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"Hat tetra\351derb\365l \+ \341ll\363 mozgathat\363 alakzat, egy " }{TEXT 493 14 "kaleidociklus\n " }{TEXT 311 12 "K\351sz\355tette: " }{TEXT 312 18 "dr. Szilassi Lajos " }{TEXT 313 62 "\n SZTE JGYTF Matematika Tansz\351k \nE-mail: " }{TEXT 314 26 "szilassi@jgytf.u-szeged.hu" }}{EXCHG {PARA 0 "" 0 "" {TEXT 258 8 "Feladat:" }{TEXT -1 194 " Tekints\374nk e gy olyan tetra\351dert, melynek k\351t szemk\366zti \351le mer\365lege s egym\341sra., \351s norm\341ltanszverz\341lisuk talppontjai e szemk \366zti \351lek felez\365pontjai. Legyen a mer\365leges oldalak - neve zz\374k \365ket " }{TEXT 473 6 "alap\351l" }{TEXT -1 16 "eknek - hoss za " }{TEXT 256 2 "2a" }{TEXT -1 14 " , t\341vols\341guk " }{TEXT 257 1 "d" }{TEXT -1 7 " ! Az " }{TEXT 261 1 "a" }{TEXT -1 5 ", \351s " } {TEXT 260 2 "d " }{TEXT -1 189 "szakaszok ar\341nya - hasonl\363s\341g t\363l eltekintve - egy\351rtelm\373en meghat\341rozza a tetra\351dert , melyet azzal is egy\351rtelm\373en megadhattunk volna, hogy lapjai c supa egybev\341g\363 egyenl\365sz\341r\372 h\341romsz\366gek.\n" }} {PARA 0 "" 0 "" {TEXT -1 486 "Illessz\374nk \366ssze hat ilyen - megfe lel\365en m\351retezett - tetra\351dert az alap\351lei ment\351n \372g y, hogy a szomsz\351dos tetra\351derek a k\366z\366s alap\351leik ment \351n elmozdulhassanak, csukl\363s szerkezetnek tekintve a k\366z\366s alap\351lt. A hat tetra\351der alkosson egy z\341rt l\341ncot, vagyi s az els\365 egyik alap\351l\351vel a hatodikhoz csatlakozz\351k! \nA \+ hat tetra\351der egy - az \366sszeillesztett alap\351lek ment\351n - mozgathat\363 konstrukci\363t alkot. \nAzt a feladatot t\373zz\374k k i c\351lul, hogy ezt a mozg\341st le\355rjuk a MAPLE eszk\366zeivel. \+ " }}{PARA 257 "" 0 "" {GLPLOT3D 497 457 457 {PLOTDATA 3 "6%-%(ANIMATEG 6gp7(-%)POLYGONSG6&7%7%$\"+6cC)G#!\")$!+?8lFB!\"*$\"\"!F37%$\"+!\\?.u) F1$\"+I%[9=\"F.F27%$\"+n$>C&=F1$!+YCb:#*F1$\"+hW[3KF17%F4F+7%$\"+F\"p' z5F.$!+T`!Gr#!#5$\"+$yT+(=F.7%F+F9FA7%F9F4FA-F(6&7%7%$!+1G7W6F.F/$\"+I )y;)>F.7%$!+X-;qVF1F7$\"+&eR$pvF1F97%FSFNFA7%FNF9FA7%F9FSFA-F(6&7%FNFS 7%$!+K(Q[q$F1FF.7%FTF7$!+&eR$pvF1Fhn7%FgoFdoF\\o7%FdoFhnF\\o7%FhnFgoF \\o-F(6&7%FdoFgo7%F:F<$!+hW[3KF17%FgoFdo7%FBFD$!+$yT+(=F.7%FdoF`pFdp7% F`pFgoFdp-F(6&7%F+F4F`p7%F4F+Fdp7%F+F`pFdp7%F`pF4Fdp7(-F(6&7%7%$\"+?+* =I#F.$!+(RvW.$F1F27%$\"+@'>x*)*F1$\"+@;(f?\"F.F27%$\"+\"Hu.\\\"F1$!+?v !))\\)F1$\"+&*RS\"e#F17%FhqFcq7%$\"+uT:m5F.$!+?HLk_FF$\"+)>Lm%=F.7%Fcq F]rFer7%F]rFhqFer-F(6&7%7%$!+5]%4:\"F.Ffq$\"+M_\\$*>F.7%$!+6)f)[\\F1F[ r$\"+Nmnr&)F1F]r7%FfsFasFer7%FasF]rFer7%F]rFfsFer-F(6&7%FasFfs7%$!+#e[ 2)HF1F`rF27%FfsFas7%$!+Z$3B8#F.FhrF27%FasFatFet7%FatFfsFet-F(6&7%7%Fbs Ffq$!+M_\\$*>F.7%FgsF[r$!+Nmnr&)F1Fat7%F`uF]uFet7%F]uFatFet7%FatF`uFet -F(6&7%F]uF`u7%F^rF`r$!+&*RS\"e#F17%F`uF]u7%FfrFhr$!+)>Lm%=F.7%F]uFiuF ]v7%FiuF`uF]v-F(6&7%FcqFhqFiu7%FhqFcqF]v7%FcqFiuF]v7%FiuFhqF]v7(-F(6&7 %7%$\"+:**p3BF.$!+:1giPF1F27%$\"+p)p]5\"F.$\"+f4,@7F.F27%$\"+7#>(z6F1$ !+]Nc\\xF1$\"+:OLV?F17%FawF\\w7%$\"+bKv`5F.$!+qTXzpFF$\"+wT:D=F.7%F\\w FfwF^x7%FfwFawF^x-F(6&7%7%$!+e*\\V:\"F.F_w$\"+wFR**>F.7%$!+X$\\`_&F1Fd w$\"+&z&=q&*F1Ffw7%F_yFjxF^x7%FjxFfwF^x7%FfwF_yF^x-F(6&7%FjxF_y7%$!+D% Q%fBF1FiwF27%F_yFjx7%$!+4l]2@F.FaxF27%FjxFjyF^z7%FjyF_yF^z-F(6&7%7%F[y F_w$!+wFR**>F.7%F`yFdw$!+&z&=q&*F1Fjy7%FizFfzF^z7%FfzFjyF^z7%FjyFizF^z -F(6&7%FfzFiz7%FgwFiw$!+:OLV?F17%FizFfz7%F_xFax$!+wT:D=F.7%FfzFb[lFf[l 7%Fb[lFizFf[l-F(6&7%F\\wFawFb[l7%FawF\\wFf[l7%F\\wFb[lFf[l7%Fb[lFawFf[ l7(-F(6&7%7%$\"+/!)=3BF.$!+l/f,XF1F27%$\"+M*4*=7F.$\"+!4#=F7F.F27%$\"+ H%)*z9*FF$!+IxUtpF1$\"+2)zWe\"F17%Fj\\lFe\\l7%$\"+KofU5F.$!+:q-ozFF$\" +&oIe!=F.7%Fe\\lF_]lFg]l7%F_]lFj\\lFg]l-F(6&7%7%$!+-S4a6F.Fh\\l$\"+[% \\*)*>F.7%$!+q'\\X4'F1F]]l$\"+opgb5F.F_]l7%Fh^lFc^lFg]l7%Fc^lF_]lFg]l7 %F_]lFh^lFg]l-F(6&7%Fc^lFh^l7%$!+'o*fH=F1Fb]lF27%Fh^lFc^l7%$!+kO>&3#F. 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R\341klikkentve aktiviz\341l \363dik a mozg\341st vez\351rl\365 kezel\365fel\374let.)" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 20 " Az alakzat megad\341sa" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 28 "A feladat \341ltal\341nos elemz\351se" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 226 "A felt\351teleknek eleget tev\365 pap\355rmodellt - vagy a fenti mozg\363 modellt vizsg\341lva \351szre vehetj\374k, hogy k\351t-k\351t \"szemk\366zti\" alap\351l egy s\355kb an mozog, \355gy ezeknek a mozg\341s\341t le\355rva l\351nyeg\351ben m egadtuk az eg\351sz konstrukci\363 mozg\341s\341t.\n" }{TEXT 259 40 "A tetra\351derek alap\351lei legyenek rendre " }{TEXT 268 1 " " } {TEXT 275 44 "A1B1 , P1Q1, A2B2, P2Q2, A3B3, P3,Q3 ." }{TEXT 269 1 "\n" }{TEXT 474 4 " Az " }{TEXT 270 4 "A1B1" }{TEXT -1 29 " ala ppal szemk\366zti alap\351l a " }{TEXT 271 4 "P2Q2" }{TEXT -1 8 " , am ely" }{TEXT 346 1 " " }{TEXT -1 34 "kezd\365 helyzet\351ben \351ppen m er\365leges " }{TEXT 347 4 "A1B1" }{TEXT -1 22 "-re, \351s innen indul va " }{TEXT 348 4 "A1B1" }{TEXT -1 167 "-el ellent\351tes ir\341nyban \+ forog.\n\332gy adjuk meg az alakzat cs\372cspontjainak a koordin\341t \341it, hogy ez a tulajdons\341g m\341r a pontok koordin\341t\341inak \+ a megad\341s\341ban is t\374kr\366z\365dj\351k." }}{PARA 0 "" 0 "" {TEXT -1 9 "Jelentse " }{TEXT 411 1 "f" }{TEXT -1 4 " az " }{TEXT 412 4 "A1B1" }{TEXT -1 15 " szakasznak az " }{TEXT 421 2 "x " }{TEXT -1 13 "tengellyel, " }{TEXT 413 1 "g" }{TEXT -1 8 " pedig a" }{TEXT 414 5 " P2Q2" }{TEXT -1 9 " szakasz " }{TEXT 415 1 "y" }{TEXT -1 51 " teng ellyel bez\341rt sz\366g\351t! \315gy a kezd\365 helyzetben " }{TEXT 416 5 "f=g=0" }{TEXT -1 5 " .Az " }{TEXT 417 4 "A1B1" }{TEXT -1 28 " s zakasz felez\365pontja legyen" }{TEXT 418 9 " (x1,y,0)" }{TEXT -1 5 " \+ , a " }{TEXT 419 4 "P2Q2" }{TEXT -1 10 " szakasz\351 " }{TEXT 420 8 "( -x2,-y)" }{TEXT -1 4 " \nAz" }}{PARA 0 "" 0 "" {TEXT -1 9 " " }{TEXT 272 201 "xa:= x1 + a*cos(f): ya:= y - a*sin(f):\n \+ xb:= x1 - a*cos(f): yb:= y + a*sin(f):\n xp:= - x2 + a*s in(g): yp:= -y - a*cos(g):\n xq:= - x2 - a*sin(g): yq:= -y + a*cos(g):" }}{PARA 0 "" 0 "" {TEXT -1 69 "v\341ltoz\363kat bevezetve \+ az egy s\355kban mozg\363 n\351gy pont koordin\341t\341it az\n " } {TEXT 273 79 "A1:=[xa, ya, 0]: \n B1:=[xb, yb, 0]:\n P2:=[xp, yp, 0 ]: \n Q2:=[xq, yq, 0]: " }{TEXT 349 1 " " }{TEXT 350 2 " \n" } {TEXT -1 75 "koordin\341t\341kkal \355rhatjuk le.\nAz alakzatot meghat \341roz\363 t\366bbi pontot ezekb\365l a" }{TEXT 262 2 " z" }{TEXT -1 60 " tengely k\366r\374li 120\260-os elforgat\341ssal kapjuk. Pl. a le gels\365 " }{TEXT 274 11 "A1 B1 P1 Q1" }{TEXT -1 31 " tetra\351der tov \341bbi k\351t cs\372csa:\n" }{TEXT 276 69 " P1:=[-xp/2, yp, -xp*sqrt (3)/2]: \n Q1:=[-xq/2, yq, -xq*sqrt(3)/2] :" }{TEXT -1 4 "\nAz " } {TEXT 263 6 "(x1,y)" }{TEXT -1 33 " pont azt a \"p\341ly\341t\" \355rj a le az " }{TEXT 395 2 "xy" }{TEXT -1 21 " s\355kban, amelyen az " } {TEXT 264 4 "A1B1" }{TEXT -1 29 " szakasz felez\365pontja mozog, " } {TEXT 265 8 "(-x2,-y)" }{TEXT -1 43 " pedig azt, amelyen a vele egy s \355kban l\351v\365 " }{TEXT 266 4 "P2Q2" }{TEXT -1 59 " felez\365pont ja. Az\351rt elegend\365 mindk\351t ponthoz ugyanazt az " }{TEXT 267 1 "y" }{TEXT -1 53 " \351rt\351ket rendeln\374nk, mert l\351nyeg\351be n csak a k\351t pont " }{TEXT 422 2 "y " }{TEXT -1 153 "koordin\341t \341ja k\366z\366tti k\374l\366nbs\351get kell meghat\341roznunk. Ezze l biztos\355tjuk, hogy a mozg\341s k\366zben az eg\351sz alakzat \"t \366megk\366z\351ppontja\" az orig\363ban maradjon. " }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }{TEXT -1 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 42 "Az alakzat cs\372cspontjainak a meghat\341roz\341 sa" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "restart;with(plots):wi th(plottools):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name chang ecoords has been redefined\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 422 " Az alakzat pontjai k\366z\366tt fenn\341ll\363 - a koordin\341t\341k m egv\341laszt\341s\341ban nem t\374kr\366z\365d\365 - kapcsolatok le \355r\341s\341ra gyakran sz\374ks\351g\374nk lesz v\351gpontjai koordi n\341t\341ival adott szakasz hossz\341nak a fel\355r\341s\341ra, valam int annak eld\366nt\351s\351re, hogy k\351t szakasz (vektor) mer\365le ges-e egym\341sra, ez\351rt egy szakasz (vektor) hossz\341nak a n\351g yzet\351re, valamint k\351t vektor skal\341rszorzat\341ra bevezet\374n k egy-egy f\374ggv\351nyt. Az ut\363bbit megoldhattuk volna a \"" } {TEXT 277 1 "l" }{TEXT -1 1 "i" }{TEXT 278 4 "nalg" }{TEXT -1 145 "\" \+ programcsomag beh\355v\341s\341val is, de emiatt az egy f\374ggv\351ny miatt nem c\351lszer\373 terheln\374nk a programot, melyre feltehet \365en \355gy is sok sz\341mol\341s v\341r.)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 132 "v:=array[1..3]:\nw:=array[1..3]:\ntn:= (v,w) -> (v[1]-w[1])^2+(v[2]-w[2])^2+(v[3]-w[3])^2:\ns:= (v,w) ->v[1]*w[1]+v[2 ]*w[2]+v[3]*w[3]: " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "Az " } {TEXT 292 58 "A1=(xa,ya,0), B1= (xb,yb,0), P2=(xp,yp,0) , Q2=(xq,yq,0 ) " }{TEXT -1 76 " pontokkal fogjuk fel\355rni az alakzat \366sszes cs \372cs\341t.\nSz\374ks\351g\374nk lesz m\351g az " }{TEXT 316 4 "A1B1 " }{TEXT -1 1 "," }{TEXT 317 5 " P1Q1" }{TEXT -1 2 ", " }{TEXT 318 5 " A2,B2" }{TEXT -1 2 " ," }{TEXT 319 4 "P2Q2" }{TEXT -1 27 " szakaszok f elez\365pontjaira." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 516 "xa:= x1+a*cos(f): ya:= y-a*sin(f):\nxb:= x1-a*cos(f): yb:= y+a*sin(f):\n xp:=-x2+a*sin(g): yp:=-y-a*cos(g):\nxq:=-x2-a*sin(g): yq:=-y+a*cos(g ):\nA1:=[xa,ya,0]: \nB1:=[xb,yb,0]:\nA2:=[-xa/2,ya,xa*sqrt(3)/2]:\nB2 :=[-xb/2,yb,xb*sqrt(3)/2]:\nA3:=[-xa/2,ya, -xa*sqrt(3)/2]:\nB3:=[-xb/2 ,yb, -xb*sqrt(3)/2]:\n\nP2:=[xp,yp,0]: \nQ2:=[xq,yq,0]: \nP1:=[-xp /2,yp,-xp*sqrt(3)/2]: \nQ1:=[-xq/2,yq,-xq*sqrt(3)/2] :\nP3:=[-xp/2,yp, xp*sqrt(3)/2]: \nQ3:=[-xq/2,yq, xq*sqrt(3)/2] :\nF1:=(A1+B1)/2:\nF3:= (A2+B2)/2:\nF2:=(P1+Q1)/2:\nF4:=(P2+Q2)/2:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "Megadjuk mag\341nak az alakzatnak a lapjait, tetra\351der enk\351nt csoportos\355tva:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 294 "T:=[[A1,B1,P1],[B1,A1,Q1],[A1,P1,Q1],[P1,B1,Q1],\n [A2,B2,P1], [B2,A2,Q1],[A2,P1,Q1],[P1,B2,Q1],\n [A2,B2,P2],[B2,A2,Q2],[A2,P2,Q2 ],[P2,B2,Q2],\n [A3,B3,P2],[B3,A3,Q2],[A3,P2,Q2],[P2,B3,Q2],\n [ A3,B3,P3],[B3,A3,Q3],[A3,P3,Q3],[P3,B3,Q3],\n [A1,B1,P3],[B1,A1,Q3] ,[A1,P3,Q3],[P3,B1,Q3]]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{SECT 0 {PARA 20 "" 0 "" {TEXT -1 1 " " }{TEXT 291 12 "Kontroll 1 : " }{TEXT 321 70 "Az alakzat \355gy megadott koordin\341t\341ib\363l \+ ad\363d\363 \366sszef\374gg\351sek vizsg\341lata." }}{EXCHG {PARA 0 " " 0 "" {TEXT 279 161 "L\351nyeg\351ben azt vizsg\341ljuk, hogy amennyi ben a fenti k\351pletekkel adjuk meg az alakzat koordin\341t\341it, az ok j\363l t\374kr\366zik-e az ebb\365l ad\363d\363an elv\341rhat\363 t ulajdons\341gokat." }}{PARA 0 "" 0 "" {TEXT 379 63 "Els\365k\351nt \+ \355rassuk ki a sz\341mol\341shoz haszn\341lt pontok koordin\341t\341i t:" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 177 "print (`\n A1: `,A1,`\n B1: `,B1,`\n P2: `,P2,`\n Q2: `,Q2,`\n ________ ___\n\n`);\nprint(`\n F1: `,F1,`\n F2: `,simplify(F2),`\n F3: `, si mplify(F3),`\n F4: `,F4,`\n ___________\n`);" }{TEXT -1 0 "" }} {PARA 12 "" 1 "" {XPPMATH 20 "6+%(|+~~A1:~G7%,&%#x1G\"\"\"*&%\"aGF'-%$ cosG6#%\"fGF'F',&%\"yGF'*&F)F'-%$sinGF,F'!\"\"\"\"!%(|+~~B1:~G7%,&F&F' F(F3,&F/F'F0F'F4%(|+~~P2:~G7%,&%#x2GF3*&F)F'-F26#%\"gGF'F',&F/F3*&F)F' -F+F?F'F3F4%(|+~~Q2:~G7%,&F " 0 "" {MPLTEXT 1 0 81 "simplify(tn(A1, B1));simplify(tn(P1,Q1));\nsimplify(tn(A2,B2));simplify(tn(P2,Q2));" } {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$)%\"aG\"\"#\"\"\" \"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$)%\"aG\"\"#\"\"\"\"\"%" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$)%\"aG\"\"#\"\"\"\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$)%\"aG\"\"#\"\"\"\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "Mind a n\351gy szakasz hossza - term\351szetese n - " }{TEXT 320 2 "2a" }{TEXT -1 3 " .\n" }}{PARA 0 "" 0 "" {TEXT -1 27 "Megvizsg\341ljuk, - egyel\365re " }{TEXT 360 5 " g=f " }{TEXT -1 29 " felt\351tel mellett - hogy az " }{TEXT 361 4 "A1B1" }{TEXT -1 4 " \351s " }{TEXT 362 5 "P2Q2 " }{TEXT -1 69 "szakaszok val\363ban elle nt\351tes ir\341nyban forognak-e mer\365leges kezd\365 ( " }{TEXT 363 5 "f=g=0" }{TEXT -1 22 " ) helyzetb\365l indulva." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 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Ezt k\351s\365bb is t\366bbsz\366r meg fogjuk tenni. \+ Most ellen\365rizz\374k is, hogy a pontjaink koordin\341t\341i a felsz abad\355t\341s el\365tt val\363ban sz\341mok, ezt k\366vet\365en v\341 ltoz\363k." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 194 "print(`\n A 1: `,A1,`\n B1: `,B1,`\n P2: `,P2,`\n Q2: `,Q2,`\n ___________\n\n `);\nunassign('x1','x2','y','a','f','g'):\nprint(`\n A1: `,A1,`\n B1 : `,B1,`\n P2: `,P2,`\n Q2: `,Q2,`\n ___________\n\n`);" }{TEXT -1 0 "" }}{PARA 12 "" 1 "" {XPPMATH 20 "6+%(|+~~A1:~G7%$\"#D\"\"!$\"+qZ'e z\"!#=F'%(|+~~B1:~G7%$\"\"&F'$!+qZ'ez\"F*F'%(|+~~P2:~G7%$!+++++:!\")$! #5F'F'%(|+~~Q2:~G7%F3$\"#5F'F'%1|+~~___________|+|+G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6+%(|+~~A1:~G7%,&%#x1G\"\"\"*&%\"aGF'-%$cosG6#%\"fGF'F ',&%\"yGF'*&F)F'-%$sinGF,F'!\"\"\"\"!%(|+~~B1:~G7%,&F&F'F(F3,&F/F'F0F' F4%(|+~~P2:~G7%,&%#x2GF3*&F)F'-F26#%\"gGF'F',&F/F3*&F)F'-F+F?F'F3F4%(| +~~Q2:~G7%,&F " 0 "" {MPLTEXT 1 0 105 "a:=10:d:=a*sqrt (3):# d:=25:\nr:=(d-d*sqrt(3)/3):\nx1:=d-r*cos(2*f):\nx2:=d+r*cos(2*f) : \ny:=r*sin(2*f):\ng:=f:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 434 "k1:= circle([d,0], r, color=red):\nk2:= circle([-d,0], r, color=r ed):\npa:=plot([xa,ya,f=0..2*Pi],color=brown):\npp:=plot([xp,yp,f=0..2 *Pi],color=blue):\nrp:=90:e:=evalf(2*Pi/rp):\nfor i from 0 to rp do \n f:=i*e:\n F[i]:=line([xa,ya],[xb,yb], color=brown):\n G[i]:=line([xp, yp],[xq,yq], color=blue):\n unassign('f'):\nod:\ndisplay(seq(display( G[i],F[i],k1,k2,pa,pp,insequence=false),i=0..rp),insequence=true,scali ng= constrained,linestyle=16);" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%(ANIMATEG6gp7* -%'CURVESG6$7$7$$!+;;5kC!\")$!#5\"\"!7$F,$\"#5F1-%'COLOURG6&%$RGBG$F1F 1F9$\"*++++\"F.-F(6$7$7$$\"#?F1F97$F9F9-F66&F8$\")#)eqkF.$\"))eqk\"F.F G-F(6$7U7$$\"+;;5kCF.F97$$\"+x\"H$eCF.$\"+$\\H]<*F07$$\"+%*G5TCF.$\"+H j`?=!\"*7$$\"+OWp7CF.$\"+i(e[p#FY7$$\"+@=btBF.$\"+k;oENFY7$$\"+_BHCBF. $\"+*o')GI%FY7$$\"+\")GplAF.$\"+@EB6]FY7$$\"+bvn)>#F.$\"+?%[0k&FY7$$\" +YKIC@F.$\"++%44='FY7$$\"+$)GuV?F.$\"+ZPzBmFY7$$\"+[pEe>F.$\"+9p@ipFY7 $$\"+]MAp=F.$\"+c<%3>(FY7$$\"+em,y#H+$!#=7$Fjt$!+2&H]<*F07$Fet$!+Hj`?=FY7$F`t$!+i(e[p# FY7$$\"+&z\\04\"F.$!+j;oENFY7$Ffs$!+(o')GI%FY7$Fas$!+=EB6]FY7$F\\s$!+E %[0k&FY7$$\"+r$)zR8F.$!+1%44='FY7$Fbr$!+_PzBmFY7$F]r$!+9p@ipFY7$$\"+m \"y[f\"F.$!+d<%3>(FY7$$\"+e\\3'o\"F.$!+IF11tFY7$$\"+fm,y(FY7$$\"+\\pEe>F.$!+6p@ipFY7$$\"+#)GuV?F.$!+[PzBmF Y7$F^p$!+,%44='FY7$Fio$!+?%[0k&FY7$$\"+#)GplAF.$!+6EB6]FY7$$\"+`BHCBF. $!+zm)GI%FY7$$\"+A=btBF.$!+b;oENFY7$Fen$!+a(e[p#FY7$FU$!+@j`?=FY7$FP$! +>%H]<*F07$FM$\"+CQ%e+'Fcu-F66&F8F:F9F9-F(6$7U7$$!+++++5F.F97$$!+RCx05 F.FR7$$!+A()*H-\"F.FW7$$!+!=290\"F.Fgn7$$!+&z\\04\"F.F\\o7$$!+k#4)R6F. 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Ebb\365l ugyanis egy\351rtelm\373en k ider\374l, hogy r\366gz\355tett " }{TEXT 425 1 "a" }{TEXT -1 15 " \351 rt\351k mellett " }{TEXT 426 9 "a*sqrt(3)" }{TEXT -1 16 " az a minim \341lis " }{TEXT 427 2 "d " }{TEXT -1 76 "\351rt\351k, amelyn\351l m \351g \"nem akad meg\" mozg\341s k\366zben az alakzat. ha enn\351l kis ebb" }{TEXT 429 3 " d" }{TEXT -1 190 " , akkor az alakzat olykor \+ \366n\341tmetsz\365v\351 v\341lik. \n\nEredm\351ny\374nkb\365l egyel \365re csak azt \341llap\355thatjuk meg, hogy az alakzat koordin\341t \341it \351s abb\363l a lapjait helyesen \355rtuk fel, de alakzatunk c sak " }{TEXT 380 7 "\"olyan\"" }{TEXT -1 45 ", mint amelyet meg szeret n\351nk konstru\341lni, de" }{TEXT 381 8 " nem az," }{TEXT -1 93 " ugy anis az alakzat m\351retei nem konstansok. 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$\"3)****>YH&=$G'Ffv$\"36Lnea^p2YF,-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+ AXESLABELSG6$Q\"f6\"Q!6\"-%%VIEWG6$;F($\"+3`=$G'!\"*%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "Ugyan\355gy annak is " }{TEXT 383 1 "f" }{TEXT -1 86 " t\365l f\374ggetlen\374l teljes\374lnie kellene, \+ hogy a tetra\351derek alapjai mer\365legesek egym\341sra." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "s(B1-A1,P1-Q1);" }{TEXT -1 0 "" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&-%$cosG6#%\"fG\"\"\"-%$sinGF'F)!$+ #" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "Egyel\365re az sem teljes \374l, hogy a tetra\351der k\351t alap\351le b\341rmely " }{TEXT 475 2 "f " }{TEXT -1 90 "eset\351n mer\365leges egym\341sra. \nEz\351rt sz abad\355tsuk fel ism\351t v\341ltoz\363inkat, \351s l\341ssunk munk \341hoz." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 119 "unassign('x1', 'x2','y','a','d','f','g'):\nprint(`\n A1: `,A1,`\n B1: `,B1,`\n P2: `,P2,`\n Q2: `,Q2,`\n ___________\n\n`);" }{TEXT -1 0 "" }}{PARA 12 "" 1 "" {XPPMATH 20 "6+%(|+~~A1:~G7%,&%#x1G\"\"\"*&%\"aGF'-%$cosG6# %\"fGF'F',&%\"yGF'*&F)F'-%$sinGF,F'!\"\"\"\"!%(|+~~B1:~G7%,&F&F'F(F3,& F/F'F0F'F4%(|+~~P2:~G7%,&%#x2GF3*&F)F'-F26#%\"gGF'F',&F/F3*&F)F'-F+F?F 'F3F4%(|+~~Q2:~G7%,&F " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "Feladatunk olyan \366sszef\374gg\351sek felt\341r\341sa lesz, m elyek alapj\341n az " }{TEXT 293 2 "x1" }{TEXT -1 1 "," }{TEXT 385 3 " x2" }{TEXT -1 4 " \351s " }{TEXT 386 2 "y " }{TEXT -1 78 "v\341ltoz \363kat, majd ezek alapj\341n az alakzat pontjainak koordin\341t\341it fel\355rjuk az " }{TEXT 387 1 "a" }{TEXT -1 3 " , " }{TEXT 388 1 "d " }{TEXT -1 4 " \351s " }{TEXT 389 1 "f" }{TEXT -1 54 " v\341ltoz\363k f\374ggv\351nyek\351nt, melyben a megjelen\355t\351s sor\341n " } {TEXT 390 2 "a " }{TEXT -1 3 "\351s " }{TEXT 391 1 "d" }{TEXT -1 29 " \+ \351rt\351ke egy-egy konkr\351t sz\341m," }{TEXT 392 2 " f" }{TEXT -1 8 " pedig " }{TEXT 393 1 "0" }{TEXT -1 4 " \351s " }{TEXT 394 4 "2*Pi " }{TEXT -1 31 " k\366z\366tt v\341ltoz\363 param\351ter lesz." }} {PARA 0 "" 0 "" {TEXT 384 15 " Els\365 egyenlet:" }{TEXT -1 61 " \n \315rjuk fel azt az \366sszef\374gg\351st, amely biztos\355tja, hogy a z " }{TEXT 287 4 "A1B1" }{TEXT -1 4 " \351s " }{TEXT 288 4 "P1Q1" } {TEXT 289 1 " " }{TEXT -1 65 "alapok mer\365legesek legyenek egym\341s ra, azaz skal\341rszorzatuk nulla:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "e1:=s(B1-A1,P1-Q1)=0;" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#e1G/,&*()%\"aG\"\"#\"\"\"-%$cosG6#%\"fGF+-%$sin G6#%\"gGF+F***\"\"%F+F(F+-F1F.F+-F-F2F+!\"\"\"\"!" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 15 "m:=solve(e1,g);" }{TEXT -1 0 "" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"mG-%'arctanG6#,$-%$tanG6#%\"fG\"\"#" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "g:=arctan(2*tan(f));" } {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG-%'arctanG6#,$-% $tanG6#%\"fG\"\"#" 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{TEXT -1 20 "L\351nyeg\351ben nem is a " }{TEXT 471 1 "g" }{TEXT -1 58 "-t, hanem annak szinusz\341t \351s koszinusz\341t kell kifejezn \374nk " }{TEXT 472 1 "f" }{TEXT -1 5 "-el. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "simplify(sin(g));\nsimplify(cos(g));" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&-%$tanG6#%\"fG\"\"\"*$-%%sq rtG6#,&F)F)*&\"\"%F))F%\"\"#F)F)F)!\"\"F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\"\"F$*$-%%sqrtG6#,&F$F$*&\"\"%F$)-%$tanG6#%\"fG\" \"#F$F$F$!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 137 "Itt c\351lszer \373 \"k\351zzel\" beleavatkoznunk, a kapott t\366rteket b\365v\355ts \374k cos(x)-el, majd az \355gy kapott kifejez\351sekre vezess\374nk b e \372j v\341ltoz\363kat:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "sg:=simplify(2*sin(f)/sqrt(cos(f)^2+4*sin(f)^2));\ncg:=simplify(co s(f)/(sqrt(cos(f)^2+4*sin(f)^2)));" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#sgG,$*&-%$sinG6#%\"fG\"\"\"*$-%%sqrtG6#,&*$)-%$cosGF )\"\"#F+!\"$\"\"%F+F+!\"\"F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#cgG *&-%$cosG6#%\"fG\"\"\"*$-%%sqrtG6#,&*$)F&\"\"#F*!\"$\"\"%F*F*!\"\"" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 259 "" 0 "" {TEXT -1 13 "Kontroll3 : " }{TEXT 430 0 "" }{TEXT 431 3 "Az " } {TEXT 432 9 "sg=sin(g)" }{TEXT 433 4 " \351s " }{TEXT 434 9 "cg=cos(g) " }{TEXT 435 25 " helyettes\355t\351s vizsg\341lata" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 " Az " }{TEXT 439 2 "sg" }{TEXT -1 4 " \351s " } {TEXT 440 3 "cg " }{TEXT -1 21 "v\341ltoz\363k l\351nyeg\351ben a" } {TEXT 441 3 " P2" }{TEXT -1 4 " \351s " }{TEXT 442 3 "Q2 " }{TEXT -1 13 "pontoknak az " }{TEXT 492 3 "F4 " }{TEXT -1 31 "pont k\366r\374li forg\341s\341t \355rj\341k le." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "plot([sg,cg],f=0..2*Pi,scaling= constrained,color=[red,blue]) ;\nplot([sg,cg,f=0..2*Pi],scaling= constrained,color=[red,blue]);" } {TEXT -1 0 "" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6' -%'CURVESG6$7in7$$\"\"!F)F(7$$\"3;`#Gi#zxZo!#>$\"3m!\\n:#\\(*e8!#=7$$ \"3i]cC&eb&p8F0$\"3Q/#eUHqsl#F07$$\"3q5)>63x`'>F0$\"3Umf(Q09'*p$F07$$ \"3YqR*pd)>hDF0$\"3K%)Q'Qg,'RYF07$$\"3Y@%HT;i7B$F0$\"3qT(3zJ#ekbF07$$ \"3Zs[E^dK,RF0$\"3Ei)))oTk>N'F07$$\"3ab^NehL]_F0$\"3)e_dW,.+d(F07$$\"3 5*QhW&\\$Hf'F0$\"3T;)y$[>)GS)F07$$\"3zS11.fpPyF0$\"3bQ_dW?VQ*)F07$$\"3 upr'fwtl7*F0$\"3TF:(y@atK*F07$$\"3!QRa&4L&f/\"!#<$\"3MJH(yrhbg*F07$$\" 3YjXwg<#)y6F[o$\"3UIz*F07$$\"36qGW%H$\\:8F[o$\"3r$G?h:Af\"**F07 $$\"3$*)e)pbO(eV\"F[o$\"3c@'p#)3Xq(**F07$$\"396:YIMRr:F[o$\"2#RRLa&*** ****F[o7$$\"3!=3SCmpuq\"F[o$\"3+Z@_S,Ww**F07$$\"3LmSEEVgQ=F[o$\"3C'zJ. 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Az els\365 k\351plet k\351t f\374ggv\351nyt \+ \355r le, melyek f\374ggetlen v\341ltoz\363ja " }{TEXT 483 1 "f" } {TEXT -1 148 " , a m\341sodik pedig az ezek \341ltal meghat\341rozott \+ param\351teres alakban fel\355rt s\355kg\366rb\351t. E k\351t dolgot a k\351s\365bbiekben is gyakran fogjuk egy\374tt szeml\351lni." }} {PARA 0 "" 0 "" {TEXT -1 188 "Eredm\351ny\374nk nem csak az, hogy l \351nyegesen egyszer\373bb alakban kaptuk meg a keresett \366sszef\374 gg\351seket, hanem az is, hogy most m\341r minden\374tt folytonos \351 s differenci\341lhat\363 f\374ggv\351nyek \355rj\341k le az" }{TEXT 436 3 " f " }{TEXT -1 3 "\351s " }{TEXT 437 2 "g " }{TEXT -1 43 "k\366 z\366tti kapcsolatot, amelyet eredetileg az " }{TEXT 438 2 "e1" } {TEXT -1 21 " egyenletben kaptunk:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "unassign('g'):simplify(e1/a^2/2);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/,&*&-%$cosG6#%\"fG\"\"\"-%$sinG6#%\"gGF*F**(\"\"#F*- F,F(F*-F'F-F*!\"\"\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 " is(cos(f)*sg-2*sin(f)*cg=0);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 155 "B\341r a kapott eredm\351ny t\366k\351letesen megnyugtat\363, az \351rdekess \351g kedv\351\351rt vizsg\341ljuk meg, hogy ha ugyanarr\363l a pontr \363l indulva egyez\365 ir\341nyba forogna k\366rbe az " }{TEXT 443 2 "f " }{TEXT -1 3 "\351s " }{TEXT 444 2 "g " }{TEXT -1 40 "sz\366g, mil yen lenne a kapcsolat k\366z\366tt\374k." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 314 "k:= circle([0,0], 1, color=red):\nrp:=360:e:=evalf(2 *Pi/rp):\nfor i from 0 to rp do \nf:=i*e:\nF[i]:=line([0,0],[cos(f),\n sin(f)], color=brown):\nG[i]:=line([0,0],[cg,sg], color=blue):\n unassign('f');\nod:\ndisplay(seq(display(G[i],F[i],k,insequence=false) ,i=0..rp),insequence=true,scaling= constrained,linestyle=16);" }{TEXT -1 0 "" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%(ANI MATEG6eal7%-%'CURVESG6$7$7$$\"\"!F-F,7$$\"+++++5!\"*F,-%'COLOURG6&%$RG BGF,F,$\"*++++\"!\")-F(6$7$F+7$$\"\"\"F-F,-F36&F5$\")#)eqkF8$\"))eqk\" F8FC-F(6$7UF<7$$\"+8q9@**!#5$\"+OBL`7FK7$$\"+6;$eo*FK$\"+t))*o[#FK7$$ \"+f[w(H*FK$\"+FbC\"o$FK7$$\"++o1j()FK$\"+Tn`<[FK7$$\"+V*p,4)FK$\"+CD& y(eFK7$$\"+tio*G(FK$\"+g5ZXoFK7$$\"+'*)RUP'FK$\"+HC80xFK7$$\"+^zEe`FK$ \"+a#zKW)FK7$$\"+=HzdUFK$\"+B0F[!*FK7$$\"+Q*p,4$FK$\"+l^c5&*FK7$$\"+UJ \"Q(=FK$\"+3D(G#)*FK7$$\"+D>0zi!#6$\"+%Gn-)**FK7$$!+m>0ziF]qF^q7$$!+YJ \"Q(=FK$\"+2D(G#)*FK7$$!+U*p,4$FK$\"+j^c5&*FK7$$!+8HzdUFK$\"+E0F[!*FK7 $$!+YzEe`FK$\"+d#zKW)FK7$$!+$*)RUP'FK$\"+JC80xFK7$$!+$G'o*G(FK$\"+]5ZX oFK7$$!+]*p,4)FK$\"+9D&y(eFK7$$!+0o1j()FK$\"+Kn`<[FK7$$!+i[w(H*FK$\"+> bC\"o$FK7$$!+8;$eo*FK$\"+l))*o[#FK7$$!+9q9@**FK$\"+IBL`7FK7$$!\"\"F-$! 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\nplot(g,f=0..2*Pi,scaling= constrained,color=[red,blue]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG*&-%'signumG6#-%$cosG6#%\"fG\"\"\"-%'a rctanG6#,$-%$tanGF+\"\"#F-" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$7W7$$\"\"!F)F(7$$\"3i]cC&eb&p8!#=$\"3aiYMh) z&*o#F-7$$\"3YqR*pd)>hDF-$\"3e!GeLD/Y#[F-7$$\"3Zs[E^dK,RF-$\"3MVRtg$GE )oF-7$$\"3ab^NehL]_F-$\"3Y>oA--5(e)F-7$$\"35*QhW&\\$Hf'F-$\"3c3y!4\"f9 y**F-7$$\"3zS11.fpPyF-$\"3J$)3e%eWe5\"!#<7$$\"3upr'fwtl7*F-$\"3h)ozXXE >?\"FI7$$\"3!QRa&4L&f/\"FI$\"3gNX*oN&**)G\"FI7$$\"3YjXwg<#)y6FI$\"3ieF \"Ri$)pO\"FI7$$\"36qGW%H$\\:8FI$\"3)GaZ+WI5W\"FI7$$\"3$*)e)pbO(eV\"FI$ \"3Hi]kql-.:FI7$$\"3/]+3VNj.:FI$\"3][$G@-xr`\"FI7$$\"396:YIMRr:FI$\"3W y\\$\\x(\\q:FI7$$\"3Z'z]kaJ%R;FI$\"3*R%f``#Qk`\"FI7$$\"3!=3SCmpuq\"FI$ \"3PVS;Q!R@]\"FI7$$\"3LmSEEVgQ=FI$\"3\"=1SNxZWV\"FI7$$\"3S`:%R;(od>FI$ \"3Ud4y*fL)p8FI7$$\"3#=uye<)G*4#FI$\"31$\\RvQWnG\"FI7$$\"3Ua[#o!GC>AFI $\"3(Q64-7I'37FI7$$\"3-YwJf&y(eBFI$\"3[L8q]#y]5\"FI7$$\"3oNrpV8H#[#FI$ \"3;!R\"eM%>#y**F-7$$\"3EzY)QW/yh#FI$\"3Y[OKNT)Gd)F-7$$\"35v)>8'\\%ou# FI$\"3a!=-g;\"yYpF-7$$\"3w$GCS?&[\")GFI$\"3m6vBtfJ\"*[F-7$$\"3Z%)='p(p 70IFI$\"3mIX.V$*H!o#F-7$$\"3sA%)\\K8\\QJFI$\"3DM?oDDe-i!#?7$$\"3c&fJlT >qF$FI$!3#)[f%3B11m#F-7$$\"39l2\"4_3wR$FI$!3S7Q+2,'G#[F-7$$\"3MOnGd![y _$FI$!3O(pMHb&RGoF-7$$\"3#*=4JU#)RiOFI$!36=+_V5tQ&)F-7$$\"3%G,f%[$HSz$ FI$!3(>(R$)p#GL\"**F-7$$\"3KoE+7#*Q@RFI$!3C&Q*p/Bl-6FI7$$\"3y(f0ii+G1% FI$!3#3nr\"HP%y?\"FI7$$\"3;fORr]')*=%FI$!3'zZlxfA/H\"FI7$$\"3R&p%*z[Lb K%FI$!3_dcV%Q`)p8FI7$$\"3M'pExBp%[WFI$!3z3^De!)\\O9FI7$$\"3oh/x$[qGe%F I$!3ME-E^Vw0:FI7$$\"3qQq'H.,hk%FI$!3^c+k.fhP:FI7$$\"3e;O;#eJ$4ZFI$!3e- >6CwEp:FI7$$\"3sU'G^sDax%FI$!3E.#*p7mCR:FI7$$\"3&)oO4o)>:%[FI$!3SR^u.1 'f]\"FI7$$\"3Ou\"4%z!e2(\\FI$!3tN/*4W>%R9FI7$$\"3*zX#[4&eg5&FI$!3)QTVS ]JgO\"FI7$$\"3n*e![/*ojB&FI$!3ITYiOo_*G\"FI7$$\"3#ob8X5I'p`FI$!3BD>`n, b-7FI7$$\"3PA\"GX$yy,bFI$!3J/$3zXSR5\"FI7$$\"39L1/jqABcFI$!39gFI$!3'4-%3h\\z[\\F-7$$\"3w)*zj%)[mYhFI$!31\"HrXkO8o#F- 7$$\"3)****>YH&=$G'FI$!3v$Gl4I<>^#!#D-%'COLOURG6&%$RGBG$\"*++++\"!\")F (F(-%+AXESLABELSG6$Q\"f6\"Q!6\"-%(SCALINGG6#%,CONSTRAINEDG-%%VIEWG6$;F ($\"+3`=$G'!\"*%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "Az \372j," }{TEXT 445 4 " sg " }{TEXT -1 3 "\351s " } {TEXT 446 2 "cg" }{TEXT -1 48 " v\341ltoz\363kat bevezetve \372jra kel l defini\341lnunk a " }{TEXT 486 1 "g" }{TEXT -1 51 "-t param\351terk \351nt tartalmaz\363 v\341ltoz\363kat \351s pontokat." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 324 "xp:=-x2+a*sg; yp:=-y-a*cg;\nxq:=- x2-a*sg; yq:=-y+a*cg;\nP2:=[xp,yp,0]: \nQ2:=[xq,yq,0]: \nP1:=[-xp /2,yp,-xp*sqrt(3)/2]: \nQ1:=[-xq/2,yq,-xq*sqrt(3)/2] :\nP3:=[-xp/2,yp, xp*sqrt(3)/2]: \nQ3:=[-xq/2,yq, xq*sqrt(3)/2] :\nF2:=(P1+Q1)/2:\nF4:= (P2+Q2)/2:\nprint(`\n P1: `,P1,`\n Q1: `,Q1,`\n P2: `,P2,`\n Q2: ` ,Q2,`\n ___________\n\n`);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xpG,&%#x2G!\"\"*&*(\"\"#\"\"\"%\"aGF+-%$sinG6#%\"fGF +F+*$-%%sqrtG6#,&*$)-%$cosGF/F*F+!\"$\"\"%F+F+F'F+" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#ypG,&%\"yG!\"\"*&*&%\"aG\"\"\"-%$cosG6#%\"fGF+F+*$ -%%sqrtG6#,&*$)F,\"\"#F+!\"$\"\"%F+F+F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xqG,&%#x2G!\"\"*&*(\"\"#\"\"\"%\"aGF+-%$sinG6#%\"fGF +F+*$-%%sqrtG6#,&*$)-%$cosGF/F*F+!\"$\"\"%F+F+F'F'" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#yqG,&%\"yG!\"\"*&*&%\"aG\"\"\"-%$cosG6#%\"fGF+F+*$ -%%sqrtG6#,&*$)F,\"\"#F+!\"$\"\"%F+F+F'F+" }}{PARA 12 "" 1 "" {XPPMATH 20 "6+%(|+~~P1:~G7%,&%#x2G#\"\"\"\"\"#*&*&%\"aGF(-%$sinG6#%\" fGF(F(*$-%%sqrtG6#,&*$)-%$cosGF/F)F(!\"$\"\"%F(F(!\"\"F<,&%\"yGF<*&*&F ,F(F8F(F(*$-F36#F5F(FF " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 294 17 "M\341sodik egyenlet:" } {TEXT -1 68 "\nA tetra\351der egy\351rtelm\373 megad\341s\341hoz az al ap\351lei k\366z\366tti t\341vols\341got, " }{TEXT 448 4 " d -" } {TEXT -1 83 "t vezett\374k be param\351terk\351nt. Ez ad lehet\365s \351get arra, hogy kapcsolatot keress\374nk az " }{TEXT 449 2 "x1" } {TEXT -1 1 "," }{TEXT 450 3 " x2" }{TEXT -1 5 ", \351s " }{TEXT 451 1 "y" }{TEXT -1 17 " v\341ltoz\363k k\366z\366tt." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 18 "e2:=tn(F1,F2)=d^2;" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#e2G/,(*$),&%#x1G\"\"\"*&#F+\"\"#F+%#x2GF+!\" \"F.F+F+*&\"\"%F+)%\"yGF.F+F+*$),&*&,&F/F0*&*(F.F+%\"aGF+-%$sinG6#%\"f GF+F+*$-%%sqrtG6#,&*$)-%$cosGF?F.F+!\"$F2F+F+F0F0F+-FC6#\"\"$F+#F+F2*( FNF+,&F/F0*&*(F.F+F \+ " 0 "" {MPLTEXT 1 0 16 "m2:=solve(e2,y);" }{TEXT -1 0 "" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#m2G6$,$*$-%%sqrtG6#,**$)%#x1G\"\"#\"\"\"!\"\" *&F.F0%#x2GF0F0*$)F3F/F0F1*$)%\"dGF/F0F0F0#F0F/,$F'#F1F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "y:=m2[1];" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yG,$*$-%%sqrtG6#,**$)%#x1G\"\"#\"\"\"!\" \"*&F-F/%#x2GF/F/*$)F2F.F/F0*$)%\"dGF.F/F/F/#F/F." }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 49 "Kiv\341lasztottuk a pozit\355v gy\366k\366t., \351 rt\351k\351t \341tadva " }{TEXT 447 1 "y" }{TEXT -1 5 "-nak." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }{TEXT -1 0 "" }}}{SECT 0 {PARA 259 "" 0 "" {TEXT -1 12 "Kontroll 4: " }{TEXT 351 50 "A tetra \351der alap\351lei k\366z\366tti kapcsolat vizsg\341lata." }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "Megvizsg\341ljuk, hogy az \355gy kapott \+ " }{TEXT 296 1 "y" }{TEXT -1 111 " \351rt\351k mellett ugyanakkora-e a z \366sszes szomsz\351dos alap\351lek felez\365pontjai k\366z\366tti t \341vols\341g, valamint, hogy - a " }{TEXT 322 15 "tan(g)=2*tan(f)" } {TEXT -1 79 " \366sszef\374gg\351sb\365l ad\363d\363an - a szomsz\351d os alap\351lek val\363ban mer\365legesek-e egym\341sra." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "is(tn(F1,F2)=d^2);\nis(tn(F2,F3)=d^ 2);\nis(tn(F3,F4)=d^2);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "is(s(A1-B1,P1-Q1)=0);\nis(s(P1-Q1,A2-B2)=0);\nis(s(A2-B2,P2-Q2)= 0);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Ez megnyugtat \363 eredm\351ny." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 297 19 "Harmadik egyenlet:\n" }{TEXT -1 86 "Eddig nem haszn\341l tuk ki, hogy a tetra\351dereket csupa egyenl\365sz\341r\372 h\341romsz \366gek alkotj\341k." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "e3: =tn(A1,P1)=tn(A1,Q1);" }{TEXT -1 0 "" }}{PARA 12 "" 1 "" {XPPMATH 20 " 6#>%#e3G/,(*$),*%#x1G\"\"\"*&%\"aGF+-%$cosG6#%\"fGF+F+*&#F+\"\"#F+%#x2 GF+!\"\"*&*&F-F+-%$sinGF0F+F+*$-%%sqrtG6#,&*$)F.F4F+!\"$\"\"%F+F+F6F+F 4F+F+*$),(*$-F=6#,**$)F*F4F+F6*&F*F+F5F+F+*$)F5F4F+F6*$)%\"dGF4F+F+F+F +*&F-F+F9F+F6*&*&F-F+F.F+F+*$-F=6#F?F+F6F+F4F+F+*&#\"\"$FCF+),&F5F6*&* (F4F+F-F+F9F+F+*$-F=6#F?F+F6F+F4F+F+,(*$),*F*F+F,F+*&#F+F4F+F5F+F6F7F6 F4F+F+*$),(FGF+FSF6FTF6F4F+F+*&FZF+),&F5F6*&*(F4F+F-F+F9F+F+*$-F=6#F?F +F6F6F4F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "m3:=solve(e3 ,x2):\nsimplify(m3[1]);\nsimplify(m3[2]);" }{TEXT -1 0 "" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,$*&,(%#x1G\"\"\"*(\"\"%F'F&F')-%$tanG6#%\"fG\" \"#F'F'*$-%%sqrtG6#,$*&,&F'F'*&F)F'F*F'F'F',&*$)F&F/F'\"\"$*&F)F')%\"d GF/F'!\"\"F'F?F'F'F'F6F?#F'F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&, (%#x1G\"\"\"*(\"\"%F'F&F')-%$tanG6#%\"fG\"\"#F'F'*$-%%sqrtG6#,$*&,&F'F '*&F)F'F*F'F'F',&*$)F&F/F'\"\"$*&F)F')%\"dGF/F'!\"\"F'F?F'F?F'F6F?#F'F /" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 126 "A sz\363ba j\366het\365 k \351t gy\366k k\366z\374l v\341lasszuk ki a nagyobbat. Ha d\366nt\351s \374nk helytelennek bizonyul, erre a pontra vissza kell t\351rn\374nk. " }}{PARA 0 "" 0 "" {TEXT -1 47 "Ism\351t \351lve egyszer\373bb fel \355r\341s lehet\365s\351g\351vel az " }{TEXT 323 11 "x2:=m3[1] " } {TEXT -1 22 "helyettes\355t\351s helyett " }{TEXT 299 3 " x2" }{TEXT -1 35 " -t fejezz\374k ki kiss\351 eleg\341nsabban:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "x2:=(x1+sqrt((4*d^2-3*x1^2)/(1+4*tan(f)^2 )))/2;" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x2G,&%#x1G #\"\"\"\"\"#*&F'F(-%%sqrtG6#*&,&*$)F&F)F(!\"$*&\"\"%F()%\"dGF)F(F(F(,& F(F(*&F4F()-%$tanG6#%\"fGF)F(F(!\"\"F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 82 "Itt egy kicsit seg\355teni kellett a MAPLE -nek. Ellen \365rizz\374k , hogy j\363- e az \341t\355r\341s:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 37 "simplify(((m3[1]-x1/2)/(x2-x1/2))^2);" } {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 161 "Nem a legegyszer\373bb m\363dj\341t v \341lasztottuk az ellen\365rz\351snek, de a MAPLE \363vatosan b\341nik az egyszer\373s\355t\351sekkel, n\351ha kev\351sb\351 l\341tja \341t, mint mi. (N\351ha viszont jobban.)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Ezt a behelyettes\355t\351st elv\351gezve " }{TEXT 356 1 "y" } {TEXT -1 10 " m\341r csak " }{TEXT 357 2 "x1" }{TEXT -1 12 " valamint \+ a " }{TEXT 358 2 "d " }{TEXT -1 3 "\351s " }{TEXT 359 2 "f " }{TEXT -1 23 "param\351terek f\374ggv\351nye:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "print(` y =`,y);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%%~y~=G,$*$-%%sqrtG6#,**$)%#x1G\"\"#\"\"\"!\"\"*&F,F.,& F,#F.F-*&F2F.-F'6#*&,&F*!\"$*&\"\"%F.)%\"dGF-F.F.F.,&F.F.*&F:F.)-%$tan G6#%\"fGF-F.F.F/F.F.F.F.*$)F1F-F.F/*$F;F.F.F.F2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 298 18 "Negyedik egyenlet:" }{TEXT -1 24 "\nHaszn\341ljuk k i, hogy az " }{TEXT 300 4 "A1P1" }{TEXT -1 21 " szakasz kifejezhet\365 " }{TEXT 324 2 "d " }{TEXT -1 8 "-vel \351s " }{TEXT 325 1 "a" } {TEXT -1 80 " -val. Hossz\341nak n\351gyzete a Pithagorasz t\351tel k \351tszeri alkalmaz\341s\341b\363l ad\363d\363an " }{TEXT 401 12 "d^ 2 + 2 a^2 " }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "e4:=simplify(tn(A1,P1))=d^2+2*a^2;" }{TEXT -1 0 "" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#e4G/,$*&,:*(%#x1G\"\"\"%\"aGF+)-%$cosG6#%\"fG\" \"$F+\"\"***\"#7F+F*F+F,F+F.F+!\"\"*(F5F+)F,\"\"#F+)F.F9F+F+**F2F+F,F+ F-F+-%%sqrtG6#*&*&,&*$)F*F9F+F2*&\"\"%F+)%\"dGF9F+F6F+F:F+F+,&*$F:F+F2 FEF6F6F+F6**FEF+F,F+F.F+F " 0 "" {MPLTEXT 1 0 17 "m4:=solve(e4,x1):" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "simplify(m4[1]);\nsimplify(m4[2]); \nsimplify(m4[3]);\nsimplify(m4[4]);" }{TEXT -1 0 "" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,$*&*(%\"dG\"\"\"-%%sqrtG6#\"\"$F'-F)6#,$*&)-%$cosG6# %\"fG\"\"#F',&*$F0F'F+\"\"%!\"\"F'F9F'F'F1F9#F'F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&*(%\"dG\"\"\"-%%sqrtG6#\"\"$F'-F)6#,$*&)-%$cosG6#% \"fG\"\"#F',&*$F0F'F+\"\"%!\"\"F'F9F'F'F1F9#F9F+" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,$*&**,>*$-%%sqrtG6#,&*$)-%$cosG6#%\"fG\"\"#\"\"\"!\"$ \"\"%F3F3\"%C5*(\"%[?F3-%$sinGF0F3F.F3F3*(\"$)GF3F(F3)F.\"#7F3F3*(\"%n QF3F(F3)F.\"#5F3!\"\"*(\"&+V\"F3F(F3)F.\"\")F3F3*(\"$S&F3F9F3)F.\"#8F3 F3*(\"&3I#F3F(F3)F.\"\"'F3FC*(\"&/^\"F3F9F3)F.\"\"$F3FC*(\"&)GWF3F9F3) F.\"\"(F3FC*(\"&;!QF3F9F3)F.\"\"&F3F3*(\"&#RDF3F9F3)F.\"\"*F3F3*(\"%(f 'F3F9F3)F.\"#6F3FC*(\"%7pF3F(F3F-F3FC*(\"&w\"=F3F(F3)F.F5F3F3F3%\"dGF3 -F)6#FSF3-F)6#*(,0\"$c#F3*&\"$W\"F3F=F3F3*&\"%/*$-%%sqrtG6#,&*$)-%$cosG6#%\"fG\"\"#\"\"\"! \"$\"\"%F3F3\"%C5*(\"%[?F3-%$sinGF0F3F.F3F3*(\"$)GF3F(F3)F.\"#7F3F3*( \"%nQF3F(F3)F.\"#5F3!\"\"*(\"&+V\"F3F(F3)F.\"\")F3F3*(\"$S&F3F9F3)F.\" #8F3F3*(\"&3I#F3F(F3)F.\"\"'F3FC*(\"&/^\"F3F9F3)F.\"\"$F3FC*(\"&)GWF3F 9F3)F.\"\"(F3FC*(\"&;!QF3F9F3)F.\"\"&F3F3*(\"&#RDF3F9F3)F.\"\"*F3F3*( \"%(f'F3F9F3)F.\"#6F3FC*(\"%7pF3F(F3F-F3FC*(\"&w\"=F3F(F3)F.F5F3F3F3% \"dGF3-F)6#FSF3-F)6#**,0\"$c#F3*&\"$W\"F3F=F3F3*&\"%/*&\"#@F3FboF3FC**\"#?F3F9F3FRF3F(F3F3*&\"#CF3F-F3F3**\"# ;F3F9F3F.F3F(F3FCFaqFCF3F3F3**F.F3,(*&F9F3FRF3FS*&F5F3F(F3FC*(FenF3F(F 3F-F3F3F3,2*(F9F3FZF3F(F3\"#g**\"#kF3F9F3F.F3F(F3F3*&\"#OF3FFF3F3*&\"$ 6\"F3FNF3FC**\"$G\"F3F9F3FRF3F(F3FC*&\"$c\"F3FboF3F3F[rF3*&F\\pF3F-F3F CF3FioF3FC#F3FS" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 134 "Az els\365 pi llant\341sra ijeszt\365 megold\341sok k\366z\374l (valamelyik) kett \365 igen kellemes \366sszef\374gg\351ss\351 szel\355d\374lt, amit to v\341bb egyszer\373s\355thet\374nk." }}{PARA 0 "" 0 "" {TEXT 402 58 "I tt h\355vjuk fel a figyelmet arra, hogy n\351ha nem kapjuk meg " } {TEXT 355 5 "m4[1]" }{TEXT 403 47 " -ben a sz\341munkra haszn\341lhat \363 gy\366k\366t. Ugyanis " }{TEXT 352 3 "e4 " }{TEXT 404 2 "az" } {TEXT 353 4 " x1 " }{TEXT 405 58 "v\341ltoz\363ra n\351zve negyedfok \372 egyenlet, melynek a gy\366keit az " }{TEXT 354 2 "m " }{TEXT 406 146 "vektor tetsz\365leges sorrendben tartalmazza, \355gyvagy fel\355 rjuk mind a n\351gyet, vagy \"ki kell tapogatnunk\", hogy ezek k\366z \374l melyikre lesz sz\374ks\351g\374nk." }{TEXT 301 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "x1:=1/3*d*sqrt(3)*sqrt((4-3*cos(f)^ 2));" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G,$*(%\"dG \"\"\"-%%sqrtG6#\"\"$F(-F*6#,&*$)-%$cosG6#%\"fG\"\"#F(!\"$\"\"%F(F(#F( F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "simplify(x2);\nsimpli fy(y);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*(%\"dG\"\" \"-%%sqrtG6#\"\"$F&-F(6#,&*$)-%$cosG6#%\"fG\"\"#F&!\"$\"\"%F&F&#F&\"\" '*(#F&F4F&F'F&-F(6#,$*&*&)F%F4F&)F0F6F&F&,&F.F*F6!\"\"FCFCF&F&" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&-%%sqrtG6#\"\"$\"\"\"-F&6#*&*()%\" dG\"\"#F))-%$cosG6#%\"fGF0F),&!\"\"F)*$F1F)F)F)F),&F8F(\"\"%F7F7F)#F)F 0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "B\341r m\341r megkaptuk az \+ " }{TEXT 407 2 "x1" }{TEXT -1 3 " , " }{TEXT 408 2 "x2" }{TEXT -1 4 " \+ \351s " }{TEXT 409 1 "y" }{TEXT -1 129 " v\341ltoz\363kat a k\355v\341 nt param\351terekkel kifejezve, eredm\351nyeinket \355rjuk fel \372jb \363l, min\351l egyszer\373bb \351s k\366nnyebben kezelhet\365 form \341ban." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 109 "Itt nem k\366nny\373 \351szrevenni az egyszer\373s\355t\351si lehet\365s\351get. (A MAPLE \+ nem vette \351szre.) Mi azonban \351szrevehetj\374k:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "simplify(x1*x2/(2*d^2/3));" }{TEXT -1 0 " " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&*&-%%sqrtG6#,&*$)-%$cosG6#%\"f G\"\"#\"\"\"!\"$\"\"%F1F1,&*&%\"dGF1F&F1F1*&\"\"$F1-F'6#,$*&*&)F6F0F1) F,F3F1F1,&F*F8F3!\"\"FAFAF1F1F1F1F6FA#F1F3" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "Ebb\365l az alakb\363l - sz\341munkra - m\341r l\341tszik , hogy " }{TEXT 452 14 "x1*x2=2*d^2/3 " }{TEXT -1 16 " Ez\351rt c\351l szer\373 " }{TEXT 453 2 "x2" }{TEXT -1 20 "-t \355gy defini\341lnunk: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "x2:=2*d*sqrt(3)/sqrt(-3 *cos(f)^2+4)/3;" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x 2G,$*&*&%\"dG\"\"\"-%%sqrtG6#\"\"$F)F)*$-F+6#,&*$)-%$cosG6#%\"fG\"\"#F )!\"$\"\"%F)F)!\"\"#F8F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 487 1 "y" } {TEXT -1 45 " -t ugyancsak fel\355rhatjuk egyszer\373bb alakban:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "y:=d*sqrt(3)*sin(f)*cos(f)/2 /sqrt(-3*cos(f)^2+4);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%\"yG,$*&**%\"dG\"\"\"-%%sqrtG6#\"\"$F)-%$sinG6#%\"fGF)-%$cosGF0F)F )*$-F+6#,&*$)F2\"\"#F)!\"$\"\"%F)F)!\"\"#F)F:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 231 "B\341r ezekkel az \372j defin\355ci\363kkal \372jra d efini\341ltuk a pontok koordin\341t\341it is, azonban hasznos lehet \+ \372jb\363l megadni \365ket, kihaszn\341lva a k\355n\341lkoz\363 egysz er\373s\355t\351si lehet\365s\351geket, melyeket r\351szben a MAPLE r \351szben mi v\351gezhet\374nk el \"k\351zzel\":" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "simplify(xa );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*(%\"dG\"\"\"-%%sqrtG6#\"\"$F& -F(6#,&*$)-%$cosG6#%\"fG\"\"#F&!\"$\"\"%F&F&#F&F**&%\"aGF&F0F&F&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "xa:=1/3*d*sqrt(3)*sqrt(-3*co s(f)^2+4)+a*cos(f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xaG,&*(%\"dG \"\"\"-%%sqrtG6#\"\"$F(-F*6#,&*$)-%$cosG6#%\"fG\"\"#F(!\"$\"\"%F(F(#F( F,*&%\"aGF(F2F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "simpli fy(ya);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&*&-%$sin G6#%\"fG\"\"\",&*(%\"dGF*-%%sqrtG6#\"\"$F*-%$cosGF(F*F**(\"\"#F*%\"aGF *-F/6#,&*$)F2F5F*!\"$\"\"%F*F*!\"\"F*F**$-F/6#F9F*F>#F*F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "ya:=1/2*sin(f)*(d*sqrt(3)*cos(f))/( sqrt(-3*cos(f)^2+4))-a*sin(f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y aG,&*&**%\"dG\"\"\"-%%sqrtG6#\"\"$F)-%$sinG6#%\"fGF)-%$cosGF0F)F)*$-F+ 6#,&*$)F2\"\"#F)!\"$\"\"%F)F)!\"\"#F)F:*&%\"aGF)F.F)F=" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "simplify(xb);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*(%\"dG\"\"\"-%%sqrtG6#\"\"$F&-F(6#,&*$)-%$cosG6#%\"f G\"\"#F&!\"$\"\"%F&F&#F&F**&%\"aGF&F0F&!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "xb:=1/3*d*sqrt(3)*sqrt(-3*cos(f)^2+4)-a*cos(f); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xbG,&*(%\"dG\"\"\"-%%sqrtG6#\" \"$F(-F*6#,&*$)-%$cosG6#%\"fG\"\"#F(!\"$\"\"%F(F(#F(F,*&%\"aGF(F2F(!\" \"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "simplify(yb);" } {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&*&-%$sinG6#%\"fG\" \"\",&*(%\"dGF*-%%sqrtG6#\"\"$F*-%$cosGF(F*F**(\"\"#F*%\"aGF*-F/6#,&*$ )F2F5F*!\"$\"\"%F*F*F*F*F**$-F/6#F9F*!\"\"#F*F5" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 65 "yb:=1/2*sin(f)*(d*sqrt(3)*cos(f))/(sqrt(-3*cos (f)^2+4))+a*sin(f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ybG,&*&**%\" dG\"\"\"-%%sqrtG6#\"\"$F)-%$sinG6#%\"fGF)-%$cosGF0F)F)*$-F+6#,&*$)F2\" \"#F)!\"$\"\"%F)F)!\"\"#F)F:*&%\"aGF)F.F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "simplify(xp);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,&*&%\"dG\"\"\"-%%sqrtG6#\"\"$F(F(*(F,F(%\"aGF(-%$s inG6#%\"fGF(!\"\"F(*$-F*6#,&*$)-%$cosGF1\"\"#F(!\"$\"\"%F(F(F3#!\"#F, " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "xp:=2/3*(-d*sqrt(3)+3*a *sin(f))/(sqrt(-3*cos(f)^2+4));" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xpG,$*&,&*&%\"dG\"\"\"-%%sqrtG6#\"\"$F*!\"\"*(F.F*% \"aGF*-%$sinG6#%\"fGF*F*F**$-F,6#,&*$)-%$cosGF4\"\"#F*!\"$\"\"%F*F*F/# F>F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "simplify(yp);" } {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&*&-%$cosG6#%\"fG\" \"\",&*(%\"dGF*-%%sqrtG6#\"\"$F*-%$sinGF(F*F**&\"\"#F*%\"aGF*F*F*F**$- F/6#,&*$)F&F5F*!\"$\"\"%F*F*!\"\"#F?F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "yp:=-1/2*cos(f)*(d*sqrt(3)*sin(f)+2*a)/(sqrt(-3*cos(f )^2+4));" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ypG,$*&* &-%$cosG6#%\"fG\"\"\",&*(%\"dGF,-%%sqrtG6#\"\"$F,-%$sinGF*F,F,*&\"\"#F ,%\"aGF,F,F,F,*$-F16#,&*$)F(F7F,!\"$\"\"%F,F,!\"\"#FAF7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "simplify(xq);" }{TEXT -1 0 "" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,&*&%\"dG\"\"\"-%%sqrtG6#\"\"$F(F( *(F,F(%\"aGF(-%$sinG6#%\"fGF(F(F(*$-F*6#,&*$)-%$cosGF1\"\"#F(!\"$\"\"% F(F(!\"\"#!\"#F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "xq:=-2/ 3*(d*sqrt(3)+3*a*sin(f))/(sqrt(-3*cos(f)^2+4));" }{TEXT -1 0 "" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xqG,$*&,&*&%\"dG\"\"\"-%%sqrtG6#\" \"$F*F**(F.F*%\"aGF*-%$sinG6#%\"fGF*F*F**$-F,6#,&*$)-%$cosGF3\"\"#F*! \"$\"\"%F*F*!\"\"#!\"#F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "simplify(yq);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&* &-%$cosG6#%\"fG\"\"\",&*(%\"dGF*-%%sqrtG6#\"\"$F*-%$sinGF(F*F**&\"\"#F *%\"aGF*!\"\"F*F**$-F/6#,&*$)F&F5F*!\"$\"\"%F*F*F7#F7F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "yq:=-1/2*cos(f)*(d*sqrt(3)*sin(f)-2 *a)/(sqrt(-3*cos(f)^2+4));" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#yqG,$*&*&-%$cosG6#%\"fG\"\"\",&*(%\"dGF,-%%sqrtG6#\" \"$F,-%$sinGF*F,F,*&\"\"#F,%\"aGF,!\"\"F,F,*$-F16#,&*$)F(F7F,!\"$\"\"% F,F,F9#F9F7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 110 "Ezzel l\351nyeg\351ben befejezt\374k az \+ alakzat pontjainak a megad\341s\341t, m\341r csak egy alaposabb ellen \365rz\351sre van sz\374ks\351g" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 259 "" 0 "" {TEXT -1 12 "Kontroll 5: " } {TEXT 326 3 "Az " }{TEXT 454 19 "A1, B1, P1, Q1 " }{TEXT 455 31 "p ontok p\341ly\341j\341nak a vizsg\341lata." }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 113 "El\365sz\366r minden p\341ly\341t megvizsg\341lunk olyan konkr\351t param\351ter \351rt\351k mellett, amelyben a kezd\365 (f= 0) helyzetben az " }{TEXT 468 5 "A1,A2" }{TEXT -1 4 " \351s " }{TEXT 469 3 "A3 " }{TEXT -1 143 "pontok \351ppen az orig\363ba ker\374lnek. \+ \nJavasoljuk m\341s param\351ter \351rt\351kekkel is ennek a vizsg\341 latnak az elv\351gz\351s\351t. (Pl legyen az \372jabb vizsg\341latban \+ " }{TEXT 466 4 "a=10" }{TEXT -1 5 " , " }{TEXT 467 4 "d=25" }{TEXT -1 5 " .) 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jJ:7()F[ilF2-F(6$7$7$Fheq$!+0$zMX*F\\eq7$FceqF_j]lFE-F(6$7$7$F`fq$!+zM ?A_F[il7$Fefq$\"*].AA&F\\eqFNFVFdhlFcanFhapFgcqF\\dq7+-F(6$7$7$F,$!*&Q Kz*)F\\o7$F/$!+D>m*[%F\\oF2-F(6$7$7$F]\\^l$!+&*********F\\eq7$Fb\\^lFC FE-F(6$7$7$FC$!+brz$p#F\\o7$F>$\"\"%F\\eqFNFVFdhlFcanFhapFgcqF\\dq-%*L INESTYLEG6#\"#;-%(SCALINGG6#%,CONSTRAINEDG" 1 2 0 16 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" "Curve 5" "Curve 6" "Curve 7" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "A fenti anim\341ci\363b\363l az sejthet\365, hogy az itt megfigyel t " }{TEXT 463 4 "A1B1" }{TEXT -1 4 " \351s " }{TEXT 464 4 "P2Q2" } {TEXT -1 78 " szakaszok felez\365pontjainak a t\341vols\341ga \341llan d\363. Vizsg\341ljuk meg ezt a sejt\351st:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 57 "unassign('a','d','f'):\nsimplify(tn((A1+B1)/2,(P2+Q 2)/2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$)%\"dG\"\"#\"\"\"\"\"$ " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Val\363ban ennek a szakasznak a hossza " }{TEXT 465 11 "d*sqrt(3). " }{TEXT -1 169 "Lehets\351ges, \+ hogy ezt kihaszn\341lva egyszer\373bben megkaphattuk volna a fenti \+ \366sszef\374gg\351seinket. Az eredm\351ny azonban a feladat egy\351rt elm\373 volt\341b\363l ad\363d\363an ugyanez lett volna." }}{PARA 0 " " 0 "" {TEXT -1 157 "Miel\365tt megjelen\355ten\351nk az alakzatunkat, - megszabadulva param\351tereink konkr\351t \351rt\351k\351t\365l - p r\363b\341ljuk meg algebrailag is ellen\365rizni sz\341m\355t\341saink helyess\351g\351t." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 " un assign('a','d','f'):" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{SECT 0 {PARA 20 "" 0 "" {TEXT -1 1 "K" }{TEXT 328 11 "ontroll 6 :" }{TEXT 327 1 " " }{TEXT 330 0 "" }{TEXT 329 42 " A tetra\351derek \351 lhosszainak a kisz\341m\355t\341sa." }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 144 "Vizsg\341l\363d\341sunknak egyetlen c\351lja van: az, hogy a k apott tetra\351der-\351lek megfelel\365 hossz\372ak-e, vagyis igaz-e, \+ hogy az alapok hossz\341nak a n\351gyzete " }{TEXT 332 5 "4*a^2" } {TEXT -1 12 ", a sz\341rak\351" }{TEXT 333 2 " d" }{TEXT -1 1 "^" } {TEXT 334 8 "2+2*a^2 " }{TEXT -1 2 ". " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "Az ala\351lek hossza:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 140 "print(` \nA1-B1 = `,simplify(tn(A1,B1)),`\nP1-Q1 = \+ `,simplify(tn(P1,Q1)),`\nA2-B2 = `,simplify(tn(A2,B2)),`\nP2-Q2 = ` ,simplify(tn(P2,Q2)));" }{TEXT -1 0 "" }}{PARA 12 "" 1 "" {XPPMATH 20 "6*%,~|+A1-B1~=~~G,$*$)%\"aG\"\"#\"\"\"\"\"%%+|+P1-Q1~=~~GF$%+|+A2-B2~ =~~GF$%+|+P2-Q2~=~~GF$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "A tetra \351der sz\341rainak hossza:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 403 "print(`\nA1-P1 = `,simplify(tn(A1,P1)),`\nA1-Q1 = `,simplify( tn(A1,Q1)),`\nB1-P1 = `,simplify(tn(B1,P1)),`\nB1-Q1 = `,simplify(tn (B1,Q1)),`\nA2-P1 = `,simplify(tn(A2,P1)),`\nA2-Q1 = `,simplify(tn(A 2,P1)),`\nB2-P1 = `,simplify(tn(B2,P1)),`\nB2-Q1 = `,simplify(tn(B2, P1)),`\nA2-P2 = `,simplify(tn(A2,P2)),`\nA2-Q2 = `,simplify(tn(A2,Q2 )),`\nB2-P2 = `,simplify(tn(B2,P2)),`\nB2-Q2 = `,simplify(tn(B2,Q2)) );" }{TEXT -1 0 "" }}{PARA 12 "" 1 "" {XPPMATH 20 "6:%+|+A1-P1~=~~G,&* $)%\"dG\"\"#\"\"\"F)*&F(F))%\"aGF(F)F)%+|+A1-Q1~=~~GF$%+|+B1-P1~=~~GF$ %+|+B1-Q1~=~~GF$%+|+A2-P1~=~~GF$%+|+A2-Q1~=~~GF$%+|+B2-P1~=~~GF$%+|+B2 -Q1~=~~GF$%+|+A2-P2~=~~GF$%+|+A2-Q2~=~~GF$%+|+B2-P2~=~~GF$%+|+B2-Q2~=~ ~GF$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 98 "A tov\341bbi h\341rom tet ra\351der adatainak a megvizsg\341l\341sa felesleges, hiszen azokat az els\365 h\341romnak az " }{TEXT 470 2 "xy" }{TEXT -1 37 " s\355kra vo natkoz\363 t\374kr\366z\351s\351vel kaptuk." }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 20 "" 0 "" {TEXT 331 13 "Kontr oll 7 : " }{TEXT 410 38 "A kapott alakzat els\365 megjelen\355t\351s e." }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "A m\341r kisz\341m\355\355to tt koordin\341ta-\351rt\351kekkel \372b\363l megadjuk a poli\351dert j elent\365 T m\341trixot." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 619 "A1:=[xa,ya,0]: \nB1:=[xb,yb,0]:\nA2:=[-xa/2,ya,xa*sqrt(3)/2]:\nB 2:=[-xb/2,yb,xb*sqrt(3)/2]:\nA3:=[-xa/2,ya, -xa*sqrt(3)/2]:\nB3:=[-xb/ 2,yb, -xb*sqrt(3)/2]:\n\nP2:=[xp,yp,0]: \nQ2:=[xq,yq,0]: \nP1:=[-x p/2,yp,-xp*sqrt(3)/2]: \nQ1:=[-xq/2,yq,-xq*sqrt(3)/2] :\nP3:=[-xp/2,yp , xp*sqrt(3)/2]: \nQ3:=[-xq/2,yq, xq*sqrt(3)/2] :\nT:=[[[A1,B1,P1],[B1 ,A1,Q1],[A1,P1,Q1],[P1,B1,Q1]],\n [[A2,B2,P1],[B2,A2,Q1],[A2,P1,Q1] ,[P1,B2,Q1]],\n [[A2,B2,P2],[B2,A2,Q2],[A2,P2,Q2],[P2,B2,Q2]],\n \+ [[A3,B3,P2],[B3,A3,Q2],[A3,P2,Q2],[P2,B3,Q2]],\n [[A3,B3,P3],[B3,A 3,Q3],[A3,P3,Q3],[P3,B3,Q3]],\n [[A1,B1,P3],[B1,A1,Q3],[A1,P3,Q3],[ P3,B1,Q3]]]:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 132 "A megjelen\355t \351s sor\341n mindenk\351ppen numerikus \351rt\351kekkel kell dolgozn unk, ez\351rt l\351nyegesen ler\366vid\355tj\374k a sz\341mol\341st, h a a l\351p\351sk\366zt ( " }{TEXT 494 1 "e" }{TEXT -1 71 " ) a proces szor aritmetik\341j\341t haszn\341lva tizedes t\366rtk\351nt sz\341m \355tjuk ki." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 224 "a:=10: d:= 10*sqrt(3):\nrp:=90:e:=evalhf(2*Pi/rp):\nfor k from 0 to rp do\n f :=k*e:\n R[k]:=polygonplot3d(T):\n unassign('f'):\nod:\ndispla y3d(seq(R[k],k=0..rp),insequence=true,scaling= constrained,orientation =[50,120]);" }{TEXT -1 0 "" }}{PARA 13 "" 1 "" {GLPLOT3D 400 300 300 {PLOTDATA 3 "6%-%(ANIMATEG6gp7(-%)POLYGONSG6&7%7%$\"+++++?!\")$\"\"!F0 F/7%F/F/F/7%$\"+++++5F.$!+++++5F.$\"+330KF. 7%FauF[vFcv7%F[vFfuFcv-F(6&7%7%$!+'yb%45F.Fdu$\"+5(G%[F^p Fiu$\"+*45MK$FjoF[v7%FdwF_wFcv7%F_wF[vFcv7%F[vFdwFcv-F(6&7%F_wFdw7%$!+ ,Grt;F.F^vF/7%FdwF_w7%$!+/O!\\@#F.FfvF/7%F_wF_xFcx7%F_xFdwFcx-F(6&7%7% F`wFdu$!+5(G%[F.7%F[yF gyF[z7%FgyF^yF[z-F(6&7%FauFfuFgy7%FfuFauF[z7%FauFgyF[z7%FgyFfuF[z7(-F( 6&7%7%$\"+5D,T?F.$\"*h&z4zF^pF/7%$\"*$3tr%)F^p$\"+xL@\\\\F^pF/7%$\"+<$ *Q_uF^p$!+%3.t?\"F.$\"+%p\"z!H\"F.7%F_[lFjz7%$\"+\\xYO6F.$\"+0z$GL'F^p $\"+$))>%o>F.7%FjzFd[lF\\\\l7%Fd[lF_[lF\\\\l-F(6&7%7%$!+bi]?5F.F][l$\" +%oovw\"F.7%$!*TleB%F^pFb[l$\"+:TtOtFjoFd[l7%F]]lFh\\lF\\\\l7%Fh\\lFd[ lF\\\\l7%Fd[lF]]lF\\\\l-F(6&7%Fh\\lF]]l7%$!+jyZ!\\\"F.Fg[lF/7%F]]lFh\\ 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"Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 300 "Az \355gy elk\351sz\374lt alakzat tov\341bbi vizsg\341lat\341hoz m\341r nem lesz sz\374ks\351g \+ a fenti sz\341m\355t\341sokra, melyeket egyel\365re m\351g \"fejben ta rt\" a MAPLE, ez\351rt ments\374k \341t eredm\351nyeinket a k\366vetke z\365 programr\351szletbe (vagy egy \372j MAPLE f\341jlba) \351s kezdj \374k el\366lr\365l a munk\341t, imm\341r csak a rajzokra, anim\341ci \363kra koncentr\341lva." }}}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 26 "Az alakzat megjelen\355t \351se \n" }{TEXT 344 88 "(Akit csak a v\351geredm\351ny \351rdekel, e zt a r\351szt az el\365z\365ekt\365l f\374ggetlen\374l is futtathatja. )" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart; with(plots):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been redefined\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 818 "xa:=1/3*d* sqrt(3)*sqrt(-3*cos(f)^2+4)+a*cos(f):\nya:=1/2*sin(f)*(d*sqrt(3)*cos(f ))/(sqrt(-3*cos(f)^2+4))-a*sin(f):\nxb:=1/3*d*sqrt(3)*sqrt(-3*cos(f)^2 +4)-a*cos(f):\nyb:=1/2*sin(f)*(d*sqrt(3)*cos(f))/(sqrt(-3*cos(f)^2+4)) +a*sin(f):\nxp:=2/3*(-d*sqrt(3)+3*a*sin(f))/(sqrt(-3*cos(f)^2+4)):\nyp :=-1/2*cos(f)*(d*sqrt(3)*sin(f)+2*a)/(sqrt(-3*cos(f)^2+4)):\nxq:=-2/3* (d*sqrt(3)+3*a*sin(f))/(sqrt(-3*cos(f)^2+4)):\nyq:=-1/2*cos(f)*(d*sqrt (3)*sin(f)-2*a)/(sqrt(-3*cos(f)^2+4)):\nA1:=[xa,ya,0]: \nB1:=[xb,yb,0 ]:\nA2:=[-xa/2,ya,xa*sqrt(3)/2]:\nB2:=[-xb/2,yb,xb*sqrt(3)/2]:\nA3:=[- xa/2,ya, -xa*sqrt(3)/2]:\nB3:=[-xb/2,yb, -xb*sqrt(3)/2]:\n\nP2:=[xp,yp ,0]: \nQ2:=[xq,yq,0]: \nP1:=[-xp/2,yp,-xp*sqrt(3)/2]: \nQ1:=[-xq/2 ,yq,-xq*sqrt(3)/2] :\nP3:=[-xp/2,yp, xp*sqrt(3)/2]: \nQ3:=[-xq/2,yq, x q*sqrt(3)/2] :\nszin:=[red,green,blue,yellow,magenta,cyan]:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 20 "" 0 " " {TEXT -1 38 "A hat tetra\351der egy\374tt, \351s k\374l\366n-k\374l \366n" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 104 "Az alakzat lapjait tetra \351derenk\351nt adjuk meg, ezzel el\351rhetj\374k, hogy ne az \366ssz es jelenjen meg a rajzon." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 306 "T:=[[[A1,B1,P1],[B1,A1,Q1],[A1,P1,Q1],[P1,B1,Q1]],\n [[A2,B2,P 1],[B2,A2,Q1],[A2,P1,Q1],[P1,B2,Q1]],\n [[A2,B2,P2],[B2,A2,Q2],[A2, P2,Q2],[P2,B2,Q2]],\n [[A3,B3,P2],[B3,A3,Q2],[A3,P2,Q2],[P2,B3,Q2]] ,\n [[A3,B3,P3],[B3,A3,Q3],[A3,P3,Q3],[P3,B3,Q3]],\n [[A1,B1,P3] ,[B1,A1,Q3],[A1,P3,Q3],[P3,B1,Q3]]]:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 306 3 "1. " }{TEXT -1 38 "Az alakzat, a MAPLE sz\355neit haszn\341lva: " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 92 "Megengedhetj\374k, hogy a \"k ezd\365 helyzet ne f=0-r\363l induljon, hanem b\341rhonnan. 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A jobb \341ttekinthet\365s\351g kedv\351\351rt kialak\355thatjuk \372gy a ko nstrukci\363t, hogy az els\365 \351s harmadik tetra\351dernek csak az \+ \351lv\341za, vagy k\351t-k\351t lapja legyen l\341that\363. 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" }{TEXT 342 4 "a=10" }{TEXT -1 3 " , " }{TEXT 343 5 "d=30 " }{TEXT -1 88 ", vagy b\341rmilyen m\341s \+ param\351ter \351rt\351kkel. Ezzel v\341lik igaz\341n szeml\351letess \351 a konstrukci\363." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }{TEXT -1 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {SECT 0 {PARA 20 "" 0 "" {TEXT -1 48 "A tetra\351derek lapjait m\341s- m\341s sz\355nnel sz\355nezz\374k:" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 441 "Aki (pl. pap\355rb\363l) elk\351sz\355ti ennek a mozg\363 alakzat nak a modellj\351t, \351rdekes modellt \341ll\355that el\365 azzal, ho gy az egym\341shoz ill\365 lapokra egy-egy - l\351nyeg\351ben h\341rom rombuszb\363l \341ll\363 - k\351pet applik\341l. Ezek a k\351pek ugya nis az egyik oldalr\363l n\351zve \341llnak csak helyesen, az ellenkez \365 oldalr\363l szeml\351lve olyannak l\341tjuk, mint ha a k\351pet a lkot\363 rombuszokat rossz sorrendbe raktuk volna \366ssze egy k\351pp \351. Ehhez a munk\341hoz ny\372jt seg\355ts\351get ez a \"jelenet\": " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 342 "unassign('a','d','f'): \nS:=[[[A1,B1,P1],[A2,B2,P1],[A2,B2,P2],\n [A3,B3,P2],[A3,B3,P3],[ A1,B1,P3]],\n [[B1,A1,Q1],[B2,A2,Q1],[B2,A2,Q2],\n [B3,A3,Q2], [B3,A3,Q3],[B1,A1,Q3]],\n [[A1,P1,Q1],[A2,P1,Q1],[A2,P2,Q2],\n \+ [A3,P2,Q2],[A3,P3,Q3],[A1,P3,Q3]],\n [[P1,B1,Q1],[P1,B2,Q1],[P2,B 2,Q2],\n [P2,B3,Q2],[P3,B3,Q3],[P3,B1,Q3]]]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 268 " a:=10:d:=10*sqrt(3):\nrp:=90:e:=evalhf(2*Pi /rp):\nfor k from 0 to rp do\n f:=k*e:\n R[k]:=display3d(seq(po lygonplot3d(S[i],color=szin[i]),i=1..4),\n insequence= false):\no d:\ndisplay3d(seq(R[k],k=0..rp),insequence=true,scaling= constrained,o rientation=[50,120]);" }{TEXT -1 0 "" }}{PARA 13 "" 1 "" {GLPLOT3D 400 300 300 {PLOTDATA 3 "6%-%(ANIMATEG6gp7&-%)POLYGONSG6)7%7%$\"+++++? !\")$\"\"!F0F/7%F/F/F/7%$\"+++++5F.$!+++++5F.$\"+330KFfpF]v$ 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