la sztva szabadon." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 143 "y;x:=9; y;\nlapok:=polygonplot3d(L(x,y)):\ndisplay3d(kocka,lapok,felirat,scali ng= constrained,title=`CA = DA`,orientation=[50,70]);\nunassign('x'): " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "C\351lszer\373. " }{TEXT 299 1 "y" }{TEXT -1 29 "-t egy f\374ggv\351nnyel megadnunk:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "f:= x ->-1/2*x-5+1/2*sqrt(x^2+60*x+ 100);f(0);f(6);f(10); " }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "Rajzoljuk le az " }{TEXT 274 7 "y=f(x) " }{TEXT -1 106 "f \374ggv\351nyt, k\374l\366n megjel\366lve a geometriai tartalom szempo ntj\341b\363l fontos r\351sz\351t az \351rtelmez\351si tartom\341nynak ." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 187 "N:=rectangle([0,10],[ 10,0], color=cyan):\nF:=plot(f(t),t=-1..12,linestyle=16,color=red):\nF t:=textplot([11.5,f(11.5)+1,\"y=f(x)\"],color=red):\ndisplay(N,F,Ft,sc aling=constrained,axes=framed);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 98 "Most m\341r k\351sz\355thet\374nk egy mozg\363 k\351 pet is, x-et folyamatosan oda-vissza v\341ltoztatva 0 \351s 10 k\366z \366tt." }}{PARA 0 "" 0 "" {TEXT -1 35 "Az anim\341ci\363k\351sz\355t \351s egyik m\363dja az " }{TEXT 325 6 "anim3d" }{TEXT -1 117 " elj \341r\341son alapszik. A m\341sik, amit most alkalmazunk: egy ciklussa l el\365\341ll\355t 11 k\351pet, (i megy 0-t\365l 10-ig.), majd az " } {TEXT 265 16 "insequence=true " }{TEXT -1 173 "utas\355t\341ssal gondo skodunk arr\363l, hogy a rajzok egym\341s ut\341n jelenjenek meg. A k \351peket egy k\366zb\374ls\365 \351rt\351kr\365l ind\355tva futtatjuk oda-vissza, \355gy folyamatoss\341 tehet\365 az anim\341ci\363. " }} {PARA 0 "" 0 "" {TEXT -1 145 "\332gy kaphatn\341nk las\372bb, jobban k \366vethet\365 k\351psorozatot, ha n\366veln\351nk a l\351p\351ssz\341 mot. Ez megterheli a sz\341m\355t\363g\351p\374nk mem\363ri\341j\341t, \351s a merevlemezt is. " }}{PARA 0 "" 0 "" {TEXT -1 64 "A lapokat is m\351t \372jra defini\341ltuk, az \351lv\341zas kock\341t is belet\351 ve." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1120 "for i from 0 to 10 do\nx:=i:y:=f(x):\nlapok[i]:=polygonplot3d([\n [[ 10, 1 0, 10],[-10, 10, 10]],\n [[-10, 10, 10],[-10,-10, 10]], \n [[-10,-10, 10],[ 10,-10, 10]],\n [[ 10, -10, 10],[ 10, 10, 10]],\n [[ 10, 10,-10],[-10, 10,-10]] ,\n [[-10, 10,-10],[-10,-10,-10]],\n [[-10 ,-10,-10],[ 10,-10,-10]],\n [[ 10,-10,-10],[ 10, 10,-10] ],\n [[ 10, 10, 10],[ 10, 10,-10]],\n [[ 1 0,-10, 10],[ 10,-10,-10]],\n [[-10,-10, 10],[-10,-10,-10 ]],\n [[-10, 10, 10],[-10, 10,-10]],\n [[ x, y, 10], [ y, -x, 10],[ -x, -y, 10],[ -y, x, 10]],\n [[ y, 10, x],[ \+ x, 10, -y],[ -y, 10, -x],[ -x, 10, y]],\n [[ 10, x, y],[ 10 , y, -x],[ 10, -x, -y],[ 10, -y, x]],\n [[ x, y, 10],[ 10, \+ x, y],[ y, 10, x]],\n [[ x, y, 10],[ y, 10, x],[ -y, \+ x, 10]]]):od:\n\nf_mozi:=display3d( seq(lapok[i],i=5..10),\n \+ seq(lapok[10-i],i=0..10),\n seq(lapok[ i],i=0..5),\n scaling=constrained,insequence=true,orientation=[5 0,70]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "f_mozi;" }{TEXT -1 5 "\nItt " }{TEXT 275 4 "x=10" }{TEXT -1 107 " estben a n\351gyzete khez kapcsol\363d\363 h\341romsz\366gek a n\351gyzettel egy\374tt egy \+ szab\341lyos nyolcsz\366get fognak alkotni." }}}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "Most ugyan\355gy a djuk meg annak a felt\351tel\351t, hogy az " }{TEXT 276 5 "CA=DA" } {TEXT -1 13 " teljes\374lj\366n:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "unassign('x','y'):CA;DA;\nm:=solve(CA=DA,y);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "A kapott megold\341st i sm\351t " }{TEXT 277 1 "x" }{TEXT -1 25 " f\374ggv\351nyek\351nt \355r juk fel:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "g:= x ->-10*(x- 10)/(x+10);g(0);g(6);g(10);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 159 "G:=plot(g(t),t=-1..12,linestyle=16,color=magenta): \nGt:=textplot([11.5,g(11.5)+1,\"y=g(x)\"],color=magenta):\ndisplay(N, F,Ft,G,Gt,scaling=constrained,axes=framed);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "M \341ris j\366het az \372jabb mozi:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "for i from 0 to 10 do\nx:=i:y:=g(x):" }{TEXT -1 26 " \+ (Csak itt van k\374l\366nbs\351g.)" }{MPLTEXT 1 0 1085 "\nlapok[i]:=po lygonplot3d([\n [[ 10, 10, 10],[-10, 10, 10]],\n \+ [[-10, 10, 10],[-10,-10, 10]],\n [[-10,-10, 10],[ 10,-10, 10]],\n [[ 10,-10, 10],[ 10, 10, 10]],\n \+ [[ 10, 10,-10],[-10, 10,-10]],\n [[-10, 10,-10], [-10,-10,-10]],\n [[-10,-10,-10],[ 10,-10,-10]],\n \+ [[ 10,-10,-10],[ 10, 10,-10]],\n [[ 10, 10, 10] ,[ 10, 10,-10]],\n [[ 10,-10, 10],[ 10,-10,-10]],\n \+ [[-10,-10, 10],[-10,-10,-10]],\n [[-10, 10, 10 ],[-10, 10,-10]],\n [[ x, y, 10],[ y, -x, 10],[ -x, -y, 10], [ -y, x, 10]],\n [[ y, 10, x],[ x, 10, -y],[ -y, 10, -x],[ \+ -x, 10, y]],\n [[ 10, x, y],[ 10, y, -x],[ 10, -x, -y],[ 10 , -y, x]],\n [[ x, y, 10],[ 10, x, y],[ y, 10, x]],\n [[ x, y, 10],[ y, 10, x],[ -y, x, 10]]]):od:\n\ng_mozi:=dis play3d( seq(lapok[i],i=5..10),\n seq(lapok[10-i],i=0 ..10),\n seq(lapok[i],i=0..5),\n scaling=const rained,insequence=true,orientation=[50,70]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "g_mozi;" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 59 "M\351g egy lehet\365s\351g\374nk van arra, hogy egyenl \365sz\341r\372 legyen az " }{TEXT 278 3 "ACD" }{TEXT -1 13 " h\341rom sz\366g: " }{TEXT 279 5 "DA=DC" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "unassign('x','y'):DC;DA;\nsolve(DC=DA,y);" }{TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "h:=x ->2*sqrt(-25+5*x);h(5); h(7);h(10);\n" }{TEXT -1 2 "A " }{TEXT 280 1 "h" }{TEXT -1 34 " f\374g gv\351ny \351rtelmez\351si tartom\341nya " }{TEXT 281 3 "x=5" }{TEXT -1 15 " -n\351l kezd\365dik." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 158 "H:=plot(h(t),t=-1..12,linestyle=16,color=blue):\nHt:=textplot([ 11.5,h(11.5)+1,\"y=h(x)\"],color=blue):\ndisplay(N,F,Ft,G,Gt,H,Ht,scal ing=constrained,axes=framed);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 29 "Ez a harmadik mozi, melyben " }{TEXT 282 5 "DA=DC" } {TEXT -1 2 ": " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "for i fro m 0 to 10 do\nx:=10-i/2:y:=h(x):" }{TEXT -1 26 " (Csak itt van k\374l \366nbs\351g.)" }{MPLTEXT 1 0 1086 "\nlapok[i]:=polygonplot3d([\n \+ [[ 10, 10, 10],[-10, 10, 10]],\n [[-10, 10, 10 ],[-10,-10, 10]],\n [[-10,-10, 10],[ 10,-10, 10]],\n \+ [[ 10,-10, 10],[ 10, 10, 10]],\n [[ 10, 10,-1 0],[-10, 10,-10]],\n [[-10, 10,-10],[-10,-10,-10]],\n \+ [[-10,-10,-10],[ 10,-10,-10]],\n [[ 10,-10,- 10],[ 10, 10,-10]],\n [[ 10, 10, 10],[ 10, 10,-10]],\n \+ [[ 10,-10, 10],[ 10,-10,-10]],\n [[-10,-10, 10],[-10,-10,-10]],\n [[-10, 10, 10],[-10, 10,-10]],\n \+ [[ x, y, 10],[ y, -x, 10],[ -x, -y, 10],[ -y, x, 10]],\n [[ y, 10, x],[ x, 10, -y],[ -y, 10, -x],[ -x, 10, y]],\n [[ 1 0, x, y],[ 10, y, -x],[ 10, -x, -y],[ 10, -y, x]],\n [[ x, y, 10],[ 10, x, y],[ y, 10, x]],\n [[ x, y, 10],[ y, \+ 10, x],[ -y, x, 10]]]):\nod:\n\nh_mozi:=display3d( seq(lapok[ i],i=8..10),\n seq(lapok[10-i],i=0..10),\n \+ seq(lapok[i],i=0..8),\n scaling=constrained,insequence= true,orientation=[50,70]):" }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "h_mozi;" }{TEXT -1 9 "\nItt az " }{TEXT 326 3 "x=5" } {TEXT -1 111 " estben a szab\341lyos h\341romsz\366gekhez kapcsol\363d \363 h\341romsz\366gek azokkal egy\374tt egy szab\341lyos hatsz\366get fognak alkotni" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "A h\341rom jelenetet egy\374tt is \+ megszeml\351lhetj\374k:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 " mozi:=display3d(f_mozi,g_mozi,h_mozi,scaling= constrained,insequence=t rue,orientation=[50,70]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "mozi;" }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "V\351g\374l ker ess\374k meg azt az x \351rt\351ket, amelyre az " }{TEXT 327 3 "ACD" } {TEXT -1 21 " h\341romsz\366g szab\341lyos:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "unassign('x');f(x);g(x);h(x);" }{TEXT -1 0 "" }} {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 " B\341rmely k\351t f\374ggv\351ny metsz\351spontj\341nak ugyanannak kel l lennie:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "x1:=solve(f(x) =g(x));\nx2:=solve(f(x)=h(x));\nx3:=solve(g(x)=h(x));\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "Az r\341n\351z\351sre is l\341tszik, hogy " }{TEXT 283 5 "x2=x3" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "is(x2=x3);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "Azt azonban a MAPLE nem tudja ilyen hat\341rozottan meg \341lap\355tani, hogy " }{TEXT 284 6 "x1=x2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "is(x1=x3);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 83 "A k\351t kifejez\351s azonos volt\341nak az igazol \341s\341ra tov\341bbi k\355s\351rleteket kell v\351gezn\374nk:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "simplify(x1-x2);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "simplify(x2/x1);" } {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 81 "Itt jegyezz\374k \+ meg, hogy a MAPLE \372jabb verzi\363i m\341r itt is a j\363 eredm\351n yt mutatj\341k." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "factor(x 1-x2);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "fa ctor(x1/x2);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "A \+ szorzatt\341 alak\355t\341s seg\355tett." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "Pr\363b\341ljuk megkeresni " }{TEXT 285 1 "y" }{TEXT -1 39 " -ra a lehet\365 legegyszer\373bb kifejez\351st:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "f(x1);f(x2);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "g(x1);g(x2);" }{TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "h(x1);h(x2);" }{TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "A megjelen\355t\351shez k\366 zel\355t\365 \351rt\351kekkel elegend\365 dolgoznunk." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 639 "x:=evalf(x1);y:=evalf(h(x1));\nlap ok:=polygonplot3d([\n [[ x, y, 10],[ y, -x, 10],[ -x, -y, 10 ],[ -y, x, 10]],\n [[ y, 10, x],[ x, 10, -y],[ -y, 10, -x], [ -x, 10, y]],\n [[ 10, x, y],[ 10, y, -x],[ 10, -x, -y],[ \+ 10, -y, x]],\n [[ x, y, 10],[ 10, x, y],[ y, 10, x]],\n \+ [[ x, y, 10],[ y, 10, x],[ -y, x, 10]]]):\nfelirat:=textp lot3d([[ x, y, 10,`A`],\n [ 10, x, y,`B`],\n \+ [ y, 10, x,`C`],\n [ -y, x, \+ 10,`D`]],color=BLACK):\ndisplay3d(kocka,lapok,felirat,scaling= constra ined,title=`Achimedesi kocka v\341za`,orientation=[30,80]);" }{TEXT -1 0 "" }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 22 "Az arkhim\351d\351szi kocka\n" }{TEXT 257 16 "Az i m\351nt kapott " }{TEXT 286 1 "x" }{TEXT 287 4 " \351s " }{TEXT 288 2 "y " }{TEXT 289 67 "\351rt\351kekkel fel\351p\355thetj\374k az arkhim \351deszi kock\341t.\nEhhez c\351lszer\373 a " }{TEXT 290 8 "restart \+ " }{TEXT 291 72 "paranccsal mindent kit\366r\366lni a mem\363ri\341b \363l, majd; k\366zvetlen\374l megadni az " }{TEXT 292 2 "x " }{TEXT 293 3 "\351s " }{TEXT 294 1 "y" }{TEXT 295 212 " k\351plet\351t, a par am\351tereket \372gy v\341lasztva, hogy a programnak min\351l kevesebb ki\351rt\351kel\351st kelljen v\351geznie, ezzel jelent\365sen meggyo rs\355thatjuk a fut\341s\341t.\nAz al\341bbi programr\351szlet \366n \341ll\363 MAPLE f\341jlk\351nt is futtathat\363." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 210 "restart;\nwith(plots):\nx := 5+1/5*(5/3*(26+ 6*sqrt(33))^(1/3)-40/3*1/((26+6*sqrt(33))^(1/3))-5/3)^2;\ny := 10/3*(( 26+6*sqrt(33))^(2/3)-8-(26+6*sqrt(33))^(1/3))/((26+6*sqrt(33))^(1/3)); \n\nx0:= evalf(x);y0:=evalf(y);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 421 "K:=[[[ 10, 10, 10],[-10, 10, 10]],\n [[-10, 10, 10],[-10,- 10, 10]],\n [[-10,-10, 10],[ 10,-10, 10]],\n [[ 10,-10, 10],[ 10 , 10, 10]],\n [[ 10, 10,-10],[-10, 10,-10]],\n [[-10, 10,-10],[- 10,-10,-10]],\n [[-10,-10,-10],[ 10,-10,-10]],\n [[ 10,-10,-10], [ 10, 10,-10]],\n [[ 10, 10, 10],[ 10, 10,-10]],\n [[ 10,-10, 10 ],[ 10,-10,-10]],\n [[-10,-10, 10],[-10,-10,-10]],\n [[-10, 10, \+ 10],[-10, 10,-10]]]:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1841 "L :=(x,y) ->[\n[[ x, y, 10],[ -y, x, 10],[ -x, -y, 10],[ y, -x, \+ 10]],\n[[ -x,-10, -y],[ -y,-10, x],[ x,-10, y],[ y,-10, -x]],\n[[ \+ -y, -x,-10],[ x, -y,-10],[ y, x,-10],[ -x, y,-10]],\n[[-10, -y, -x ],[-10, x, -y],[-10, y, x],[-10, -x, y]], \n[[ 10, x, y],[ 10, - y, x],[ 10, -x, -y],[ 10, y, -x]],\n[[ y, 10, x],[ x, 10, -y],[ - y, 10, -x],[ -x, 10, y]],\n[[ x, y, 10],[ 10, x, y],[ y, 10, x] ],\n[[ -x, 10, y],[ -y, x, 10],[-10, y, x]],\n[[ y,-10, -x],[ x, -y,-10],[ 10, -x, -y]], \n[[ -y, -x,-10],[-10, -y, -x],[ -x,-10, -y]] ,\n[[ y, -x, 10],[ 10, -y, x],[ x,-10, y]],\n[[-10, -x, y],[ -y,- 10, x],[ -x, -y, 10]], \n[[ 10, y, -x],[ x, 10, -y],[ y, x,-10]], \n[[ -y, 10, -x],[ -x, y,-10],[-10, x, -y]]\n\n,[[ x, y, 10],[ -y , x, 10],[ y, 10, x]]\n,[[ y, 10, x],[ -x, 10, y],[ -y, x, 10]] \n\n,[[-10, y, x],[-10, -x, y],[ -x, -y, 10]] \n,[[ -y, x, 10],[ \+ -x, -y, 10],[-10, y, x]]\n\n,[[ -x, -y, 10],[ y, -x, 10],[ -y,-10, \+ x]] \n,[[ -y,-10, x],[ x,-10, y],[ y, -x, 10]]\n\n,[[ y, -x, 10] ,[ x, y, 10],[ 10, -y, x]] \n,[[ 10, x, y],[ 10, -y, x],[ x, y , 10]] \n\n,[[ -y, -x,-10],[ x, -y,-10],[ y,-10, -x]]\n,[[ y,-10, - x],[ -x,-10, -y],[ -y, -x,-10]] \n\n,[[ x, -y,-10],[ y, x,-10],[ 10 , y, -x]]\n,[[ 10, -x, -y],[ 10, y, -x],[ x, -y,-10]] \n\n,[[ -x, \+ y,-10],[ -y, -x,-10],[-10, -y, -x]]\n,[[-10, -y, -x],[-10, x, -y],[ - x, y,-10]]\n \n,[[ y, x,-10],[ -x, y,-10],[ -y, 10, -x]]\n,[[ x, \+ 10, -y],[ -y, 10, -x],[ y, x,-10]] \n\n\n,[[-10, x, -y],[-10, y, \+ x],[ -x, 10, y]]\n,[[ -y, 10, -x],[ -x, 10, y],[-10, x, -y]]\n\n,[[ -10, -y, -x],[-10, -x, y],[ -x,-10, -y]] \n,[[ -x,-10, -y],[ -y,-10, \+ x],[-10, -x, y]]\n\n,[[ x,-10, y],[ y,-10, -x],[ 10, -x, -y]]\n,[ [ 10, -y, x],[ 10, -x, -y],[ x,-10, y]]\n\n,[[ 10, y, -x],[ 10, x , y],[ x, 10, -y]]\n,[[ y, 10, x],[ x, 10, -y],[ 10, x, y]] \n \n ]:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "polygonplot3d([L(x0,y0)],scaling=CO NSTRAINED);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 108 "Az \351lv\341zas kocka adatait az\351rt tett\374k k \374l\366n m\341trixba, hogy el tudjuk k\374l\366n\355teni a kiindul \341sul vett kock\341t\363l." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "x1:=10*sqrt(2)-10:\npolygonplot3d([L(x1,x1)],scaling=CONSTRAINED );" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "Az, hogy a p oli\351dert " }{TEXT 296 2 "x " }{TEXT -1 3 "\351s " }{TEXT 297 1 "y" }{TEXT -1 62 " f\374ggv\351nyek\351nt adtuk meg, lehet\365s\351get ad \+ arra, hogy b\341rmilyen " }{TEXT 298 6 "( x,y)" }{TEXT -1 31 " \351rt \351kre azonal megkapjuk az " }{TEXT 328 1 "x" }{TEXT -1 4 " \351s " }{TEXT 329 1 "y" }{TEXT -1 96 " \341ltal meghat\341rozott poli\351dert . Ez \351rdekes lehet\365s\351geket k\355n\341l a konstrukci\363 tov \341bbi elemz\351s\351re." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "polygonplot3d(L(6,5),scaling=CONSTRAINED);" }{TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "polygonplot3d(L(7,0),scaling =CONSTRAINED);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 28 "A \"balos\" arkhim\351d \351szi kocka" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 386 "Az arkhim\351d\351szi kock\341nak l\351t ezik egy \"jobbos\" \351s egy \"balos\" v\341ltozata, melyek egym\341s ba egy s\355kra vonatkoz\363 t\374lr\366z\351ssel (\351s t\351beli ir \341ny\355t\341start\363 mozg\341sokkal) vihet\365k \341t egym\341sba. Ez az egyetlen olyan kock\341b\363l sz\341rmaztathat\363 arkhim\351d \351szi f\351lig szab\341lyos poli\351der, melynek nincs szimmerias \355kja (\355gy centr\341lisan sem szimmetrikus). A m\341sik v\341ltoz athoz egy \372j lap-m\341trixot kell k\351sz\355ten\374nk." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1839 "G :=(x,y) ->[\n[[ x, y, 10],[ \+ -y, x, 10],[ -x, -y, 10],[ y, -x, 10]],\n[[ -x,-10, -y],[ -y,-10, x ],[ x,-10, y],[ y,-10, -x]],\n[[ -y, -x,-10],[ x, -y,-10],[ y, x ,-10],[ -x, y,-10]],\n[[-10, -y, -x],[-10, x, -y],[-10, y, x],[-10 , -x, y]], \n[[ 10, x, y],[ 10, -y, x],[ 10, -x, -y],[ 10, y, -x] ],\n[[ y, 10, x],[ x, 10, -y],[ -y, 10, -x],[ -x, 10, y]],\n[[ x, y, 10],[ 10, x, y],[ y, 10, x]],\n[[ -x, 10, y],[ -y, x, 10],[ -10, y, x]],\n[[ y,-10, -x],[ x, -y,-10],[ 10, -x, -y]], \n[[ -y, \+ -x,-10],[-10, -y, -x],[ -x,-10, -y]],\n[[ y, -x, 10],[ 10, -y, x],[ \+ x,-10, y]],\n[[-10, -x, y],[ -y,-10, x],[ -x, -y, 10]], \n[[ 10, \+ y, -x],[ x, 10, -y],[ y, x,-10]], \n[[ -y, 10, -x],[ -x, y,-10],[- 10, x, -y]],\n\n[[ x, y, 10],[ -x, 10, y],[ y, 10, x]],\n[[ -x, \+ 10, y],[ -y, x, 10],[ x, y, 10]], \n\n[[-y, x, 10],[-10, -x, y] ,[ -x, -y, 10]], \n[[-10, -x, y],[ -y, x, 10],[ -10, y, x]],\n\n[[ \+ -x, -y, 10],[ x, -10,y],[ -y,-10, x]], \n[[ x,-10, y],[ y, -x, 1 0],[ -x, -y, 10]],\n\n[[ y, -x, 10],[ 10, x, y],[ 10, -y, x]], \n[ [ 10, x, y],[ x, y, 10],[ y, -x, 10]], \n\n[[ x, -y,-10],[ -x, - 10,-y],[ y,-10, -x]],\n[[ -x,-10, -y],[ -y, -x,-10],[ x,-y, -10]], \+ \n\n[[ y, x,-10],[ 10, -x, -y],[ 10, y, -x]],\n[[ 10, -x, -y],[ x, -y,-10],[ y, x, -10]], \n\n[[ -y, -x,-10],[ -10, x,-y],[-10, -y, -x ]],\n[[-10, x, -y],[ -x, y,-10],[-y, -x, -10]],\n \n[[ -x, y,-10],[ x, 10, -y],[ -y, 10, -x]],\n[[ x, 10, -y],[ y, x,-10],[ -x, y, -1 0]], \n\n[[-10, y, x],[ -y, 10, -x],[ -x, 10, y]],\n[[ -y, 10, -x], [-10, x, -y],[-10, y, x]],\n\n[[-10, -y, -x],[ -y,-10, x],[ -x,-10 , -y]], \n[[ -y,-10, x],[-10, -x, y],[-10, -y, -x]],\n\n[[ y,-10, - x],[ 10, -x, -y],[ 10, -y, x]],\n[[ 10, -y, x],[ y, -10, -x],[ x,-1 0, y]],\n\n[[ 10, y, -x],[ y, 10, x],[ x, 10, -y]],\n[[ y, 10, \+ x],[ 10, x, y],[ 10, y, -x]] \n ]:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "polygonplot3d([K,G(y0,x0)],scaling=CONSTRAINED);" } {TEXT -1 0 "" }}}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 126 "A fenti eredm \351nyeket c\351lszer\373 egy \372j programba be\351p\355ten\374nk, am ely az arkhim\351d\351szi kock\341val kapcsolatos anim\341ci\363kat mu tatja be." }}}}}{MARK "0" 0 }{VIEWOPTS 1 1 1 3 2 1804 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }