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{SECT 0 {SECT 0 {PARA 3 "" 0 "" {TEXT -1 34 "Z\341rt g\366rb\351k a ne
gyedik dimenzi\363ban" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 170 "Megmutat
om, hogyan lehet egy Maple sz\341m\355t\341sb\363l egy 3 dimenzi\363s \+
forgathat\363 \341br\341t l\351trehozni. A Maple sz\341m\355t\341s ala
pja egy polinom lesz, ami egy algebrai g\366rb\351t hat\341roz meg." }
}{PARA 0 "" 0 "" {TEXT -1 241 "Milyen j\363 lenne, ha lenne lehet\365s
\351g\374nk olyan webes prezent\341ci\363t l\351trehozni, amelyben egy
 eg\351rkattint\341ssal k\351pesek vagyunk az objektumot k\366rben\351
zni, elforgatni, \351s nem kell hozz\341 a Maple. Erre mutatok meg egy
 m\363dszert, a Maple VRML exportj\341t." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart;" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 27 "Az e
gyenl\365s\351gek fel\341ll\355t\341sa" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "Eq:=x^2-y^2=0;" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#EqG/,&*$)%\"xG\"\"#\"\"\"F+*$)%\"yGF*F+!\"\"\"\"!" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "Ezt az egyenletet kicsit \341tre
ndezve kapjuk a k\366vetkez\365 egyenl\365s\351get" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 13 "Eq1:=x^2=y^2;" }{TEXT -1 0 "" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%$Eq1G/*$)%\"xG\"\"#\"\"\"*$)%\"yGF)F*" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "Az Eq1 egyenl\365s\351get terjessz
\374k ki a komplex sz\341mokra, legyen ennek a neve Eq2!" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "subs(x=a+b*I,y=c+d*I,Eq1);" }{TEXT 
-1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$),&%\"aG\"\"\"*&%\"bGF(^#
F(F(F(\"\"#F(*$),&%\"cGF(*&%\"dGF(F+F(F(F,F(" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 14 "Eq2:=evalc(%);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%$Eq2G/,(*$)%\"aG\"\"#\"\"\"F+*(^#F*F+F)F+%\"bGF+F+*
$)F.F*F+!\"\",(*$)%\"cGF*F+F+*(F-F+F5F+%\"dGF+F+*$)F7F*F+F1" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 80 "az immagin\341rius (k\351pzetes) r
\351szeket v\341lasszuk sz\351t egy \341tmeneti (tmp) v\341ltoz\363ba.
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "tmp:=map(z->select(has,
z,I),Eq2);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$tmpG/*
(^#\"\"#\"\"\"%\"aGF)%\"bGF)*(F'F)%\"cGF)%\"dGF)" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 67 "a val\363s, \351s a k\351pzetes r\351szekb\365l k
\366vetkeznek Eq3 \351s Eq4 egyenletek:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 13 "Eq3:=Eq2-tmp;" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$Eq3G/,&*$)%\"aG\"\"#\"\"\"F+*$)%\"bGF*F+!\"\",&*$)%
\"cGF*F+F+*$)%\"dGF*F+F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 
"Eq4:=tmp/(2*I);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$
Eq4G/*&%\"aG\"\"\"%\"bGF(*&%\"cGF(%\"dGF(" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 319 "Eq3 \351s Eq4 most azok az egyenl\365s\351gek az a,b,c,d
 v\341ltoz\363kban, amelyekb\365l mi kiindultunk. K\366vetkez\351sk
\351pp valamilyen alakzatot hoznak l\351tre a 4 dimenzi\363s t\351rben
, amit az egys\351gg\366mbbel akarunk elmetszeni. Az\351rt az egys\351
gg\366mbbel, mert ezt a metszetet fogjuk a 3D-s t\351rre vet\355teni. \+
Ebb\365l a k\366vetkez\365 Eq5 egyenl\365s\351ghez jutunk:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "Eq5:=a^2+b^2+c^2+d^2=1;" }{TEXT -1 
0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$Eq5G/,**$)%\"aG\"\"#\"\"\"F+
*$)%\"bGF*F+F+*$)%\"cGF*F+F+*$)%\"dGF*F+F+F+" }}}}{SECT 1 {PARA 4 "" 
0 "" {TEXT -1 28 "Az egyenl\365s\351gek ki\351rt\351kel\351se" }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 211 "Most a 3 egyenl\365s\351get (Eq3,
 Eq4, Eq5) egyszerre fogjuk ki\351rt\351kelni, hogy a helyzetet saj
\341t szem\374nkkel l\341ssuk. De el\365tte Eq4-et (a k\351pzetes r
\351szek egyenl\365s\351g\351t) emelj\374k n\351gyzetre, mert \355gy k
\366nnyebb vele dolgozni." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
16 "map(x->x^2,Eq4);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#/*&)%\"aG\"\"#\"\"\")%\"bGF'F(*&)%\"cGF'F()%\"dGF'F(" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 68 "most mindegyikre h\355vjuk meg a solve el
j\341r\341st, \355gy egy list\341t kapunk." }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 38 "res:=solve(\{Eq3,%,Eq5\},\{b^2,c^2,d^2\});" }{TEXT 
-1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$resG<%/*$)%\"cG\"\"#\"\"
\"*$)%\"aGF*F+/*$)%\"bGF*F+,&F,!\"\"#F+F*F+/*$)%\"dGF*F+F3" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 53 "Ebb\365l a k\366vetkez\365 3 egyenl\365s
\351g (E1,E2,E3) k\366vetkezik:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 21 "E1:=res[3]+(a^2=a^2);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#E1G/,&*$)%\"dG\"\"#\"\"\"F+*$)%\"aGF*F+F+#F+F*" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "E2:=res[1];" }{TEXT -1 0 "" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#E2G/*$)%\"cG\"\"#\"\"\"*$)%\"aGF)
F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "E3:=res[2]-res[3]+(b^
2=b^2);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#E3G/,&*&
\"\"#\"\"\")%\"bGF(F)F)*$)%\"dGF(F)!\"\"*$F*F)" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 144 "E1 egy k\366r egyenlete, E2-b\365l k\366vetkezik, h
ogy c=a vagy c=-a, ugyan\355gy E3-b\363l d=b vagy d=-b. Eq4-b\365l ped
ig (c=a \351s d=b) vagy (c=-a \351s d=-b), mert" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 4 "Eq4;" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/*&%\"aG\"\"\"%\"bGF&*&%\"cGF&%\"dGF&" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 227 "Ezzel 4 dimenzi\363s t\363ruszokhoz [a,b,a,b] \+
\351s [a,b,-a,-b] jutottunk, amelyek \351ppen a keresett megold\341sok
. Ez az a k\351t k\366r, ami a 4. dimenzi\363ba van most be\341gyazva.
 A tov\341bbi vizsg\341lathoz sz\374ks\351g\374nk van a k\366r\366k pa
ram\351teres alakj\341ra:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
58 "K1:=[r*sin(alpha),r*cos(alpha),r*sin(alpha),r*cos(alpha)];" }
{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#K1G7&*&%\"rG\"\"\"-
%$sinG6#%&alphaGF(*&F'F(-%$cosGF+F(F&F-" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 60 "K2:=[r*sin(alpha),r*cos(alpha),-r*sin(alpha),-r*cos(a
lpha)];" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#K2G7&*&%
\"rG\"\"\"-%$sinG6#%&alphaGF(*&F'F(-%$cosGF+F(,$F&!\"\",$F-F1" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Ahol a sug\341r a k\366vetkez\365:
 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "r:=1/sqrt(2);" }{TEXT 
-1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG,$*&\"\"#!\"\"F'#\"\"
\"F'F*" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "Vet\355t\351s a 3D-s \+
val\363s vil\341gra" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 172 "Ennek az a
 c\351lja, hogy a kapott alakzatot 3D-ben meg tudjuk jelen\355teni, eh
hez vizsg\341ljuk azokat a pontokat, amelyek biztosan az egys\351gg
\366mbre illeszkednek, pl. egy v pontot:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 13 "v:=[2,0,0,0];" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"vG7&\"\"#\"\"!F'F'" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 57 "Innen mindent levet\355t\374nk a f\374gg\365legesen \341l
l\363 vet\355t\365s\355kra." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
10 "[1,b,c,d];" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&\"
\"\"%\"bG%\"cG%\"dG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 136 "Egy w pon
tot (az alakzat vet\355tett k\351p\351nek egy pontj\341t) megkapunk, h
a vessz\374k a w-v k\374l\366nbs\351get, \351s az els\365 koordin\341t
\341t 1-esre cser\351lj\374k:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 27 "P:=w->(w-v)[2..4]/(w-v)[1];" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"PGf*6#%\"wG6\"6$%)operatorG%&arrowGF(*&&,&9$\"\"\"%
\"vG!\"\"6#;\"\"#\"\"%F0&F.6#F0F2F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 13 "P([a,b,c,d]);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#*&7%%\"bG%\"cG%\"dG\"\"\",&\"\"#!\"\"%\"aGF(F+" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "Most vet\355ts\374k mindk\351t k
\366r\374nket!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "PK1:=P(K1
);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$PK1G*&7%,$*&#
\"\"\"\"\"#F**&F+F)-%$cosG6#%&alphaGF*F*F*,$*&F)F**&F+F)-%$sinGF/F*F*F
*F'F*,&F+!\"\"*&F)F*F3F*F*F7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 17 "PK1:=expand(PK1);" }{TEXT -1 0 "" }}{PARA 12 "" 1 "" {XPPMATH 
20 "6#>%$PK1G7%,$*&#\"\"\"\"\"#F)*(F*F(-%$cosG6#%&alphaGF),&F*!\"\"*&F
(F)*&F*F(-%$sinGF.F)F)F)F1F)F),$*&F(F)*(F*F(F4F)F0F1F)F)F&" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "PK2:=P(K2);" }{TEXT -1 0 "" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$PK2G*&7%,$*&#\"\"\"\"\"#F**&F+F)-%$
cosG6#%&alphaGF*F*F*,$*&#F*F+F**&F+F)-%$sinGF/F*F*!\"\",$*&#F*F+F*F,F*
F7F*,&F+F7*&F)F*F4F*F*F7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 
"PK2:=expand(PK2);" }{TEXT -1 0 "" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>
%$PK2G7%,$*&#\"\"\"\"\"#F)*(F*F(-%$cosG6#%&alphaGF),&F*!\"\"*&F(F)*&F*
F(-%$sinGF.F)F)F)F1F)F),$*&#F)F*F)*(F*F(F4F)F0F1F)F1,$*&#F)F*F)F+F)F1
" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "Az eredm\351ny megrajzolva \+
a Maple-lel" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 374 "A tubeplot f\374gg
v\351nnyel mindk\351t g\366rb\351t (a k\366reinket) valamilyen szinten
 3 dimenzi\363ss\341 v\341ltoztathatjuk. (A k\366r\366kb\365l gy\373r
\373k lesznek) Ehhez be kell t\366lten\374nk a plots csomagot, mert en
nek az egyik elj\341r\341sa a tubeplot. A tubeplot (cs\365rajzol\363) \+
egy \"cs\366vet\" illeszt a g\366rbe vonal\341ra, amit\365l vastags
\341ga lesz. A cs\365 sugar\341t, keresztmetszet, beoszt\341sait is pa
ram\351terekben adhatjuk meg." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 12 "with(plots):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name c
hangecoords has been redefined\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 43 "R1:=tubeplot(PK1,alpha=0..2*Pi,radius=0.1):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "R2:=tubeplot(PK2,alpha=0..2*
Pi,radius=0.1):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "Most szeml\351
lj\374k meg mindk\351t gy\373r\373t egy\374tt. " }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 17 "display([R1,R2]);" }{TEXT -1 0 "" }}{PARA 13 "
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aal" 1 2 0 1 10 0 2 1 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1
" "Curve 2" }}}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 18 "Export\341l\341s
 VRML-be" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 680 "A Maple-ben ezt az \+
\341br\341t egyszer\373en megfogjuk az eg\351rrel, \351s minden oldal
\341r\363l megn\351zhetj\374k. De, ha egy weblapra illesztj\374k, csak
 ezt a kimerev\355tett k\351pet l\341thatjuk. Most \351rt\374nk el oda
, hogy egy val\363s, 3D-ben megtekinthet\365 \341br\341nk legyen, amit
 egy webb\366ng\351sz\365 meg tud jelen\355teni. Az els\365 m\363dszer
, ahogy azt kor\341bban eml\355tettem, a VRML (Virtual Reality Modelin
g Language) k\366rnyezet. Ehhez a Maple egy export\341l\341si lehet
\365s\351get k\355n\341l, amit a k\366vetkez\365k\351ppen haszn\341lha
tunk ki. El\365sz\366r is ezt a display-jel megjelen\355tett \341br
\341t beletessz\374k egy v\341ltoz\363ba, legyen ennek a neve: R. Ezut
\341n bet\366ltj\374k a plottools csomag vrml export-utas\355t\341s
\341t, majd az \341br\341nkat egy vrml file-ba \355juk (gyuru.wrl)" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "R:=%:" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 306 "A % jel a legut\363bbi utas\355t\341st jelenti, aza
z ezt a sort \355gy is \355rhattuk volna: R:=display([R1,R2]): Az\351r
t tett\374nk kett\365spontot a v\351g\351re, hogy ne jelen\355tsen meg
 semmit. Ha pontosvessz\365t tett\374nk volna, az alakzat koordin\341t
\341inak sereg\351t olvashatn\341nk t\366bb k\351perny\365n kereszt
\374l, de ez most a sz\341munkra felesleges." }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 21 "with(plottools,vrml);" }{TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "vrml(R,`gyuru.wrl`);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 286 "Ezzel l\351trehoztuk a gyuru.wrl file-t,
 ami a Maple munkak\366nyvt\341r\341ba ker\374lt, ott tal\341lunk r
\341. Ha nem oda akarjuk tenni, az el\351r\351si utat megadhatjuk a vr
ml elj\341r\341s m\341sodik param\351ter\351ben. P\351ld\341ul, ha a g
orbe\\vrml mapp\341ban szeretn\351nk a gyuru.wrl file-t l\351trehozni,
 \355gy n\351z ki az utas\355t\341sunk:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 32 "vrml(R,`/gorbe/vrml/gyuru.wrl`);" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 159 "Fig
yelj\374nk arra, hogy a mapp\341k elv\341laszt\341sa nem a \\, (AltGr+
q) hanem a / (Shift+6) jellel t\366rt\351nik, az el\351r\351si utat pe
dig ` jelek (AltGr+7) k\366z\351 kell tenn\374nk. " }}}}}}{MARK "0" 0 
}{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }