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30..30,grid=[36,2],color=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "display(hengf,axes=framed,style=hidden,orientation=[30,45],sca ling=CONSTRAINED);" }}{PARA 13 "" 0 "" {TEXT 231 0 "" }}{PARA 13 "" 1 "" {TEXT 231 0 "" }{TEXT 231 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 206 30 "#gomb es henger metszesvonala:" }{MPLTEXT 1 0 3 " \n" } {MPLTEXT 1 207 34 "#Viviani gorbe, ezt is rarajzoljuk" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 89 "vg:=spacecurve([2*r*cos(t)*cos(t),2*r*cos(t)* sin(t),2*r*sin(t)],t=0..2*Pi,color=green):\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "display([gombf,hengf,vg],axes=NONE,style=patch,orient ation=[30,45],scaling=CONSTRAINED);" }}{PARA 13 "" 1 "" {TEXT 231 0 "" }{TEXT 231 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 127 "display ([gombf,hengf,vg],axes=NONE,style=PATCHNOGRID,orientation=[30,45],scal ing=CONSTRAINED,lightmodel='light4',shading=ZHUE);" }}{PARA 13 "" 1 "" {TEXT 231 0 "" }{TEXT 231 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 208 22 "#fekvo henger: sugar=r" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 35 "vheng:=[r*(cos(u)+1),v,r*sin(u)]:\n" }{MPLTEXT 1 0 67 "vhengf:=plo t3d(vheng,u=0..2*Pi,v=-30..30,grid=[36,2],color=grey):\n" }{MPLTEXT 1 0 81 "display(vhengf,axes=framed,style=hidden,orientation=[30,45],scal ing=CONSTRAINED);" }}{PARA 13 "" 1 "" {TEXT 231 0 "" }{TEXT 231 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 209 20 "#ket henger athatasa" } {MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 102 "display([hengf,vhengf],orientat ion=[30,45],style=PATCHNOGRID,lightmodel='light2',scaling=CONSTRAINED) ;" }}{PARA 13 "" 1 "" {TEXT 231 0 "" }{TEXT 231 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 210 52 "#a fekvo henger sugarat m-szeresre novel juk, 1 " 0 "" {MPLTEXT 1 213 59 "#torusz: a vezergorbe (koz epkor) sugara=a, a meridiankore=b" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 398 "a:=12: b:=8:\nvezerg:=spacecurve([a*cos(u),a*sin(u),0],u=0..2*Pi, color=blue):\nmeridian:=spacecurve([a+b*cos(v),0,b*sin(v)],v=0..2*Pi,c olor=red):\ncenterm:=spacecurve([a+0.3*cos(v),0,0.3*sin(v)],v=0..2*Pi, color=red):\ncenterv:=spacecurve([0.2*cos(u),0.2*sin(u),0],u=0..2*Pi,c olor=blue):\ndisplay([vezerg,meridian,centerm,centerv],axes=framed,sty le=hidden,orientation=[30,45],scaling=CONSTRAINED);\n" }{MPLTEXT 1 0 58 "tor:=[(a+b*cos(v))*cos(u),(a+b*cos(v))*sin(u),b*sin(v)]:\n" } {MPLTEXT 1 0 64 "torf:=plot3d(tor,u=0..2*Pi,v=0..2*Pi,grid=[36,36],col or=pink):\n" }{MPLTEXT 1 0 113 "display([torf,vezerg,meridian,centerv, centerm],axes=framed,style=hidden,orientation=[30,45],scaling=CONSTRAI NED);" }}{PARA 13 "" 1 "" {TEXT 231 0 "" }{TEXT 231 0 "" }}{PARA 13 "" 1 "" {TEXT 231 0 "" }{TEXT 231 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 214 26 "#torusz es henger athatasa" }{MPLTEXT 1 0 2 "\n" } {MPLTEXT 1 0 100 "display([torf,hengf],orientation=[30,45],style=PATCH NOGRID,lightmodel='light2',scaling=CONSTRAINED);" }}{PARA 13 "" 1 "" {TEXT 231 0 "" }{TEXT 231 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 215 15 "#nagyobb torusz" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 17 "an:=15 : bn:=10:\n" }{MPLTEXT 1 0 64 "torn:=[(an+bn*cos(v))*cos(u),(an+bn*cos (v))*sin(u),bn*sin(v)]:\n" }{MPLTEXT 1 0 67 "tornf:=plot3d(torn,u=0..2 *Pi,v=0..2*Pi,grid=[36,36],color=brown):\n" }{MPLTEXT 1 0 80 "display( tornf,axes=framed,style=hidden,orientation=[30,45],scaling=CONSTRAINED );" }}{PARA 13 "" 1 "" {TEXT 231 0 "" }{TEXT 231 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 216 69 "#torusz es henger athatasa \n#2 duplapo nttal (itt kozos az erintosik)" }{MPLTEXT 1 0 106 "\ndisplay([tornf,vh engf],orientation=[35,40],style=PATCHNOGRID,lightmodel='light2',scalin g=CONSTRAINED);\n" }}{PARA 13 "" 1 "" {TEXT 231 0 "" }{TEXT 231 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 217 40 "#torusz es a nagyobbik v izszintes henger" }{MPLTEXT 1 0 104 "\ndisplay([tornf,mhengf],orientat ion=[90,35],style=PATCHNOGRID,lightmodel='light2',scaling=CONSTRAINED) ;" }}{PARA 13 "" 1 "" {TEXT 231 0 "" }{TEXT 231 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "display([torf,vhengf],orientation=[117,63 ],style=PATCHNOGRID,lightmodel='light3',scaling=CONSTRAINED);" }} {PARA 13 "" 1 "" {TEXT 231 0 "" }{TEXT 231 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 218 62 "#toruszt belulrol erinto sik: origon at a v1 \+ es v2 vektorokkal" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 45 "v1:=[sqrt(an *an-bn*bn),0,bn]: v2:=[0,bn,0]:\n" }{MPLTEXT 1 0 296 "sik:=evalm(s*v1+ t*v2):\nA:=[0,0,0]:\negyenesa:=spacecurve(evalm(A+t*v1),t=0..1,color=b lue):\negyenesb:=spacecurve(evalm(A+t*v2),t=0..1,color=green):\nsikf:= plot3d(sik,s=-2..2,t=-3..3,grid=[2,2],color=khaki):\ndisplay([sikf,egy enesa,egyenesb],axes=framed,orientation=[108,56],scaling=CONSTRAINED); " }}{PARA 13 "" 1 "" {TEXT 231 0 "" }{TEXT 231 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 219 35 "#ez a sik a toruszt 2 korben metszi" } {MPLTEXT 1 0 104 "\ndisplay([torf,sikf],orientation=[108,56],style=PAT CHNOGRID,lightmodel='light3',scaling=CONSTRAINED);\n" }}{PARA 13 "" 1 "" {TEXT 231 0 "" }{TEXT 231 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 220 41 "#a sik eltolasa d*vek vektorral, -10 " 0 "" {MPLTEXT 1 222 19 "#hen ger sikmetszete" }{MPLTEXT 1 0 115 "\ndisplay([sikf,hengf],axes=framed ,style=PATCHNOGRID,orientation=[120,50],lightmodel='light2',scaling=CO NSTRAINED);" }}{PARA 13 "" 1 "" {TEXT 231 0 "" }{TEXT 231 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 223 21 "#valtozo alaku torusz" } {MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 143 "animate3d([(q*an+bn*cos(v))*cos (u),(q*an+bn*cos(v))*sin(u),bn*sin(v)],u=0..2*Pi,v=0..2*Pi,q=bn/an..5* bn/an,grid=[24,24],scaling=CONSTRAINED);\n" }{MPLTEXT 1 224 55 "#a kep re ra kell kattintani, es az animaciot lejatszani" }}{PARA 13 "" 1 "" {TEXT 231 0 "" }{TEXT 231 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 225 27 "#valtozo torusz sikmetszete" }{MPLTEXT 1 0 229 "\nnagysikf:=pl ot3d(sik,s=-6..6,t=-6..6,grid=[2,2],color=yellow):\ndisplay([animate3d ([(q*an+bn*cos(v))*cos(u),(q*an+bn*cos(v))*sin(u),bn*sin(v)],u=0..2*Pi 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