ࡱ>  q rbjbjt+t+ AAr7c]LLLL@@@d`3E`(,,,AAAA5BBC$FHCM@,k+N,,,Cq=LLq=q=q=,\L @AJLLLL,Aq=rq=A^ @A$kj1h5\APramen konika zadan pomou jedne dvostruke i dviju jednostrukih realnih to aka MILJENKO LAPAINE Conic section pencil given by a double point and two single real points ABSTRACT The algorithm is presented for the determination of coefficients in the equation of the type IV conic section pencils when the pencil is given by three base points one of which is a double point. In order to work with the points in infinity, the homogeneous coordinates are introduced. The approach is illustrated by several examples. Key words: conic section pencil, homogeneous coordinates, computer graphics Pramen konika zadan pomou jedne dvostruke i dviju jednostrukih realnih to aka SA}ETAK Prikazan je algoritam za odreivanje koeficijenata u jednad~bi pramena konika tipa IV kad je pramen zadan s tri to ke od kojih je jedna dvostruka. Da bi se moglo raditi s neizmjerno dalekim to kama uvedene su homogene koordinate. Postupak je ilustriran s nekoliko primjera. Klju ne rije i: pramen konika, homogene koordinate, ra unalna grafika 1. UVOD Pramen konika openito je odreen s etiri realne i razli ite to ke A, B, C i D. Pramen konika tipa IV (` uri i Sachs, 1995, 1997) karakteriziran je svojstvom da se temeljne to ke C i D podudaraju. Tom dvostrukom to kom prolazi zajedni ka tangenta svih konika pramena. Grafi ki prikaz pramena konika mo~e se u inkovito ostvariti primjenom ra unala i priklju enog ureaja za crtanje. Matemati ka osnova razvijenog softvera opisana je u radu (Lapaine 1997). U ovome radu daje se algoritam za ra unanje koeficijenata u jednad~bi pramena konika tipa IV, ako je pramen zadan s tri to ke (od kojih je jedna dvostruka) i zajedni kom tangentom svih konika pramena. Postupak se temelji na primjeni homogenih koordinata kako bi se i neizmjerno daleke to ke ravnine mogle analiti ki obuhvatiti posve analogno kao i to ke u kona nosti. 2. PRAMEN KONIKA Neka su  EMBED Equation.2  (2.1)  EMBED Equation.2  (2.2) jednad~be dviju konika. Za proizvoljni  EMBED Equation.2  sasta-vimo izraz  EMBED Equation.2  (2.3) Polinom H je oblika  EMBED Equation.2  (2.4) gdje smo ozna ili  EMBED Equation.2  (2.5) Za svaki pojedini  EMBED Equation.2 , izraz  EMBED Equation.2  (2.6) je jednad~ba konike ako je barem jedan od brojeva a, b i c razli it od nule. Ako je a = b = c = 0 tada se radi o specijalnim, ali jednostavnim slu ajevima (Lapaine, 1997). Za zadane realne brojeve  EMBED Equation.2  i  EMBED Equation.2  skup svih konika obuhvaenih jednad~bom (2.6) naziva se pramenom konika. Konike pomou kojih je pramen definiran i kojima odgovaraju jednad~be  EMBED Equation.2  i  EMBED Equation.2  nazivaju se osnovnim konikama pramena. Za svaki vrsti  EMBED Equation.2  jednad~ba (2.4) predstavlja jednu krivulju iz pramena ili u specijalnom slu aju prazan skup. Jedan na in odreivanja tipa konike s mogunoau grafi kog prikazivanja pomou ra unala objaanjen je u radu (Lapaine i Jovi i 1996). 3. PRAMEN KONIKA ZADAN POMOU JEDNE DVOSTRUKE I DVIJU JEDNOSTRUKIH REALNIH TO AKA Ako su zadane etiri to ke  EMBED Equation.2  i = 1,2,3,4 u ravnini, od kojih ni koje tri nisu kolinearne, te ako je  EMBED Equation.2  (3.1) jednad~ba pravca  EMBED Equation.2  tada je jednad~bom  EMBED Equation.2  (3.2) predo en pramen konika kojem su to ke  EMBED Equation.2  temeljne. Sve konike pramena prolaze to kama  EMBED Equation.2  (Cesarec 1957).  Slika 1. Pramen konika tipa IV Neka su sada zadane tri nekolinearne to ke A, B i C i neka je t bilo koji pravac koji prolazi to kom C, ali ne sadr~i to ke A i B. Ozna imo pravac kroz to ke A i B sa g, pravac kroz to ke A i C sa  EMBED Equation.2 , te pravac kroz to ke B i C sa  EMBED Equation.2  (vidi sliku 1). Ako su  EMBED Equation.2  (3.3) jednad~be navedenih pravaca, tada je  EMBED Equation.2  (3.4) jednad~ba pramena konika kojem su A, B i C temeljne to ke, a t zajedni ka tangenta svih konika pramena. Zaista, to ka A je temeljna to ka pramena jer le~i na pravcima g i  EMBED Equation.2 , to ka B je temeljna jer le~i na pravcima g i  EMBED Equation.2 , a to ka C je temeljna jer je sjeciate pravaca t i  EMBED Equation.2  Za svaki zadani  EMBED Equation.2  jednad~ba (3.4) predstavlja jednu koniku pramena. Za to ku (x,y) koja je zajedni ka proizvoljnoj konici pramena i pravcu t(x,y)=0 mora biti  EMBED Equation.2  (3.5) ato vrijedi samo za to ku koja je istovremeno na pravcima t i  EMBED Equation.2  ili t i  EMBED Equation.2  Dakle, radi se o to ki C. Odatle zaklju ujemo da je pravac t zajedni ka tangenta svih konika pramena jer sa svakom konikom ima samo jednu zajedni ku to ku. Ta zajedni ka to ka C naziva se dvostrukom temeljnom to kom pramena. Mo~e se takoer rei i da je pramen zadan s etiri to ke od kojih su dvije pale zajedno. 4. HOMOGENE KARTEZIJEVE KOORDINATE U RAVNINI Pod homogenim kartezijevim koordinatama to ke P(x,y) podrazumijevamo ureenu trojku realnih brojeva  EMBED Equation.2  za koje je  EMBED Equation.2  (4.1) Iz definicije slijedi da homogene koordinate  EMBED Equation.2  nisu jednozna no odreene pomou nehomogenih (x,y), jer ih joa mo~emo pomno~iti bilo kojim realnim brojem razli itim od nule, a da ipak daju iste vrijednosti x i y. Tako npr. za x=3, y=2 imamo  EMBED Equation.2  ali i  EMBED Equation.2   EMBED Equation.2 itd. Ako je  EMBED Equation.2  a  EMBED Equation.2  i  EMBED Equation.2  razli ito od nule, tada je  EMBED Equation.2  i  EMBED Equation.2  tj. ta je to ka neizmjerno daleka to ka ravnine i to u smjeru spojnice ishodiata koordinatnog sustava x,y s to kom  EMBED Equation.2 . I baa zbog toga, ato su neizmjerno daleke to ke ravnine sada karakterizirane time da im je homogena koordinata  EMBED Equation.2  jednaka nuli, uvodimo homogene koordinate. Tako mo~emo sada te osobite, neizmjerno daleke to ke ravnine analiti ki svladavati posve jednako kao i to ke u kona nosti (Cesarec 1957). Na primjer, neizmjerno daleka to ka osi x ima homogene koordinate (0,1,0), a ona na osi y (0,0,1). Uobi ajena oznaka za homogene koordinate je  EMBED Equation.2 , ali u ovome radu e se rabiti  EMBED Equation.2  jer se radi o ureenoj trojki brojeva, a ne o produ~enom razmjeru. Osim toga, takav zapis je identi an onome koji se upoptrebljava pri programiranju. Trojka (0,0,0) nema zna enja i isklju uje se iz razmatranja. 5. JEDNAD}BA PRAMENA KONIKA ZADANOG POMOU JEDNE DVOSTRUKE I DVIJU JEDNOSTRUKIH REALNIH TO AKA I HOMOGENE KOORDINATE Neka su s pomou homogenih koordinata zadane tri to ke  EMBED Equation.2   EMBED Equation.2  (5.1)  EMBED Equation.2  Mo~e se vidjeti da se jednad~ba pravca g koji prolazi to kama A i B mo~e napisati u obliku  EMBED Equation.2  (5.2) gdje smo ozna ili  EMBED Equation.2   EMBED Equation.2  (5.3)  EMBED Equation.2  Potpuno analogno imamo za pravac  EMBED Equation.2  koji prolazi to kama C i A:  EMBED Equation.2  (5.4) gdje smo ozna ili  EMBED Equation.2   EMBED Equation.2  (5.5)  EMBED Equation.2  kao i za pravac  EMBED Equation.2  koji prolazi to kama C i B:  EMBED Equation.2  (5.6) uz oznake  EMBED Equation.2   EMBED Equation.2  (5.7)  EMBED Equation.2  Pravac t prolazi to kom C, a njegov smjer neka odreuje jedna pomona to ka T s homogenim koordinatama  EMBED Equation.2  (5.8) Jednad~ba pravca t koji prolazi to kama C i T glasi tada  EMBED Equation.2  (5.9) gdje smo ozna ili  EMBED Equation.2   EMBED Equation.2  (5.10)  EMBED Equation.2  U 3. poglavlju ustanovili smo da je  EMBED Equation.2  (3.4) jednad~ba pramena konika kojem su A, B i C temeljne to ke, a t zajedni ka tangenta svih konika pramena. Pomou relacija (5.1) (5.10) mo~e se izvesti da je  EMBED Equation.2 oblika  EMBED Equation.2  (5.11) uz oznake  EMBED Equation.2  EMBED Equation.2  (5.12) Analogno gt je oblika  EMBED Equation.2  (5.13) uz oznake  EMBED Equation.2  (5.14) Pomou izvedenih formula mo~emo sastaviti program za ra unalo koji na temelju zadanih to aka A, B, C i T odreuje koeficijente u jednad~bi pramena  EMBED Equation.2  6. PRIMJERI Prethodna razmatranja izvedena su po etkom 1990-ih radi izrade crte~a za rad Klassifikationstheorie der Kegelscnittbuschel vom Typ IV der Isotropen Ebene, II, V. ` uri i H. Sachsa koji je objavljen 1997. Za prou avanje geometrije izotropne ravnine mo~e se upotrijebiti npr. monografija H. Sachsa (1987). Budui da ra unala po etkom 1990-ih nisu imala ugraeno poznavanje geometrije izotropne ravnine, a koliko mi je pozanto, nemaju ni danas, trebalo se nekako snai, odnosno pojedine jednad~be "prevesti" na jezik geometrije euklidske ravnine. U tu je svrhu, na temelju u ovome radu prikazanih formula, bio sastavljen odgovarajui potprogram za ra unalo u Basicu, koji polazei od zadanih homogenih koordinata to aka A, B, C i T odreuje koeficijente u jednad~bi pripadnog pramena konika. Postupak se temelji na primjeni homogenih koordinata kako bi se i neizmjerno daleke to ke ravnine mogle analiti ki obuhvatiti analogno kao i one u kona nosti. To ka C je dvostruka to ka, a to ka T je pomona to ka, koja zajedno s to kom C definira zajedni ku tangentu svih konika pramena. Nakon ato su bili izra unani koeficijenti u jednad~bi pramena, primjena odgovarajueg programa omoguila je grafi ko prikazivanje pramena (Lapaine 1997). Postupak se odvija na taj na in da se najprije za svaku krivulju pramena izra unaju koordinate niza uzastopnih to aka. Pritom se gustoa to aka uzdu~ pojedine krivulje i gustoa krivulja u pramenu mogu interaktivno regulirati. Gustoa to aka uzdu~ pojedine krivulje bira se tako da se pri iscrtavanju ne primijeti izlomljenost linije, ali da se istovremeno prevelikom gustoom ne preoptereuje memorija i vrijeme izvoenja. Gustoa krivulja u pramenu odreuje se odabirom koraka parametra na taj na in da na slici ne bude previae linija i time slika ne itljiva i na djelovima precrna, ali da se istovremeno prika~u sve karakteristi ne krivulje pramena. Nakon ato smo zadovoljni s prikazom na ekranu monitora, slika se sprema u .DXF zapisu kako bismo je mogli u itati u AutoCAD i dalje ureivati. Tu se prvenstveno misli na opis slike, eventualno brisanje suvianih elemenata te zadavanje boje, odnosno debljine onih linija koje na slici ~elimo posebno istaknuti. Primjeri koji slijede ozna eni su u skladu s radom (` uri i Sachs, 1997). Primjer 1 Pramen tipa IV3  Slika 2. Pramen tipa IV3 Zadane su to ke u homogenim koordinatama A(1,4,4), B(1,4,0), dvostruka to ka C(1,0,0) i pomona to ka T(1,0,1). Jednad~ba pramena je:  EMBED Equation.2  Primjer 2 Pramen tipa IV4,1,1  Slika 3. Pramen tipa IV4,1,1 Zadane su to ke u homogenim koordinatama A(0,3,1), B(0,1,4), dvostruka to ka C(1,0,0) i pomona to ka T(1,1,0). Jednad~ba pramena je:  EMBED Equation.2  Primjer 3 Pramen tipa IV4,2 Zadane to ke u homogenim koordinatama: A(1,1,2), B(1,3,6), dvostruka to ka C(0,1,0) i pomona to ka T(1,1,0). Jednad~ba pramena je:  EMBED Equation.2   Slika 4. Pramen tipa IV4,2 Primjer 4 Pramen tipa IV4,2k Zadane to ke u homogenim koordinatama: A(1,1,1), B(1, 1, 1), dvostruka to ka C(0,1,0) i pomona to ka T(1,1,0). Jednad~ba pramena je:  EMBED Equation.2   Slika 5. Pramen tipa IV4,2k Primjer 5 Pramen tipa IV5,1 Zadane to ke u homogenim koordinatama: A(0,1,0), B(0,1,2), dvostruka to ka C(1,0,0) i pomona to ka T(1,0,1). Jednad~ba pramena je:  EMBED Equation.2   Slika 6. Pramen tipa IV5,1 Primjer 6 Pramen tipa IV7 Zadane to ke u homogenim koordinatama: A(0,1,1), B(1, 0.5,1.5), dvostruka to ka C(1,0,0) i pomona to ka T(1,1,0). Jednad~ba pramena je:  EMBED Equation.2   Slika 7. Pramen tipa IV7 Primjer 7 Pramen tipa IV8 Zadane to ke u homogenim koordinatama: A(0,0,1), B(1,2,2), dvostruka to ka C(1,0,0) i pomona to ka T(1,1,0). Jednad~ba pramena je:  EMBED Equation.2   Slika 8. Pramen tipa IV8 Primjer 8 Pramen tipa IV9k Zadane to ke u homogenim koordinatama: A(1,2,0), B(1, 2,0), dvostruka to ka C(0,0,1) i pomona to ka T(1,0,1). Jednad~ba pramena je:  EMBED Equation.2   Slika 9. Pramen tipa IV9k Primjer 9 Pramen tipa IV10 Zadane to ke u homogenim koordinatama: A(0,1,1), B(1,0,2), dvostruka to ka C(0,1,0) i pomona to ka T(1,1,0). Jednad~ba pramena je:  EMBED Equation.2   Slika 10. Pramen tipa IV10 Primjer 10 Pramen tipa IV13 Zadane to ke u homogenim koordinatama: A(1,0,0), B(1,1,3), dvostruka to ka C(0,1,0) i pomona to ka T(0,0,1). Jednad~ba pramena je:  EMBED Equation.2   Slika 11. Pramen tipa IV13 LITERATURA CESAREC, R. (1957): Analiti ka geometrija linearnog i kvadratnog podru ja, I dio. `kolska knjiga, Zagreb. LAPAINE, M. (1997): Grafi ki prikaz pramena konika pomou ra unala. KoG 2, 43-47. LAPAINE, M., JOVI I, D. (1996): Grafi ki prikazi konika pomou ra unala. KoG 1, 19-26. SACHS, H. (1987): Ebene Isotrope Geometrie, Friedr. Vieweg & Sohn, Braunschweig, Wiesbaden. ` URI- UDOVAN, V., SACHS, H. (1995): Klassifikationstheorie der Kegelschnittbuschel vom Typ IV der Isotropen Ebene, I. Rad HAZU 470, Matemati ke znanosti, Svezak 12, 119-137. ` URI- UDOVAN, V., SACHS, H. (1997): Klassifikationstheorie der Kegelschnittbuschel vom Typ IV der Isotropen Ebene, II. Rad HAZU 472, Matemati ke znanosti, Svezak 13, 27-53. Autor: Doc. dr. sc. Miljenko Lapaine, dipl. in~. mat. 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D>v FMicrosoft Equation 3.0 DS Equation Equation.39q.IkI G(x,y)=ObjInfo6Equation Native 7_975601951F %d1h %d1hOle <a 2 x 2 +2b 2 xy+c 2 y 2 +2d 2 x+2e 2 y+f 2 =0,L4H@4E  .PIC =LMETA ?CompObjGfObjInfoI1  @& & MathTypePSymbol2- 2 `@mSymbol- 2 ` CRO_Dutch-Normal- 2 `R' &  "System-f~ 2 0f~ FMicrosoft Equation 3.0 DS Equation Equation.39q.IDsI "RL {h {D  .1  @&` &Equation Native J4_975601971 .F@,d1h@,d1hOle KPIC LLMETA NHCompObj\fObjInfo^Equation Native _ MathTypepCRO_Dutch-Normal- 2 `@H: 2 `gx 2 `Ty 2 `uF 2 `Q x 2 `> y 2 `G7 2 `x 2 `yCRO_Dutch-Normal- 2 `( 2 `Z,` 2 `R) 2 ` ( 2 `D ,` 2 `< ) 2 `R( 2 `,` 2 `) 2 `.`Symbol- 2 `= 2 `t+Symbol- 2 `m &  "System-F FMicrosoft Equation 3.0 DS Equation Equation.39q.lXII H(x,y)=F(x,y)+G(x,y)  .L%{|h%{  .1  @`"& " & MathTypePCRO_Dutch-Normal-_975602005"F z=d1hFd1hOle bPIC !$cLMETA e 2 @H: 2 gx 2 My 2 max 2  bxy 2 cy 2 dx 2 ey 2  f~CRO_Dutch-Normal- 2 ( 2 Z,` 2 K) 2 !,`Symbol- 2 = 2  + 2 + 2 + 2 + 2 G+ CRO_Dutch-Normal- 2 3 2p 2 R2pCRO_Dutch-Normal- 2 5 2 2 T2 2 )2 &  "System- FMicrosoft Equation 3.0 DS Equation Equation.39qCompObj#%ufObjInfo&wEquation Native x_976529835)FCNd1hjWd1h.ҘkI(~I H(x,y)=ax 2 +2bxy+cy 2 +2dx+2ey+f,L  .1  @&@ &Ole {PIC (+|LMETA ~CompObj*,f MathTypeCRO_Dutch-Normal- 2 @+a 2 @a 2 @a 2 @Vf~ 2 @f~ 2 @0f~Symbol- 2 @= 2 @+ 2 @= 2 @+ CRO_Dutch-Normal- 2 H1p 2 2p 2 r1p 2 2pSymbol- 2 @(m 2 @RmCRO_Dutch-Normal- 2 @~ ,` 2 @X .` 2 @' .` 2 @ .` 2 @ ,` 2 @.` &  "System- FMicrosoft Equation 3.0 DS Equation Equation.39qObjInfo-Equation Native _975602141 50Ffd1hfd1hOle ֠$IDgI a=a 1 +a 2 ,...,f=f 1 +f 2   .L4H@4^9  .1  @& &PIC /2LMETA CompObj13fObjInfo4 MathTypePSymbolx- 2 `@mSymbol- 2 ` CRO_Dutch-Normal- 2 `R' &  "System- FMicrosoft Equation 3.0 DS Equation Equation.39q.XI$I "RL4T@4O  .1   & & MathTypePCRO_Dutch-Normal-Equation Native 4_9756021627Fuwd1huwd1hOle PIC 69LMETA CompObj8:fObjInfo;Equation Native  2 `@H: 2 `gx 2 `My 2 `F 2 `^ x 2 `D y 2 `G7 2 `x 2 `yCRO_Dutch-Normal- 2 `( 2 `Z,` 2 `K) 2 ` ( 2 `Q ,` 2 `B ) 2 `p( 2 `,` 2 `)Symbol- 2 `= 2 `+ 2 `_=Symbol- 2 `&mCRO_Dutch-Normal- 2 `)0 &  "System- FMicrosoft Equation 3.0 DS Equation Equation.39q.hI(~I H(x,y)=F(x,y)+G(x,y)=0-L4l @4D  .1  & & MathTypePCRO_Dutch-Normal-     / "!#%$'&)(+*-,0.2M13465879:;=<>?@ABCEDGFHIJKNLQkOPRSUTWVYX[Z\]^`_acbdefhgijlnmopqsrtuvwxyz{|}~_975602216X>Ffd1hfd1hOle PIC =@LMETA h 2 `+a 2 `[b 2 `f~ 2 ` a 2 `D b 2 `f~ CRO_Dutch-Normal- 2 1p 2 /1p 2 1p 2  2p 2 $ 2p 2 2pCRO_Dutch-Normal- 2 `,` 2 `,` 2 `.` 2 `t.` 2 `C.` 2 `,` 2 `, ,` 2 `t ,` 2 ` ,` 2 `.` 2 `u.` 2 `D.` 2 `,` 2 `E,` &  "System-F FMicrosoft Equation 3.0 DS Equation Equation.39q.ҰXIPI a 1 ,b 1 ,...,f 1 ,a CompObj?AfObjInfoBEquation Native _975602225EF@d1h`d1h2 ,b 2 ,...,f 2 ,L4H@469  .1  @& & MathTypePSymbolx- 2 `@Ole PIC DGLMETA CompObjFHfmSymbol- 2 ` CRO_Dutch-Normal- 2 `R' &  "System- FMicrosoft Equation 3.0 DS Equation Equation.39qObjInfoIEquation Native 4_975602232CQLF d1h d1hOle .II "RL 4@ 4(  .1   & & MathTypePCRO_Dutch-Normal- 2 `@F 2 `x 2 ` yPIC KNLMETA hCompObjMOfObjInfoPCRO_Dutch-Normal- 2 `~( 2 `,` 2 `)Symbol- 2 `j=CRO_Dutch-Normal- 2 `*0 &  "System-%SF  FMicrosoft Equation 3.0 DS Equation Equation.39q.,II F(x,y)=0L 4@ 4P  .1   & &Equation Native H_975602257SFd1hd1hOle PIC RULMETA hCompObjTVfObjInfoWEquation Native H MathTypePCRO_Dutch-Normal- 2 ` G7 2 `x 2 `yCRO_Dutch-Normal- 2 `x( 2 ` ,` 2 `)Symbol- 2 `d=CRO_Dutch-Normal- 2 `$0 &  "System-(i)-(iii     !$')*+,-./0123469:=?@ABCDEGJMOPQRSTUVWXY[^_bdefghjoqrstuvwy~ FMicrosoft Equation 3.0 DS Equation Equation.39q.,IxI G(x,y)=0L4H@_975602268J_ZFd1hd1hOle PIC Y\LMETA 4N  .1  @& & MathTypePSymbol- 2 `@mSymbol- 2 ` CRO_Dutch-Normal- 2 `R' &  "System-t6p. CompObj[]fObjInfo^Equation Native 4_981968595'faFd1he1h FMicrosoft Equation 3.0 DS Equation Equation.39q.,I0I "RL*4@*4^j   .Ole PIC `cLMETA (CompObjbd f1  &@ & MathTypePTimes New Roman- 2 `MP 2 `)x 2 `y Times New Roman- 2 i> 2 i> 2 i>Times New Roman- 2 `( 2 `d,` 2 `[) & "System-,`  FMicrosoft Equation 3.0 DS Equation Equation.39qP$cI,gI P i (x i ,y i ) L4W @ObjInfoe"Equation Native #l_981968682hFe1he1hOle %PIC gj&LMETA ((CompObjik5fObjInfol74i  .1  & & MathTypePTimes New Roman- 2 `Mg 2 `a 2 `Ux 2 `b 2 `I y 2 ` c 2 `cik 2 `ck Times New Roman- 2 &ik>c 2 gik>c 2 Mik>c 2 > ik>cSymbol- 2 `o= 2 `_+ 2 `S + 2 ` = 2 `(Times New Roman- 2 `0 2 `,` 2 `0,` & "System- 2 ` FMicrosoft Equation 3.0 DS Equation Equation.39qEquation Native 8_981968736<oFNe1h`v(e1hOle ;PIC nq<LѠ}IH~I g ik =a ik x+b ik y+c ik =0,i`"k,L40@4i  .1  &@ &META >CompObjprFfObjInfosHEquation Native IP MathTypePTimes New Roman- 2 `MP 2 `P Times New Roman- 2 i> 2 CkcTimes New Roman- 2 `,` & "System- FMicrosoft Equation 3.0 DS Equation Equation.39q4xII P i P k ,0L14 @14VI [ .1   &  &_981968795vF@?9e1h@?9e1hOle KPIC uxLLMETA N MathTypePTimes New Roman|- 2 `Mg 2 `.g 2 `g 2 `jg Times New Roman|- 2 12pp 2 34pp 2 Z13pp 2 C 24ppTimes New Roman|- 2 ` 0Symbol- 2 `{+ 2 ` =Symbol- 2 `m & "System- FMicrosoft Equation 3.0 DS Equation Equation.39qtkIoI g 12 g 34 +g 13 g 24 =0CompObjwyZfObjInfoz\Equation Native ]_981968854t}F Je1h@Qe1hOle `PIC |aLMETA chCompObj~ifL4@4*  .1  &@ & MathTypePTimes New Roman|- 2 `MP Times New Roman|- 2 i> & "System- FMicrosoft Equation 3.0 DS Equation Equation.39qxI\I P iL{4h@ObjInfokEquation Native l8_981968889F rbe1h rbe1hOle mPIC nLMETA pCompObjxfObjInfoz{4*  .1  @& & MathTypePTimes New Roman|- 2 `MP Times New Roman|- 2 i>Times New Roman|- 2 `) & "System-"System- FMicrosoft Equation 3.0 DS Equation Equation.39qkIPI P iL4@4I  .Equation Native {8_981969044{F;se1hb|e1hOle |PIC }LMETA hCompObjfObjInfoEquation Native 81  & & MathTypePTimes New Roman|- 2 `Mg Times New Roman|- 2 1p & "System- FMicrosoft Equation 3.0 DS Equation Equation.39q$cI,gI g 1L4,@4i  .1  & & MathTypePTimes New Roman|- 2 `Mg Times New Roman|_981969060F+e1h+e1hOle PIC LMETA h- 2 &2p & "System- FMicrosoft Equation 3.0 DS Equation Equation.39qxI4yI g 2DEL4 @CompObjfObjInfoEquation Native 8_981969071Fe1he1hOle PIC LMETA CompObjf4V \ .1  @& & MathTypePTimes New Roman- 2 `3tk 2 `g 2 ` gSymbol- 2 `= 2 ` = 2 ` =Times New Roman- 2 `A0 2 `60 2 ` 0 Times New Roman- 2  1pTimes New Roman- 2 `,` 2 `,` & "System- FMicrosoft Equation 3.0 DS Equation Equation.39qObjInfoEquation Native _981969209F`e1h`e1hOle рII t=0,g=0,g 1 =0ig 2 =0LU 4@U 4 = .1  ` &  &PIC LMETA CompObjfObjInfo MathTypePTimes New Roman- 2 `Mg 2 `g 2 `gtk Times New Roman- 2 1p 2 2pTimes New Roman- 2 `h0Symbol- 2 `+ 2 `;=Symbol- 2 `m & "System- FMicrosoft Equation 3.0 DS Equation Equation.39qH`II g 1 g 2 +gt=00L4@Equation Native d_981969259F@e1h@e1hOle PIC L4~X  .1  & & MathTypePTimes New Roman|- 2 `Mg Times New Roman|- 2 1p & "System- FMicrosoft Equation 3.0 DS EqMETA hCompObjfObjInfoEquation Native 8uation Equation.39q,I8I g 1 L4,@4\  .1  & &_981969284F Oe1h Oe1hOle PIC LMETA h MathTypePTimes New Roman{- 2 `Mg Times New Roman{- 2 &2p & "System- FMicrosoft Equation 3.0 DS Equation Equation.39q`II g 2CompObjfObjInfoEquation Native 8_981969415F e1he1hOle PIC LMETA CompObjfL_4 @_4F]  .1  & & MathTypePTimes New Roman{- 2 `Mg 2 `Og Times New Roman{- 2 1p 2 (2pTimes New Roman{- 2 `( 2 `).` & "System- FMicrosoft Equation 3.0 DS Equation Equation.39q<$cI,gI g 1 (gObjInfoEquation Native X_981969426Ff1hf1hOle  2 ).Lnl  .1  & @ & MathTypePSymbol- 2 @m & "SystemPIC LMETA (CompObjfObjInfo-"System- FMicrosoft Equation 3.0 DS Equation Equation.39qxI4yI L 4@Equation Native ,_981969477Frf1hrf1hOle PIC LMETA CompObjfObjInfoEquation Native    "')*+,-./01247:<=>?@ABCDEFGILMPRSTUVWXYZ[]`cefghijklmnopqsvwz|}~ 4G l .1   &`  & MathTypePTimes New Roman|- 2 `Mg 2 `Ox 2 `y 2 `g 2 `.x 2 `y Times New Roman|- 2 1p 2 2pTimes New Roman|- 2 `N 0 2 `( 2 `,` 2 `o) 2 `( 2 `,` 2 `N ) 2 ` ,`Symbol- 2 `! = & "System-  FMicrosoft Equation 3.0 DS Equation Equation.39qdII g 1 (x,y)g 2 (x,y)=0,L4@4C  .1  & & MathTypePTimes New Roman|- 2 `Mg Times New Roman|_981969505F+f1h+f1hOle  PIC  LMETA  h- 2 1p & "System- FMicrosoft Equation 3.0 DS Equation Equation.39qkII g 1L{4h@CompObjfObjInfoEquation Native 8_981969510F  .1  ` &  & MathType "-6V Times New Roman|- 2 Mx 2 zx 2 x 2 META ;(CompObjHfObjInfoJEquation Native Kmy 2 zx 2 x Times New Roman|- 2 {op 2  opSymbol- 2 V= 2 v= Times New Roman|- 2 1p 2  2pTimes New Roman|- 2 z,` 2  .` & "System-| FMicrosoft Equation 3.0 DS Equation Equation.39qфxII x=x 1 x 0 ,y=x 2 x 0 .L4`@_981969672F of1h of1hOle NPIC OLMETA Q4~r 2 .1  & & MathTypePTimes New Roman|- 2 `3( 2 `A,` 2 `%,` 2 `=)Times New Roman|- 2 `x 2 `x 2 `x Times New Roman|- 2 op Times New Roman|- 2 1p 2 2p & "System-1p 2 2p & FMicrosoft Equation 3.0 DS Equation Equation.39qPDIxI (x 0 ,xCompObj\fObjInfo^Equation Native _l_981969749Ff1hf1h 1 ,x 2 )L4 @4nu  .1  @& & MathTypePTimes New Roman|- 2 `Mx 2 `x 2 ` x Ole aPIC bLMETA dHCompObjrfTimes New Roman|- 2  opSymbol- 2 `= 2 `6= 2 ` =Times New Roman|- 2 ` 1 2 `c3 2 ` 2 Times New Roman|- 2 J1p 2  2pTimes New Roman|- 2 `,` 2 ` ,` 2 `,` & "System-,` 2 `,`  FMicrosoft Equation 3.0 DS Equation Equation.39qt$cI,gI x 0 =1,x 1 =3,x 2 =2,ObjInfotEquation Native u_981969841FZf1hZf1hOle xL]4 @]4&n  .1  & & MathTypePTimes New Roman|- 2 `Mx 2 `x 2 ` x Times New Roman|- 2  oPIC yLMETA {hCompObjfObjInfopSymbol- 2 `= 2 ` = 2 ` = Symbol- 2 ={Times New Roman|- 2 `;2 2 `7 6 2 `4 Times New Roman|- 2 1p 2  2pTimes New Roman|- 2 `,` 2 ` ,` & "System-,`  FMicrosoft Equation 3.0 DS Equation Equation.39qPzI{I x 0 =2,x 1 =6,L]4 @Equation Native l_981969853F#f1hĹf1hOle PIC L]4&n  .1  & & MathTypePTimes New Roman|- 2 `Mx 2 `x 2 ` x Times New Roman|- 2  opSymbol- 2 `= 2 ` = 2 ` = META hCompObjfObjInfoEquation Native DSymbol- 2 ={Times New Roman|- 2 `;2 2 `7 6 2 `4 Times New Roman|- 2 1p 2  2pTimes New Roman|- 2 `,` 2 ` ,` & "System-,`  FMicrosoft Equation 3.0 DS Equation Equation.39q($sIl}I x 2 =4L4@4u  ._981969867Ff1hf1hOle PIC LMETA 1  &@ & MathTypePTimes New Roman|- 2 `Mx Times New Roman|- 2  opSymbol- 2 `=Times New Roman|- 2 `;0 2 `,` & "System-CompObjfObjInfoEquation Native D_981969879 FVf1h@~f1h FMicrosoft Equation 3.0 DS Equation Equation.39q(8Equation Native 9_982254505AFWg1hWg1h&d$cI,gI A=(x 0A ,x 1A ,x 2A )L+*WT+*W  .1   @&&& &Ole ;PIC @C<LMETA >HCompObjBDTf MathType`Times New Roman- 2 `[A 2 `]x 2 `7x 2 `x 2 `k B 2 `zx 2 `?x 2 `x 2 `YC 2 `x 2 `_ x 2 `#x Times New Roman- 2 op 2 xA 2 = A 2 8op 2 B 2 xB 2 op 2 =B 2 Oop 2 C 2 !C 2 X$CSymbol- 2 `= 2 `= 2 `=Times New Roman- 2 `( 2 `,` 2 `4,` 2 ` ),` 2 `( 2 `,` 2 `<,` 2 `),` 2 `( 2 `,` 2 `e",` 2 `@%).` Symbol- 2 A Times New Roman- 2 1p 2  2p 2 1p 2  !1p 2 #2p & "System- FMicrosoft Equation 3.0 DS Equation Equation.39q&hphI|I B=(x 0ObjInfoEVEquation Native W_982254548?HFg1hg1hOle ZB ,x 1B ,x 2B )L+*WT+*W  .1   @&&& & MathType`Times New Roman-PIC GJ[LMETA ]HCompObjIKsfObjInfoLu 2 `[A 2 `]x 2 `7x 2 `x 2 `k B 2 `zx 2 `?x 2 `x 2 `YC 2 `x 2 `_ x 2 `#x Times New Roman- 2 op 2 xA 2 = A 2 8op 2 B 2 xB 2 op 2 =B 2 Oop 2 C 2 !C 2 X$CSymbol- 2 `= 2 `= 2 `=Times New Roman- 2 `( 2 `,` 2 `4,` 2 ` ),` 2 `( 2 `,` 2 `<,` 2 `),` 2 `( 2 `,` 2 `e",` 2 `@%).` Symbol- 2 A Times New Roman- 2 1p 2  2p 2 1p 2  !1p 2 #2p & "System- FMicrosoft Equation 3.0 DS Equation Equation.39q&lII C=(x 0C ,x 1C ,x 2C )Equation Native v_981970759#[OFg1hg1hOle yPIC NQzL.L {Dh { ? .1  @ &`  & MathTypepTimes New Roman- 2 `Mg 2 `x 2 `%g 2 `WMETA |CompObjPRfObjInfoSEquation Native xg Times New Roman- 2 4xc 2 yc 2 7zWSymbol- 2 `+ 2 `+ 2 `+ =Times New Roman- 2 `X 0 2 ` ,` & "System-0 FMicrosoft Equation 3.0 DS Equation Equation.39q\II g x x+g y y+g z =0,LF'<DF'<  ._981982490VFg1hg1hOle PIC UXLMETA 1  #&`#m & MathType "-SqgqS g SgS g SvgvS!g!Times New Roman- 2 Mg 2 x 2 Ox 2 x 2 Ox 2  g 2 Dx 2 x 2 0x 2 x 2 `g 2 x 2 hx 2 x 2 Tx Times New Roman- 2 4xc 2 B 2 A 2 4mop 2 GB 2 4 op 2 GA 2  yc 2 }B 2 %A 2 4op 2 GB 2 4op 2 G8A 2 @zW 2 B 2  A 2 GB 2 G ASymbol- 2 /= 2  = 2 - 2 4= Times New Roman- 2 m2p 2  2p 2 1p 2 1p 2 v1p 2  1p 2 4r2p 2 4 2pTimes New Roman- 2  ,` 2 {,` 2 #.` & "System- 2 GB 2 G CompObjWYfObjInfoZEquation Native _981982572pb]Fzg1hzg1h FMicrosoft Equation 3.0 DS Equation Equation.39qѨ$cI,gI g x =x 2B x 2A x 0B x 0A ,Ole PIC \_LMETA CompObj^`fLF'<DF'<  .1  #&`#m & MathType "-SqgqS g SgS g SvgvS!g!Times New Roman- 2 Mg 2 x 2 Ox 2 x 2 Ox 2  g 2 Dx 2 x 2 0x 2 x 2 `g 2 x 2 hx 2 x 2 Tx Times New Roman- 2 4xc 2 B 2 A 2 4mop 2 GB 2 4 op 2 GA 2  yc 2 }B 2 %A 2 4op 2 GB 2 4op 2 G8A 2 @zW 2 B 2  A 2 GB 2 G ASymbol- 2 /= 2  = 2 - 2 4= Times New Roman- 2 m2p 2  2p 2 1p 2 1p 2 v1p 2  1p 2 4r2p 2 4 2pTimes New Roman- 2  ,` 2 {,` 2 #.` & "System- 2 GB 2 G  FMicrosoft Equation 3.0 DS Equation Equation.39qѬzI{I g y ="x 1B x 1A xObjInfoaEquation Native _981982641dFg1hg1hOle  0B x 0A ,LF'<DF'<  .1  #&`#m & MathType "-SqgqPIC cfLMETA CompObjegfObjInfohS g SgS g SvgvS!g!Times New Roman- 2 Mg 2 x 2 Ox 2 x 2 Ox 2  g 2 Dx 2 x 2 0x 2 x 2 `g 2 x 2 hx 2 x 2 Tx Times New Roman- 2 4xc 2 B 2 A 2 4mop 2 GB 2 4 op 2 GA 2  yc 2 }B 2 %A 2 4op 2 GB 2 4op 2 G8A 2 @zW 2 B 2  A 2 GB 2 G ASymbol- 2 /= 2  = 2 - 2 4= Times New Roman- 2 m2p 2  2p 2 1p 2 1p 2 v1p 2  1p 2 4r2p 2 4 2pTimes New Roman- 2  ,` 2 {,` 2 #.` & "System- 2 GB 2 G  FMicrosoft Equation 3.0 DS Equation Equation.39qѤoIXcI g z =x 1B x 1A x 2B x 2A .L4@4  .Equation Native _981970962kF h1h h1hOle PIC jmLMETA hCompObjlnfObjInfooEquation Native 81  & & MathTypePTimes New Roman|- 2 `Mg Times New Roman|- 2 1p & "System- FMicrosoft Equation 3.0 DS Equation Equation.39q      !"#$%&'()*+,-./0124789<>?@ABCDEFGHIJKLMNOPQRSTVYZ[^`abcdefghijklmnopqrstuvx{|}cII g 1LTT  .1   & C & MathTypeTimes New Roman{- 2 }Mg 2 };x 2 }g 2 }_981971024iTrFh1h ?h1hOle PIC qtLMETA Hg Times New Roman{- 2 xc 2 yc 2 H zW Times New Roman{- 2 1p 2 K1p 2 1pTimes New Roman{- 2 }_ 0Symbol- 2 }A+ 2 }+ 2 }6 =Times New Roman{- 2 }h ,` & "System-Times New Roman{ FMicrosoft Equation 3.0 DS Equation Equation.39qрItpI g 1  x x+g 1  y y+g 1z =0,CompObjsufObjInfovEquation Native _981982933yF+h1h+h1hOle PIC x{LMETA CompObjz|3fL''  .1  $&#C & MathType "-]]] ] ]u]u]7]7]]]!]!Times New Roman|- 2 Mg 2 x 2 x 2 x 2 x 2  g 2 x 2 &x 2 x 2 x 2 g 2 Kx 2 x 2 7x 2 x Times New Roman|- 2 xc 2 TA 2 C 2 =oAp 2 =PoCp 2 Q yc 2 A 2 9C 2 =qoAp 2 =oCp 2 zW 2 tA 2  C 2 =A 2 = C Times New Roman|- 2 1p 2 2p 2 J2p 2  1p 2 d1p 2 1p 2 W1p 2 1p 2  1p 2 =2p 2 =} 2pSymbol- 2 = 2 L= 2 - 2 =Times New Roman|- 2 ( ,` 2 ,` 2 v#.` & "System-2  FMicrosoft Equation 3.0 DS Equation Equation.39qѬ|I\I g 1x =x 2A x 2C xObjInfo}5Equation Native 6_9819829078F/4h1h;h1hOle : 0A x 0C ,L''  .1  $&#C & MathType "-]]PIC ;LMETA =CompObjUfObjInfoW] ] ]u]u]7]7]]]!]!Times New Roman|- 2 Mg 2 x 2 x 2 x 2 x 2  g 2 x 2 &x 2 x 2 x 2 g 2 Kx 2 x 2 7x 2 x Times New Roman|- 2 xc 2 TA 2 C 2 =oAp 2 =PoCp 2 Q yc 2 A 2 9C 2 =qoAp 2 =oCp 2 zW 2 tA 2  C 2 =A 2 = C Times New Roman|- 2 1p 2 2p 2 J2p 2  1p 2 d1p 2 1p 2 W1p 2 1p 2  1p 2 =2p 2 =} 2pSymbol- 2 = 2 L= 2 - 2 =Times New Roman|- 2 ( ,` 2 ,` 2 v#.` & "System-2  FMicrosoft Equation 3.0 DS Equation Equation.39qEquation Native X_981982777FDh1hDh1hOle \PIC ]LѰcII g 1y ="x 1A x 1C x 0A x 0C ,L''  .META _CompObjwfObjInfoyEquation Native z1  $&#C & MathType "-]]] ] ]u]u]7]7]]]!]!Times New Roman|- 2 Mg 2 x 2 x 2 x 2 x 2  g 2 x 2 &x 2 x 2 x 2 g 2 Kx 2 x 2 7x 2 x Times New Roman|- 2 xc 2 TA 2 C 2 =oAp 2 =PoCp 2 Q yc 2 A 2 9C 2 =qoAp 2 =oCp 2 zW 2 tA 2  C 2 =A 2 = C Times New Roman|- 2 1p 2 2p 2 J2p 2  1p 2 d1p 2 1p 2 W1p 2 1p 2  1p 2 =2p 2 =} 2pSymbol- 2 = 2 L= 2 - 2 =Times New Roman|- 2 ( ,` 2 ,` 2 v#.` & "System-2  FMicrosoft Equation 3.0 DS Equation Equation.39qѨIdhI g 1z =x 1A x 1C x 2A x 2C .L4,@_981983001wFb]h1hb]h1hOle ~PIC LMETA h4!  .1  & & MathTypePTimes New Roman|- 2 `Mg Times New Roman|- 2 &2p & "System- FMicrosoft Equation 3.0 DS EqCompObjfObjInfoEquation Native 8_982254608F+nh1h+nh1huation Equation.39q0IoI g 2L{\h{  .1  @` &  &Ole PIC LMETA HCompObjf MathTypepTimes New Roman{- 2 `Mg 2 `Px 2 `g 2 `Og Times New Roman{- 2 xc 2  yc 2  zW Times New Roman{- 2 &2p 2 z2p 2 ( 2pTimes New Roman{- 2 ` 0Symbol- 2 `Z+ 2 `+ 2 ` =Times New Roman{- 2 ` .` & "System- 2 ` .`  FMicrosoft Equation 3.0 DS Equation Equation.39q&pphI8I g 2x x+ObjInfoEquation Native _981983101F`~h1h h1hOle g 2y y+g 2z =0,L&&&  .1  @#&#C & MathType "-]]PIC LMETA CompObjfObjInfo] ] ]]]]]v]v]f"]f"Times New Roman|- 2 Mg 2 +x 2 x 2 8x 2 x 2 4 g 2 x 2 |x 2 x 2 ox 2 g 2 x 2 O x 2 x 2 < x Times New Roman|- 2 xc 2 sB 2 C 2 =oBp 2 =woCp 2  yc 2 )B 2 C 2 =oBp 2 =-oCp 2 @zW 2 B 2 b!C 2 =B 2 =v!C Times New Roman|- 2 &2p 2 2p 2 q2p 2  2p 2 1p 2 *1p 2 2p 2 v1p 2  1p 2 =r2p 2 = 2pSymbol- 2 = 2 = 2 - 2 4=Times New Roman|- 2 O ,` 2 ,` 2 ".` & "System-2  FMicrosoft Equation 3.0 DS Equation Equation.39qEquation Native _981983137F@h1h@h1hOle PIC LѬI0I g 2x =x 2B x 2C x 0B x 0C ,L&&&  .META CompObjfObjInfoEquation Native 1  @#&#C & MathType "-]]] ] ]]]]]v]v]f"]f"Times New Roman|- 2 Mg 2 +x 2 x 2 8x 2 x 2 4 g 2 x 2 |x 2 x 2 ox 2 g 2 x 2 O x 2 x 2 < x Times New Roman|- 2 xc 2 sB 2 C 2 =oBp 2 =woCp 2  yc 2 )B 2 C 2 =oBp 2 =-oCp 2 @zW 2 B 2 b!C 2 =B 2 =v!C Times New Roman|- 2 &2p 2 2p 2 q2p 2  2p 2 1p 2 *1p 2 2p 2 v1p 2  1p 2 =r2p 2 = 2pSymbol- 2 = 2 = 2 - 2 4=Times New Roman|- 2 O ,` 2 ,` 2 ".` & "System-2  FMicrosoft Equation 3.0 DS Equation Equation.39qѬHIԤI g 2y ="x 1B x 1C x 0B x 0C ,L&_981983170Fh1h h1hOle PIC LMETA &&  .1  @#&#C & MathType "-]]] ] ]]]]]v]v]f"]f"Times New Roman|- 2 Mg 2 +x 2 x 2 8x 2 x 2 4 g     "#$%&'()*+-0356789:;<=>?@ABCDEFGHIJKMPQRUWXYZ[\]^_`abcdefghijklmorstwyz{|}~2 x 2 |x 2 x 2 ox 2 g 2 x 2 O x 2 x 2 < x Times New Roman|- 2 xc 2 sB 2 C 2 =oBp 2 =woCp 2  yc 2 )B 2 C 2 =oBp 2 =-oCp 2 @zW 2 B 2 b!C 2 =B 2 =v!C Times New Roman|- 2 &2p 2 2p 2 q2p 2  2p 2 1p 2 *1p 2 2p 2 v1p 2  1p 2 =r2p 2 = 2pSymbol- 2 = 2 = 2 - 2 4=Times New Roman|- 2 O ,` 2 ,` 2 ".` & "System-2  FMicrosoft Equation 3.0 DS Equation 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tk 2 Gx 2 x 2 5x 2 x 2 tk 2 %x 2 x 2 x 2 x Times New Roman|- 2 xc 2 ~T| 2 C 2 = oTp| 2 =oCp 2  yc 2 RT| 2 C 2 =oTp| 2 =ioCp 2 zW 2 0T| 2 C 2 =CT| 2 =CSymbol- 2 = 2  = 2 - 2 = Times New Roman|- 2  2p 2 2p 2 1p 2 f1p 2 1p 2 L1p 2 =2p 2 =I2pTimes New Roman|- 2 b ,` 2 ;,` 2  .` & "System-- 2 = FMicrosoft Equation 3.0 DS Equation Equation.39qѨ$cI,gI t x =x 2T x 2C x 0T x 0C ,L#d#  .1   &` C &_981983458Frh1hrh1hOle SPIC TLMETA V MathType "-]]]]]]]]]]]]Times New Roman|- 2 3tk 2 Lx 2 x 2 Nx 2 x 2 - tk 2 Gx 2 x 2 5x 2 x 2 tk 2 %x 2 x 2 x 2 x Times New Roman|- 2 xc 2 ~T| 2 C 2 = oTp| 2 =oCp 2  yc 2 RT| 2 C 2 =oTp| 2 =ioCp 2 zW 2 0T| 2 C 2 =CT| 2 =CSymbol- 2 = 2  = 2 - 2 = Times New Roman|- 2  2p 2 2p 2 1p 2 f1p 2 1p 2 L1p 2 =2p 2 =I2pTimes New Roman|- 2 b ,` 2 ;,` 2  .` & "System-- 2 = FMicrosoft Equation 3.0 DS Equation Equation.39qCompObjnfObjInfopEquation Native q_982254657FF`;i1h`;i1hѬXImI t y ="x 1T x 1C x 0T x 0C ,L#dOle uPIC vLMETA xCompObjf#  .1   &` C & MathType "-]]]]]]]]]]]]Times New Roman|- 2 3tk 2 Lx 2 x 2 Nx 2 x 2 - tk 2 Gx 2 x 2 5x 2 x 2 tk 2 %x 2 x 2 x 2 x Times New Roman|- 2 xc 2 ~T| 2 C 2 = oTp| 2 =oCp 2  yc 2 RT| 2 C 2 =oTp| 2 =ioCp 2 zW 2 0T| 2 C 2 =CT| 2 =CSymbol- 2 = 2  = 2 - 2 = Times New Roman|- 2  2p 2 2p 2 1p 2 f1p 2 1p 2 L1p 2 =2p 2 =I2pTimes New Roman|- 2 b ,` 2 ;,` 2  .` & "System-- 2 = FMicrosoft Equation 3.0 DS Equation Equation.39q&ӤЇI8kI t z =x 1T x 1C x 2T x 2C .LU 4@ObjInfoEquation Native _981983749F@i1h`i1hOle PIC LMETA CompObjfObjInfoU 4V; = .1  ` &  & MathTypePTimes New Roman|- 2 `Mg 2 `g 2 `gtk Times New Roman|- 2 1p 2 2pTimes New Roman|- 2 `h0Symbol- 2 `+ 2 `;=Symbol- 2 `m & "System- FMicrosoft Equation 3.0 DS Equation Equation.39qH$cI,gI g 1 g Equation Native d_981983809F &i1h &i1hOle PIC L2 +gt=02L"X|"  .1  `&  & MathTypeTimes New Roman|- 2 `Mg 2 `g TimeMETA CompObjfObjInfoEquation Native Hs New Roman|- 2 1p 2 2p & "System- FMicrosoft Equation 3.0 DS Equation Equation.39q,xILyI g 1 g 2_982255123F7i1h 7?i1hOle PIC LMETA hL#{dh#{ - .1  @ &`  & MathTypePTimes New Roman- 2 Mgtk 2 G 2 x 2 =y 2 a 2 o x 2 <b 2 xy 2 yc 2 y 2 d 2 >x 2 4e 2 efkSymbol- 2 = 2 = 2 C + 2 L+ 2 + 2 H+ 2 +Times New Roman- 2 8( 2 ,` 2 ) Times New Roman- 2  2p 2 ? 2p 2 2p 2 72p 2 2p 2 2p 2 2p 2 2pTimes New Roman- 2 p 2 2 2 2 u2 & "System-p  FMicrosoft Equation 3.0 DS Equation Equation.39qCompObjfObjInfoEquation Native 0_971346148F^Hi1h^Hi1h&xIwI g 1 g 2 =F(x,y)==a 1 x 2 +2b 1 xy+c 1 y 2 +2d 1 x+2e 1 +f 1L=4@Ole PIC LMETA CompObjf=4~ P .1   & & MathTypeP &  FMicrosoft Equation 2.0 DS Equation Equation.29q4 G@FF +9ObjInfoEquation Native <_9822553341F'Yi1h'Yi1hOle PIC LMETA  CompObjfObjInfo L <   v .1  `& & MathType "-==~~Times New Roman- 2 _3a 2 _}g 2 _Yg 2 _ b 2 _g 2 _}g    !"#$%&'()*,/0125789:;<=>?@ABCDEFGHIJKLMNOPQRSUXYZ[\]^_`adfghijklmnopqrstuvwxyz{|}~ 2 _+g 2 _g 2 3c 2 g 2 g 2  d 2 pg 2 Lg 2 g 2 g 2 3e 2 g 2 g 2 > g 2  g 2 fk 2 g 2 g Times New Roman- 2 xc 2 xc 2 xc 2 yc 2 oyc 2 rxc 2 Xyc 2 cyc 2 xc 2 zW 2 zW 2 xc 2 Ayc 2 AzW 2 As zW 2 Aw yc 2 AzW 2 AzW Times New Roman- 2 1p 2 F1p 2 22p 2  1p 2 j1p 2 V2p 2 1p 2 2p 2 1p 2 1p 2 2p 2 1p 2 91p 2 %2p 2 1p 2 2p 2 A1p 2 A1p 2 A2p 2 A 1p 2 A 2p 2 A1p 2 A}1p 2 A[2pTimes New Roman- 2 i1 2 2 2 1 2 2 2 1 2  2 Times New Roman- 2 Z)JTimes New Roman- 2 _( 2 _#) 2 ( 2 ) 2 ( 2 () 2 .`Symbol- 2 _6= 2 _ = 2 _+ 2 = 2 m = 2 + 2 = 2 + 2 m= & "System-1 2  FMicrosoft Equation 3.0 DS Equation Equation.39q&ӸItI a 1 =g 1x g 2x b 1 =12(g 1x g 2y +g 1y g 2x )c 1 =g 1y g 2y d 1 =1Equation Native  _982255286Fii1hii1hOle PIC L2(g 1x g 2z +g 1z g 2x )e 1 =12(g 1y g 2z +g 1z g 2y )f 1 =g 1z g 2z .L#{dh#{ - .1  @ &`  & MathTypePTimes New Roman- 2 Mgtk 2 G 2 x 2 =y 2 a 2 o x 2 <b 2 xy 2 yc 2 y 2 d 2 >x 2 4e 2 efkMETA hCompObj+fObjInfo-Equation Native .Symbol- 2 = 2 = 2 C + 2 L+ 2 + 2 H+ 2 +Times New Roman- 2 8( 2 ,` 2 ) Times New Roman- 2  2p 2 ? 2p 2 2p 2 72p 2 2p 2 2p 2 2p 2 2pTimes New Roman- 2 p 2 2 2 2 u2 & "System-p  FMicrosoft Equation 3.0 DS Equation Equation.39q&(hITI gt=G(x,y)==a 2 x 2 +2b 2 xy+c 2 y 2 +2d 2 x+2e 2 y+f 22LIx I  .1  & &_982255360Fzi1hZi1hOle 3PIC 4LMETA 6h MathType "- w} }  Times New Roman- 2 _3a 2 _Hg 2 _tk 2 _ b 2 __g 2 _tk 2 _g 2 _*tk 2 3c 2 ;g 2 tk 2 d 2 g 2 tk 2 g 2 )tk 2 a3e 2 ag 2 a^tk 2 a1 g 2 a tk 2 aTfk 2 a/g 2 atk Times New Roman- 2 /xc 2 Sxc 2 Fxc 2 ryc 2 yc 2 xc 2 @ *yc 2 @ Vyc 2 @ qxc 2 @ zW 2 @ zW 2 @ xc 2 yc 2 zW 2  zW 2 6 yc 2 zW 2 %zW Times New Roman- 2 2p 2 Y 2p 2 @ 2p 2 @ 2p 2 2p 2 2pTimes New Roman- 2 i 1 2  2 2 1 2 2 2 k91 2 32Symbol- 2 _= 2 _\ = 2 _n+ 2 _= 2 = 2 = 2 + 2 a= 2 a+ 2 a=Times New Roman- 2 _( 2 ( 2 x) 2 aE( 2 a ) 2 a+.` & "System-2 FMicrosoft Equation 3.0 DS Equation Equation.39q&tpII a 2 =gCompObjTfObjInfoVEquation Native W_981990170F`i1h#i1h x t x b 2 =12(g x t y +g y t x )c 2 =g y t y d 2 =12(g x t z +g z t x )e 2 =12(g y t z +g z t y )f 2 =g z t z .LR=q"8R=q  .1  7&`7 & MathTypeTimes New Roman|-Ole bPIC cLMETA eCompObjf 2 `MF 2 `x 2 `Ty 2 `G 2 `x 2 ` y 2 ia 2 x 2 b 2 xy 2  c 2  y 2 d 2 x 2 e 2 Dy 2 fk 2 a 2 cx 2 0!b 2 "xy 2 m%c 2 &y 2 *d 2 2,x 2 (/e 2 0y 2 3fkTimes New Roman|- 2 `O( 2 `,` 2 `) 2 `( 2 `p ,` 2 ` ) 2 f( 2 _4) 2 7.`Symbol- 2 `+ 2 ` = 2 <= 2 + 2 k + 2  + 2 + 2 N+ 2 9+ 2 7+ 2 @$+ 2 (+ 2 <-+ 2 1+ 2 25=Symbol- 2 `!m 2 sm Times New Roman|- 2 t$1p 2 h2p 2 t&1p 2 tF 1p 2 h 2p 2 t~1p 2 t1p 2 tL1p 2 t2p 2 h32p 2 t!2p 2 t+&2p 2 h'2p 2 t+2p 2 t/2p 2 t32pTimes New Roman|- 2 2 2 2 2 F2 2 d 2 2 )2 2 i.2 2 _60 & "System-2  FMicrosoft Equation 3.0 DS Equation Equation.39q$cI,gI F(x,y)ObjInfo Equation Native _981990941 F i1h i1hOle +G(x,y)==a 1 x 2 +2b 1 xy+c 1 y 2 +2d 1 x+2e 1 y+f 1 ++(a 2 x 2 +2b 2 xy+c 2 y 2 +2d 2 x+2e 2 y+f 2 )=0.L{ h{R  .1  @&@ & MathTypePTimes New Roman- 2 Mxy 2 Ty 2 x 2  xSymbol- 2 PIC  LMETA (CompObj fObjInfo- 2 (+ 2  - 2 c = Times New Roman- 2 $2p 2 2pTimes New Roman- 2  4 2 0Symbol- 2 bmTimes New Roman- 2 U( 2  ) & "System- FMicrosoft Equation 3.0 DS Equation Equation.39q`$cI,gI xy"y 2 +(x 2 "4x)=00L{( hEquation Native |_981991169 Fݽi1hݽi1hOle PIC LMETA CompObjfObjInfoEquation Native {^  t .1  @@& & MathTypePTimes New Roman|- 2 34 2 13 2 U3 2 O0 Times New Roman|- 2 3p 2 2pTimes New Roman|- 2  x 2 txy 2 / y 2  ySymbol- 2 - 2 (+ 2  + 2 "=Symbol- 2 = m & "System- 2 = m  FMicrosoft Equation 3.0 DS Equation Equation.39qd$cI,gI 4x 2 "13xy+3y 2 +y=0L8{ h8{R  .1  @@& & MathTypePTimes New Roman|-_981991530Fi1hGi1hOle PIC LMETA h 2 [y 2 &y 2  xy 2 y Times New Roman|- 2 +2p 2 2pTimes New Roman|- 2 L8 2 B12 2  2 2 90Symbol- 2 .- 2 0+ 2 + 2 \ - 2  =Symbol- 2 S mTimes New Roman|- 2 F ( 2 9) & "System-(  FMicrosoft Equation 3.0 DS Equation Equation.39qp$cI,gI y 2 "8y+12+(2xy"y 2 )CompObjfObjInfoEquation Native _981991793-!Fi1hi1h=00L{ h{  .1  @&@ & MathTypePTimes New Roman- 2 [y 2 xy 2  y Ole PIC  #LMETA (CompObj"$fTimes New Roman- 2 +2p 2 2pTimes New Roman- 2 ?1 2  0Symbol- 2 .- 2 .+ 2 - 2 b =Symbol- 2 hmTimes New Roman- 2 [( 2  ) & "System-( 2   FMicrosoft Equation 3.0 DS Equation Equation.39q\$cI,gI y 2 "1+(xy"y 2 )=00Lg {hObjInfo%Equation Native x_981991997(Fi1hi1hOle PIC '*LMETA CompObj)+fObjInfo,g { F .1  @@ &  & MathTypePTimes New Roman- 2 32 2 3 0 Times New Roman- 2 2pTimes New Roman- 2  xy 2 y 2 xSymbol- 2 - 2 + 2  =Symbol- 2 !m & "System- FMicrosoft Equation 3.0 DS Equation Equation.39q@$cI,gI 2xy"y Equation Native \_981992455&;/Fj1hj1hOle PIC .1L2 +x=0L~{th~{f  .1  @ & & MathTypePTimes New Roman|- 2 33 2 2 2 2 2 1META CompObj02 fObjInfo3 Equation Native     "%(*+,-./012358;=>?@ABCDEFGHIKNQSTUVWXYZ[\]^_adfghijklnopqrstuvwxyz{|}~0 Times New Roman|- 2 2p 2 2p 2 Z2pTimes New Roman|- 2 x 2 xy 2 y 2  xy 2 y 2 rySymbol- 2 - 2 - 2  + 2 6- 2 ^+ 2 =Symbol- 2  mTimes New Roman|- 2  ( 2 1) & "System- FMicrosoft Equation 3.0 DS Equation Equation.39qш$cI,gI 3x 2 "2xy"y 2 +(xy"y 2 +2y)=0L{ h{  .1  @`&  & MathTypePTimes New Roman|- 2 Mx 2 fxy 2 xy 2 _9819926166F@"j1h@"j1hOle PIC 58LMETA ( y Times New Roman|- 2 2pTimes New Roman|- 2  2 2 p0Symbol- 2  - 2 + 2  - 2 C =Symbol- 2 TmTimes New Roman|- 2 G( 2 p ) & "System-2p FMicrosoft Equation 3.0 DS Equation Equation.39qT$cI,gI x 2 "xy+(xy"2y)=0Lp {|hCompObj79!fObjInfo:#Equation Native $p_9819930504B=F [3j1h [3j1hOle &PIC <?'LMETA )CompObj>@4fp { = .1  @` &  & MathTypePTimes New Roman|- 2 Mx 2 yxy Times New Roman|- 2 2pTimes New Roman|- 2 L4 2 Y 0Symbol- 2  - 2 c+ 2 ,=Symbol- 2 m & "System- FMicrosoft Equation 3.0 DS Equation Equation.39q<$cI,gI x 2 "4+ObjInfoA6Equation Native 7X_981993211IDF Kj1h Kj1hOle 9xy=0L{ h{  .1  @ & & MathTypePTimes New Roman|- 2 [y 2 xxy 2  y 2 PIC CF:LMETA <HCompObjEGJfObjInfoHLg ySymbol- 2 d- 2 + 2 +- 2 S + 2 =Times New Roman|- 2 2 2  2 2 &0 Times New Roman|- 2 O 2pSymbol- 2 mTimes New Roman|- 2 ( 2 &) & "System-( 2 & FMicrosoft Equation 3.0 DS Equation Equation.39qX$cI,gI y"2+(xy"y 2 +2y)=0Equation Native Mt_981993393KF\j1hej1hOle OPIC JMPLL{t h{  .1  @ & & MathTypePTimes New Roman|- 2 [y 2 4y 2 x 2  y Times New Roman|- 2 +2pTimes New Roman|META RHCompObjLN`fObjInfoObEquation Native cp- 2 Z3 2 3 2  0Symbol- 2 .- 2 >+ 2  - 2  =Symbol- 2 xmTimes New Roman|- 2 k( 2  ) 2 .` & "System-) 2  FMicrosoft Equation 3.0 DS Equation Equation.39qT$cI,gI y 2 "3y+(3x"y)=0. 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