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| Datum: | 1953 |
|---|---|
| 1. Verfasser: | |
| Format: | Artikel |
| Sprache: | Russisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
1953
|
| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/7737 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512723436044288 |
|---|---|
| author | Кованцов, Н. И. Кованцов, Н. И. |
| author_facet | Кованцов, Н. И. Кованцов, Н. И. |
| author_sort | Кованцов, Н. И. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2023-08-15T10:12:02Z |
| description | - |
| first_indexed | 2026-03-24T03:33:19Z |
| format | Article |
| fulltext |
1953 YKPAl1HCKHYf MATEMATI14ECKI1Yf >K:YPHA.n
HHCTHTYT MA TEMA THKH
T. V, Ng 3
--------------------------------- ------------------------
l(pUBble JIHHeHlJaTOrO KOMUJieKCa
H. H. KoaaHLfOB
JlHHCHT.JaTblM KOMriJI CKCOM Ha3bi Ba iOT TpexnapaMeTpHtiCCI<Oe l\1l-!Of006pa-
3 HC np51Mb!X B LfCTblpCXMepHOM JIHHCHT.JaTOM npOCTpaHCTBC. 5JBJI5151Cb pa3;:r:e
JIOM JIHI-ICHT.JaTOH -reOMCTpHH, HaT.JaJIO I<OTOpOi1 6b!Jl0 riOJlO}KCHO B 1868 r.
0JIIOI<KepOM [1], TCOpH51 I<OMriJ!CI<COB OI<a3aJiaCb pa3pa6oTaHI-IOH llCCpaBHCimO
CJia6ee, LICM .JJ:Ba .npyrHX H3 Tpex CCTCCTBCI-IHO Bbl.Ll:e.J!HlOW:HXCH pa 3.JJ:CJIO B
.I!Hl!CHl!3TOH r e oMeTpHH - TCOpH51 J!HHCHl!aTblX riOBepXHOCTCH H TeOpH51 I<OH
rpy::JH U.HH. B c51 JIHTepaTypa o I<OMnJiei<cax orpaHHT.JHBaeTC51 .HHWb OT.LI:CJJb
Hbnm fJ13B3MH B 06W:HX COLIHI-ICHH5IX riO J1HHC[IT.J3TOH fCOl\leTpHH, a Ta!OKC
He60JJbWHMH MeMyapaMH pH;:r:a COBCTCI<HX H 3apy6e/I\Hb!X f C01\1eTpOB
(C. n. cl>m-IHKOB, M. A. AI<HBHC, B. H. KopOBHH, w. Haak, E. Picard,
P. Appel H ;:r:p.).
OepBb!MH riOHHTHHMH. I<OTOpb!e Bb!.JJ:CJJHlOTCH B TeOpHH I<OMnJJei<COB,
HBJJHIOTC51 nOH5ITHH n JI OCKOH I<pHBOH I<OMnJJei<Ca S H KOHyca I<OMn.T!ei<ca.
C .n:pyroi1 cTopoHbJ, nepeceKa51 KoHyc I<OMil.ller<ca nJJOCI<OCTbJO, nepneH;:r:HI<y
.115IpHoi1 I< JJYT.JY H npOXO.LI:5IW:eH qepe3 U.CI-!Tp 8TOro JJy'Ia, Mbl noJJyT.JaeM ;:r:py
ryro nJJOCI<YJO I<pHsyro /. He JJHllleHo HHTepeca TO o6cT05ITCJJhCTBO, liTO 1\!e)[{.n:y
I<pHBH3HaMH I<pHBb! X S H { cym:eCTByeT BeCbMa npOCTOe COOTHOWCHHe, 33BH
CHlll.ee JlHWb OT T3K H33b!B3eMOH I<pHBH3Hbl I<OMnJJei<Ca, 'ITO 5I H nOI<33 b!BalO
B H3CT05IW:eH pa6oTe .
C.Tie;:r:yeT 3aMeTHTb, 'ITO I<pHBH3Ha nJIOCI<OH I<pHBOH S 6hiJJa Hai1;:r:e!-la eme
faai<OM [2], nOJJb30BaBWHMCH MCTO.JJ:OM ;:r:yaJJbl-lbiX Bei<TOj)OB, O.LI:Hai<O fa ai<
OCTaBJJHE:T H3H.JJ:CHI-IY!O HM cpopMyJJy KpHBH 3Hbl nO'ITH 6e3 npHMeHel-IHH.
B HacToHmei1 pa60Te H naxo:>I<y 3TY cpopMyJJy, I<aK H p5I.JJ: ;:r:pymx, noJJh-
3YHCh MeTo;:r:oM BHCWHHX cpopM Kap-raHa. OpH 3TOM ocHOBHOe BHHMaHHe
y;:r:eJJHeTCH ynoMHHYTOMY BbiUJe COOTHOWeHH!O Me}I<.JJ:Y KpHBH3 H3MH I<pHBb!X f
H s (cpopMyJia (III)).
B Tpe-rbeM nyr-IKTe pa6oTbi onHCbiBaeTCH npocTpaHCTBeHHaH KpHBa51, .LI:JJH
I<OTOpOH cymeCTBYCT COnpHK3CaiOW:a5ICH JJHHCH43Ta51 nOBCpXI-IOCTb BTOj)OfO
nop5I;:r:Ka, npHT.IeM I<03cpcpHU.HCHTaMH ypaBI-ICHH51 nOBCpXHOCTH HBJ15IIOTCH
HHBapHaHTbl, npHHa;:r:JJe/K3W:HC OI<peCTHOCTH BTOpOrO nop5I.JJ:I<a. 3Ta nOBepx
HOCTb o6JJa;:r:aeT TCM CBOHCTBOM, 'ITO I<aCaTeJJbHaH nJJOCKOCTb I< HCH B Ka}!{
.LI:OH TO'IKe o6pa3y!OW:eH CHMMCTpH'IHa OTI-IOCHTeJJbHO rJJamiOH HOpMaJJH JJyT.Ja
C nJJOCKOCTb!O nJJOCKOH I<pHBOH S.
MeTo;:r:oM BHemHHX cpopM KapTaHa npH npo;:r:OJJ}KCHHHX CHCTeM ypasHe
HHH Ocpacpcpa n OCJIC.LI:OBaTeJJbHO BBO.JJ:HTCH B paCCMOTpei-IHC HCI<OTOpb!C HHBa
pHaHTHb!C cpyHKU.Hl-1 JJyT.Ja, .JJ:JJH KOTOj)blX 33TCM CJJe;:r:yeT OTbiCIUIBaTb reOMCTpH-
312
'leCKOe HCTOJlKOB3HHe. B H3CTOHW,ell pa60Te, I10Jlb3YHCb ITOHHT!UIMH, CB5133H
HbiMH C TpeMH BBe,LI,eHI-IbiMH KpHBbiMH, 5I H3XO)KY reoMeTpH'IeCKHH CMbiCJl
HI-IB3pH3HTOB, rrpm·I3,LI,Jle)l<all.I,HX OKpecniOCTHM rrepBOfO H BTOpOrO I1 0pH,LI,K3.
T1 o ,LI, B 11 )K H o li p err e p. B MeTpH'IecKoli TeopHH KOMrrJieKCOB c K3)K
.LJ:biM JIY'IOM KOMIT.TJeKCa CB513biBaeTCH onpe,ll.eJieHHblll HOpM3JlbHblll TpexrpaH
HHK, p e6pa KOTOpOrO COBI13)l,aJOT COOTBeTCTBeHHO C JIY'IOM KOMITJleKCa (/;),
rJI3BHOi1 HOpM3JlbiO (It) H 6HHOpMaJibiO (~) KOMITJleKca. BepWHHa HOp
J\13JlbHOf0 TpexrpaHHHKa A conrra,LI,aeT c u.eHTpOM Jiy'la [2, 3, 4].
ECJiu HHrpHHHTe3 HMaJibHbie rrep eMew.emm HOpMaJi bHoro TpexrpaHHHKa
onpe,LI,eJIHTb paneHCTBaMu
dA = wJ1
dll = (Oikfk
(wik = - Wki• i, I~ = 1, 2, 3),
TO rpOpMbl w 2 H w 31 6y.LJ:yT CB5133llbl COOTtiOWeHHeM
r..a:e a - KpHBH 3 Ha KOMITJieKCa ([4], CTp. 417).
(1)
(2)
BHe wHee ..a:urprpe peHu.uponarme pane HCTBa (2) H ITOCJle,LI,yiOmee pa3Bep
TbiBmme no JieMMe KapTaHa rrpHBO.LJ:llT K coOTHO
meHHH~I:
W12 =x1W1 + X2W31 + XaC•>a2,
da=x2w1 +x.1 ro31 +x5w~2· (3)
w3=(ax1-x1)w1 +(ax 2-x6)Wa1 +(ax3-x6)ws2.
r..a:e X1, X 2, ... , X6 - HeKOTOpbre rpyi·IKU.HH Jiy'la.
0KpeCTHOCTb Jiy'la, orrpe..a:eJIHCMYIO KpHBH3HOkl a,
6y..a:eM H33biB3Tb OKpecnioCTbiO rrepnoro rropH,LI,Ka,
OKpeCTHOCTb, onpe..a:eJJHeMyiO HfiB3pH3HT3MH a,
X 1, X 2, X 6 - OKpeCTI·IOCTbiO BTOporo I10-
pH,LI,K3 [2]. Bee ,LI,aJibHeiiwne nccJI.e..a:onaHHH Mbi
6y,LI,eM orpaHH'IHI33Tb OKpeCTHOCTbiO BTOpOrO
rropH,LI,Ka.
r,
PHc. I.
1. K p n B a H s. lbne cnro, 'ITO B Ka)I<.LJ:OI':\: rrJioCKOCTH pacrroJiaraeTC5I
HeiWTOpaH KpH133fl, OrH63CM351 JIY'I3MH KOMITJieKCa (B CJiy'lae JlHHeHHOfO
KOMrrJieKca ara KpHBaH BbipO)K.LJ:aeTCH B TO'IKY). B qacTHOCTH, rry'IKY ITJIOCKO
cTei-i C OCb!O Ha Jly'le IWMI1JlCKCa COOTBCTCTByeT O)l,HOI1apaMeTpH'IeCKOC ce
Meii:cTBO rrJioCKHX r<pHBbiX. Hali..a:eM KpHBH3Hbi 3THX KpHBbiX.
flpOH3BOJlbH3H TO'!Ka KpHBOH S, OrH6aeMOH Jiy'laMH KOMITJieKCa, HMeeT
pa..a:uyc-BCKTOp (puc. 1) r = A+ tl;.
Ol!eBH..a:Ho, dr i 11.3, HJIH n cHJIY ypanHeHHli ( 1)
0TCIO,LI,a H3XO)l,HM
w1 +twa! = 0,
-W2 + twa2 = 0.
dr = (wa + dt)fs.
(4)
313
flOCKOJibKY S - ITJIOCK35I KpHB35I, TO )J,OJI)KHO Hl\'leTb MeCTO paBeHCTBO
(dr, d 2r, d3r) = o.
llHcpcpepeHU.HpyH paBeHCTBO ( * ), noJiytiHM
d:r= d ( w8 +dt) fs +( W8 + dt) ( Ws 1"4 + Ws2l:;),
d 5r= d 2 (w3 +dt) Is +2d (w5 +dt) (w81 7; +w5l~) +
+ ( w, +dt) [ dw3} 1 + dws:l2 +£!1st (w12l2 + W13~) + Ws•dW21ft + co2sfs)].
3JJ.eCb dwat, dwa2 H T. JJ. . - JJ.HcpcpepeHU.HaJibi n¢a¢cpoBbiX ¢opM, 6epymHeCSI
B)J.OJib KpHBOH S. 0 0)J.CT3BJI515I 3HatieHHH )J.HcpcpepeHU.H3JIOB B paBCHCTBO (**),
HJIH
w51 dro32 - w 32dw31 + £uu ( wJ1 + w~2) = 0.
TaK KaK npoeKU.HH seKTopa d2r Ha nJiocKocTb 71fz eCTb
(w3 +dt) (1<~3]1 +w32~),
TO, HCI10Jib3Y5I paBeHCTBa (2) H (4), H3H)J.eM
ws, ws! a tg e = - =a - = - - .
W 3t W2 I
TaK KaK 73 - e)J,HHHtiHbrj;'r seKTOp, TO H3 paseHCTBa (*) 3 3KJJI04aeM, 'ITO
ds = w 3 + dt.
J<pHBH3Ha k KpHBOH S orrpe)J,eJJHTCH, CJie)J.OBaTeJibHO, H3 paB.eHCTBa
(5)
(6)
k =I dl;, I= I Ws1l1 +w82h I= j.wj1 +w<l2 (a)
ds w 3 +dt 1o3 +dt
( 3 H3K li'IOJJ.YJIH y CYMMbl w 3 + dt Mbl OT6paCbiB3eM paJJ.H npOCTOTbl )J.3.'lb
HeHlllHX 0603H3tieHHH). 0epe nHCb!B35I BTOpoe paBe HCTBO H3 CHCTeMbl ( 4)
B BH)J,e
awa1 + t.wa2 = 0 (b)
H )J.HcpcpepeH U.HP YH ero o6b!tiHb!M rryTeM, ITOJIY4Hi\1
dar.o 31 +adlo51 + dl1o3 2 + td1o5 2 =0.
3aMeH5I5I 3)J.eCb t ero 3HatieHHeM, !IaH)J,CHHbJM H3 paBeHCTBa (b), onpe)J.eJIHM
JJ.Hcpcpep eHU.H3JI dt:
dt = a (wstdws2 -wa2~Wa1)-ul:ll~<~s2 da.
(1)32
flO)J.CT3BJI5IeM CIO)J,a 3H34eHHe pa 3HOCTH WatdW32- W 32 d"1 31, KOTOpoe H3H)J,eM
H3 (5):
dt = - awl 2 ( wJt + I•IJ;) + wSt con2 da .
I.V~2
flO)J.CT3BJ15I5I, H3KOHe U., 3TO 3HatieiiHC B paBe iiCTBO (a), ITOJIY'IH~l 3 H3tiCHHe
KpHBH3Hbl
(a' )
3 14
Jib paBeHCTB ( 4), 3aMeHHH Wz ero 3HatieHHeM, Haif,n:eM
t t2
£tlal =-a W3~ ' (A)l = a Wa2• (4')
BHOCH npaBbie tiaCTH B paBeHCTBa (3), noJiytiHM
(7)
CAJa= r(axl-x3)~ -(ax2-x5)£ +(axs-xG>] CAJ32.
f10,LI.CTaBJIHH, HaKOHe~, npaBbJe 'la.cTH cpopMyJI (7) H ( 4') B cpopMyJiy (a'),
IlOJIY'lHM OKOH'laTeJibHOe BbipmKei-me ,LI.JIH KpHBH3Hbl KpHBOH S B TO'lKe M:
ava~+ t2
k= ----~~-=--~~~--~~~-~----- x1 t1 +2x2f5 - (2ax5 +x,) t 2+2ax5t-a1x6 ·
(I)
ECJIH KOMnJieKC JIHHeHHbiH, TO Ka:tK,n:aH KpHBaH S BblpOM,n:aeTCH
B TO'lKy, a IlOTOMY ee KpHBH3Ha CTaHOBHTCH paBHOH 6eCKOHe'lHOCTH. f1oJia
raH no3TOMY B cpopMyJie (I) 3HaMeHaTeJib TOM,LI.eCTBeHHO paBHbiM Hymo, no
JIY'IHM ycJIOB'liH, xapaKTepH3YJ-Olli,He JIHHeHHbiH KOMnJieKC,
(8)
Mbi HCI{JIIOtiaeM H3 paccl\WTpeHHH cne~HaJibHbie KOMnJieKCbi a = 0.
<I>opMyJiy (I) MO:tKHO paccMaTpHBaTb KaK aHaJior cpopMyJihi KpHBH3Hhi
HOpMaJibHOrO Ce'lei-IHH IlOBepXHOCTH. 3KCTpeMaJibHbJe 3Ha'leHHH KpHBH3Hbl
TOr,n:a CJie,LJ.yeT paCCMaTpHBaTb KaK rJiaBHbJe KpHBH3Hbl KOMnJieKCa . .[lHcpcpe
pei:I~HpyH (I) no t H npHpammBaH Hymo pe3yJihTaT ,LI.HcpcpepeH~HposmmH,
JlOJIY'lHM 3HatieHHH f, COOTBeTCTByJ-Olli,He ,rJiaBHbiM" KpHBH3HaM,
3xd5 - 4xzt{ + (4xia2 + 2ax3 + x ,1)t3 -
(9)
CJie,LI.oBaTeJihHO, B Ka:»<,n:oM Jiy'le KOMnJieKca HMeeTCH nHTh ,rJiaBHbiX ceqe
Hui'l", T. e. IlHTb JIHHHH S, KpHBH3Ha KOTOpbiX .D:OCTHraeT 3KCTpeMaJibHbiX
3HatieHHH.
EcJIH KpHBH3Ha k ,n:JIH Ka:»<,LI.oro 3HatieHH51 t ocraeTCH o,LI.Hoi'! 11 Tow Me,
TO paBeHCTBO (9) 6y,LI.eT y,LI.oBJieTBOpHThCH JIJ-06biM 3Ha'leHHeM t. Ho B ra
KOM CJiytiae Mbl 6y,LI.eM HMeTb
X1 = 0, Xz = 0, 2axa + X.1 = 0, XG = 0, X5 = 0.
T. e. cHoBa paaeHCTBa (8). CJie,LI.oBaTeJihHO, e,LI.HHCTBeHHbiM KOMnJieKCOM, ace
JIY'IH KOTOpOrO - .,OM6HJIH'leCKHe", 51BJI5leTC51 JIHJ-JeHHbiH KOMilJieKC, y KOTO
poro KpHBH3 Hbl BCeX HOpMaJihHbiX ,CetieHHli" CTaHOBHTCH · paBHbiMH 6eCKO
HetiHOCTH.
2. K p H B a 5I l . I13BeCTHo, 'ITO qepe3 I<aM,LI.yio TO'II<Y npocTpaHCTBa
npoxo,LI.HT KoHyc npHMhiX KOMnJieKca. B qacTHOCTH, npHMOJIHHeiiHoMy pH.D.Y
TO'leK M, paCilOJIO:tKeHHbiX Ha Jiy'le KOM!lJieKCa, COOTBeTCTByeT O.D:HOnapaMe
TpHtieCKOe CCMeHCTBO TaKHX KOHYCOB. J<aM,L(biH H3 3THX KOHYCOB nepeceKaeT
315
TIJIOCKOCTb ~of;.o , COOTBCTCTBYIOW.YJO )l.3HHOMY JIY4Y (13°), no HCKOTOpoii n JIO
CKOI1: Kpwsoii: l. Haii:JJ.eM KpHBH3HY k1 ·<noii KpHBOI1.
I1poH3BOJibHa51 T04I<a Ha Jiyqe KOMnJi e Kca AM onpeJJ.CJI5ICTC51 pa,awycoM
BCKTopoM (pHC. 2)
r=X + ;.t; .
.[{JIH T04KH P nepece4eHH51 Jiy4a AM c nJIOCKOCTbiO l 1°l 2° cnpase)l.JIHBO
paBCHCTBO
r-·- -- " OTKyJJ.a
L ~-- _____...! ..
CJie)l.oBaTCJi bHO,
HMCCT BH,U
paJJ.HyC-BCKTOp p
r ;
r=A- (A -= ~o) I~ fs . (c)
13 /:
KpHBH3HY KpHsoii: l Hai"IJJ.CM no cpopMy: Ie
PHC. 2. kl =
l[d;:; dirJ 1 (ct )
jdr j3
- - - dA r - d2AF
d1r= d' A-2dl3 - - • . - Is - -- " + · · ·
I~ I: 13 1~
(TO'IKa MH o603Ha'leHbi CJiaraeMbi e, HC'le3 aJOmHe npH nepexo)l.e K npeJJ.e.TJy) .
I1epexo,UH K npe,aeJiy npH P -~ Ao, noJiy'IHM:
d"""i= dA-l 3 (dA l 3) = W 1ft + w2ls + ru3ls-Wafs= ruJ1 + w2I2,
+1111 (w12h + wuls)+ W 2 (w21.J;. +w23J~) + £rJ3 (wsJl + Wa2I2)
-2(wull +w32.1;) w3-l3 (dw8 +w1w13 + w2w13) =
= (dw1 +wzw~1 -w8w81)l1 +(dw2 +w1w12 -w8w32 )I 2 •
(pa,UH npOCTOTbl 0603H3'leHH5! 1\lbl O'f6pOCHJIH HH,UCKC 0).
CJieJJ.OBaTeJihHO, paseHCTBO (d) 6yJJ.eT HMCTb BHJJ. :
_ [ llo1(dw2 +w1w12 - w3w32)-rui(dw1 +w2 w21 -w3wSl)jl (d ')
kl - ( • + ·)"'- .
W 1 lrJ 2 ' . P=Ao
To'IKa JJ1 HBJIHCTCH .,p e6poM B03BpaTa" Konyc.a. JlJili ee OTbiCKaHHH
npHMeHHM H3BCCTHbiif npHeM H3XO:tK)l.CHHH p e6pa B03BpaTa pa3BepTbiB3JO
J.I.I:CHC5! TIOBepxHOCTH ([5], CTp. 237). f1yCTb HanpaBJI5IIOI..IJ.CH JIHHCH'l3TOH no
BepXHOCTH HBJIHeTCH JIHHHH, onHCbiBaeMaH u e HTpOM Jiy'la A , o6pa3yiOI..IJ.aH
nosepxHOCTH napaJIJieJihHa BeKTopy 73. EcJIH nosepx HOCTb HBJIHCTCH p a3Bep-
316
TblB3JOW:eikH, TO ~OJJ)KHO HMeTb MeCTO paBeHCTBO (/3 , d/3 , dA) = 0, OTKy~a
dl3 = al3 + fJdA, r,Ue a, fJ - HeKOTOpble K03cpQJHU.HeHTbl, HJIH
0TCIO~a
tus,II + Ws2l2 = a.la + fJ (w1II +u)2J; + Wsls).
W31 = (3W I,
W32 = f3w2.
(e)·
Pa~Hyc-BeKTOp ToqKH pe6pa B03BpaTa pa3sepTb!BaJOIU.eilcH nosepxuocTH
onpe~eJJHTCH B cooTBeTCTBHH c yi<a3aHHblM npneMOM cpopMyJioH:
- - 1 - - W1 - -
R=A- - I 3 = A - - I'J.
fJ wa,
Y paCCM3TpHBaeMOfO H3MH KOHyca pa~uyc-BeKTOp HOCTOHHeH H paBeH
R = Ao+tl3° (r~e t = A 0M).
CJJe)l.oBaTeJJbHO,
li,HcpcpepeHU.HpyH 3TO paBeHCTBO, Hai-i)l.el\1
dA- Wst dwl-wl dwelT-~ dfs=O
wJ1 3 W31 '
HJIH
0TCIO)l.a
(f)·
(g)
flepsoe paBeHCTBO, Me)K)l.Y npoqH~ I , CJie)l.yeT T810Ke 113 paBeHCTB (e)
BTopoe paseHCTBO nepenmueM B Bll)l.e
w 3wJ1- ( w 31 dw1 - w 1 dw3 1) = 0.
l1cnOJib3YH ero, uy)l.eM HMeTb
w1 drv2 -w2 dw1 = w1 (da W81 +a dw3,)-aW51 dwt =
=w1£<>:n da+a(wt dw31 -W31 dro1)=W1W31 da-awaw§,.
YtmTb!BaH noJJyqeHHble paseHCTBa, Mbi npHBe)l.eM cj:>opMyJJy (d') K BH)l.y:
- [I WICIJ:JI da- aw:~w5, + wl2 (w~ +w!) ll
kl - ( 2 + ")·"·
(t)l W2 2 P=.Ao
t= [- ~].
wsl_ o
0TCJO)l.a n H3 nepsoro paBeHCTBa (g) Haxo~HM
[2 t
Wl = a f<ls2 o W3l =- a £0st ( 1 0)'
317
(.ll.JIH HalJaJibHoro Jiyqa i~0) . ITo.ncraBJIHH 3TH 3HalJeHHH B paneHCTRa (3) ,
Mbl npHBe.ll.~M HX K BHJI.)' (7). 0r6paCb!BaH HHJI.eKC 0 H 3HaK MO.ll.)'JIH, Mbl
llOJI)'lJHM OKOHlJaTeJibHOe Bb!pa)!\e iiHe JI.Jl5l KpHBH3Hbl fl 1:
-x1t:l.+2x2f3-(x4 +2ax3) t 2 + 2ax5t-a2x6
k =
1 t(t2 +a~)~ ·
(II)
CpaBHHBa5! ¢opM)'Jibi (I) H (II), 3 aKJIIOlJaeM, lJTO Me)KJI.Y KpHBH3HaMH n JIO
·{:KHX KpHBbiX S H { cymeCTB)'eT BeCbMa llpOCTOe COOTHOllieHHe, 3aBHCHIUee
.J!Hllib OT KpHBH3Hbl KOMllJieKCa a H OT HOJIO)KeHHH TOlJKH M Ha Jl)'lJe,
a
(III)
dA=w1 11 +w2 I2 +w8l;;.
:DpoeKll,H5! dA Ha llJIOCKOCTb / 1!2 eCTb
w;ll + w2I2 .
. Jl"1H KOH)'Ca CllpaBeJI.JIHBbl paBeHCTBa ( CM. Bb!llie)
f2
w.l = a w 32o w2 = - tw32·
a
t '
(11)
T .ne 81 - yro.rr, o6pa3yeMbiH KaCaTeJibHOfl llJIOCKOCTbiO KOHyca C BeKTOpOM
I 1. CpaBHHBaH ¢opM)'Jibi (II) H (6), 3aK.moqaeM, lJTO
(J='=OI,
T. e. I<aCaTeJibHaH llJlOCKOCTb KOH)'Ca, HMeiOIUero BeplUHH)' B TOlJKe M, COBITU
. .ll.aeT c rrJIOCKOCTbiO KpHBOii s, KacaromeiicH Jiyqa n Toil )Ke TOlJKe. HopMaJIH
KpHBbiX { H S o6pa3)'IOT, CJie.ll.OBaTeJibHO, 11p5IMOH
yro.TI (pH C. 3) .
.[lo CHX rrop Mbl paCCl\laTpHBaJIH llJIOCKHe
KpHBb!e S H /, CB5!3aHHb!e Me)KJI.)' C060H TeM )'CJIO
BHeM, lJTO BeplUHHa KOH)'C3, Ha KOTOpOM paCITO
JiaraeTCH KpHBa5! /, COBIIa.ll.aeT C TOlJKOH, COOT
BeTCTB)'IOIUeH ITJIOCKOCTH KpHnoi1 s. Ocno6o.n.HMC5!
Terrepb OT 3TOrO )'CJIOBH5! H paCCI\10TpHM llpOH3-
--~-..=o--rn----t----'l BOJibHbie KpHBb!e l H S . fiyCTb BeplUHHa KOH)'Ca,
Ha KOTOpOM Jie)I\HT KpHBa5! {, !IaXOJI.HTC5I B TOlJKe
ii M 1 JI)'lJa, TOlJKOH )Ke, COOTBeTCTB)'IOIUeH ITJIOC
KOCTH KpHBOH S, II)'CTb 6y.neT HeKOTopa5! T OlJKa JJ.f.
Puc. 3. ITycTb t1 H t - paccTOHHH5! TOlJeK M1 H _M 02_
u.ei-ITpa .rrytJa. B 3TOM c .rryqae HOpMaJIH ·J' H n
I<pHBbiX S H [ MOryT o6pa30Bb!BaTb Me)!{JI.)' C060H rrpOH3BOJ!bH"b!H yrOJ1.
- - 7(
ECJIH nopMaJIH 11 H n napaJIJieJibHbi, TO 01 = 0 + 2 , r)l.e rrorrpe)KHeMy o -
yroJI Me)!<JI.Y ITJIOCKOCTbiO KpHBOH s H rJiaBHOi-i Hop~Ia.'IbiO K011lllJi eKca I1, fl1 -
.'3 18
yroJI Me)!{,LJ.y KaCaTeJibHOH ITJIOCKOCThiO KOHyca ( C sepiiH!HOll S TOl.JKe Ml)
11 seKTopoM 71. CJie,LJ.osareJihHO,
a t - - = tg 01 = - ctg o = - .
t1 a
(12)
3To 03HaqaeT, l.JTO TOl.JKH M 11 M 1 pacrroJiaraiOTCH no pa3Hhie cTopoHbi OT
ueHTpa. ECJin t =a, TO t1 =-a, T. e. s 3TOM CJiyqae TOl.JKH M 11 M 1 oKa-
3biBaJOTCH CHMMCTpH4HbiMH OTHOCHTCJibHO uei-npa. B 3TOM CJiyqae tg 0 =- 1,
CJIC,LJ.OSaTCJibiiO, HOpMaJIH KpHSblX S H [ OKa 3 biBaiOTCH 6HCCeKTpHCaMH STO
poro KOOp,LI.HHaTHOfO yrJia BCKTOpOS J 1 H l 2• PaseHCTBO ( 12) CHMMeTpH4HO
OTHOCHTCJibHO f H f1, a 3TO 03Hal.JaCT, 'ITO CCJIH TOl.JKa M rrepeMeCTHTCH
B TOtJKY !!!1, ~o TOl.JKa M1 rrepeMeCTHTCH s T04KY M. EcJIH )Ke t = -a, TO
BeKTOpbl Y ll n OKa3biBaiOTCH 6HCCCKTpHCaMH rrepSOfO KOOp,LI.HHaTHOfO yrJia.
EcJIH S 3 51Tb .use rrpoH3SOJibHbie TOl.JKH M 11 M 1 Ha OJLHOM 11 TOM )!{e
paCCTOHHHH OT l.I,CHTpa, TO MO)KHO ITOKa3aTb, l.JTO SCKTOpbl ·v H n CHMMe
TpH4Hbl OTHOCHTCJibHO 6HCCeKTpHC KOOp,UHHaTHhiX yrJIOB.
YJ3 cpopMyJibl (I) MQ)KHO 3aKJIIOl.JHTb, l.JTO KpHSH3H3 ITJIOCKOH KpHSOH S,
1
·Kacmorueik5I Jiyqa KOMrrJieKca s ero ue1-npe (t = 0), pasHa - . CJie.uosa-
xc
TCJibHO, mmapuaHT x6 - pa.u11yc KpHSH3Hbi 3Toli KpHsoH:. Mbi HaWJIH, TaKHM
o6pa30M, fCOMeTpH4eCKOe HCTOJIKOSaHHe HHSapHaHTa X6. l{JIH 6eCKOHCl.JHO
y,LJ.aJieHHOll TOl.JKH Jiyqa KpHSH3Ha KpHSOH S CT3HOSHTCH paSHOll HYJIIO.
l13 cpopMyJihi (I I) 3 aKJIIoqaeM, l.JTO KpHSH3Ha KpHsoli l .D.JIH 6ecKoHetJHO
)",LJ.aJieHHOi1 TOl.JKH Jiyl.Ja (T. e. ,U.TIH UHJIHH,Upa KOMITJieKCa) CTaHOSHTCH paB
f!Oll X1 . TaKoB reoMe TpnqecKui'r CMbiCJI mmapuaHTa x 1.
LfTO )K C KacaeTCH l\pHSH3Hbl KOMIT.1 CKC a a, TO ee MO)KHO HCT0.1KOSaTb
( CM. paBCHCTBO ( 12)) KaK cpe,UHee nponopUHOHaJibHOe Me)KJLY paCCTOHHH5IMH
O T uetiTpa TOl.JeK Jlf H M 1, S KOTOpblX rrapaJIJieJibHbl HOpM3JIH ITJIOCKHX Kpll
Bb!X S ll /.
Ha Kouyce KoMnJieKca pacnoJiaraeTC51 JIHHHH u e tiTpon. EcJiu 3TY JIHHHIO
C11pOCKTHpOSaTb Ha ITJIOCKOCTb 71°J2° napaJIJICJibHO seKTOpy J3°, TO KpHSH3H3
npoeKI.I.HH B TO'IKe A 0 6y.ueT pasHa KpHBH3HC KpHsoH: l s TOH )Ke TOl.JKe.
3. Bo3bMeM TPH .uocTaT04HO 6.nH3KHx J1yqa KOMITJieKca s.uoJib HeKoTo
poli KpHBOi;I. 3TH JIYtiH onpe,TJ:eJI5JT .IIH!·ICi:itJaTyiO ITOBepXHOCTb BTOporo ITO
pH,UKa, conpHKacmoruyioCH c KOMnJieKCOM. Bhi6e peM KpHBYIO TaK, l.JT06hi
K03cpcpHUHCHTbl ypaBHCHHH ITOBepXHOCTH OTHOCHTCJibHO HOpMaJibHOfO Tpex
rpaHHHKa ri[JHHa,UJl C)K3JI H OKpCCTiiOCTH BTOporo ITOp5I,LI.K3 Ji y tJa KOMITJICKCa,
T. e. onpe,UCJI5!JIHCb HIIB3pH3HTaMH a, X1, X 2, ... , Xo·
YpaBH CHHC ITOB CpXHOCTH BTOpOrO nOp5I,UKa HMCCT SH,U:
C11X2 + Cnlj 2 + C33Z 2 + 2Ct zXY + 2c13XZ + 'lC23ljZ + 2c1oX + 2CzoY --j-
+ 2c3oZ + Coo= 0. (h)
flpOH3BOJ!bH a H TOl.JKa !·Ia Jryqe KOMITJICKCa onpe,UeJ!5!CTC5! pa.UHYC0!\1-BCK
TOpOM A+ tl3. no,UCT3SJI5151 KOOp,LI.HHaTbl '3TOro BCKTopa B ypaBHCHHe (h)
II npHpaBHHBaH HYJIIO K03cpcpHl.I,HCI!Tbl npH f 2 , f H CB060JJ:Hb!H l.JJICH, ITO.TIY'IHM
AA=O, Al; = O, /;T3= 0 (i)
3!9
r,ll.e 4 epe3 AA = 0 o6o3 HalJeHo yp3BHeiuie
c 11 x2 + c 22y2 + o o o + Coo = 0 (x, lj, Z - KOOp,li.HHaTbl BeKTOpa r1),
q ep e3 13~ = 0 - yp3BI-IeHHe
CIIX32 + c 22ijo2 + o o o +Coo= 0 (X3, lj3, Z;l - I<OOp,li.HH3Tbl BCKTOp3 / 3) ,_
4epe3 _AI; = ·0 - yp3BHeHHe
CI IXX3 + Cnljlj3 + C33ZZ3 + CI2{Xlja + ljX3) + o 0 o +Coo = 0.
Jl.Hc}lc}lepei-IU,HpyH yp3BHeHH51 (i) B,'I.OJ! b HeKOTOpOH Kp!-IBOi'i, TIOJIY1IHM
A((cJ/t+w2:h+w3l s) = 0,
A (w81lr + Ws 212) + ( wJ1 + w2h + (u;Iu) lo = 0,
J;, (w:J 1~ +w;;2~) = 0,
HJIH (y4HTb!B351 p3Be HCTBa (i))
W 1 A l 1 +aw3,Al2 = 0,
uJ:J 1Al; +ws2A/; +wJtfu +ctW:n~I~ =0,
Wstltll + (tJs21tfs = Oo
(j)·
3TH p3BeHCTB3 HaM He06XO,li.HMO npO,li.H QlQle p e HlJ.HpOBaTb B,li.OJlb TOi\ }Ke
CClMOi-i KpHBOil. Jl.JIH TOrO 4T06bi HOBOe ,li.HQlc}lepCHIJ.HpOBatiHe He Bb!Be.'IO 33
OKp e CTIIOCTb BTOpOrO TIOp51,li.K3, H3,li.O 1IT06bl B p3BeHCTB3X (j) He CO,li.Cp)!<(I
JIOCb QlOpM. JierKO, O,li.H3KO, BHJJ.CTb, 'ITO 3TH QlOp Mbl '110l'YT HClJC3 11 YTb J!Hill b
B Tpex c .rly43Hx:
I) wai = wa2 = 0,
2) WI = Wa i = 0,
3) WI = W32 = 0.
J.1CCJ! e,ll.yeli'I KCl}K,ll.bii{ H3 3THX CJIY43CB.
I) KpnBaH, onpe,ll.eJIHel\1351 paBe HCTBmiH .w:n = W32 = 0, ecTb He 1ITO
HHoe, KaK JIHHHH IJ.CIITpon, pacnoJimi<eHH3 51 H3 IJ.HJIHH,ll.pe KOMnJieKca. CJi e,ll.o
BaTeJibHO, nee JIY'lH KO!ttnJieKC3 B,li.O.Jib 3Toi1 JIHHHH n3p3JIJieJibHbl. ConpHI<a
caiOill.eikH JIHHeil'laTOH TIOBepXHOCTbiO 6y ,ll.eT HeKOTOp351 IJ.H.'lHH ,ll.pH4 e CK35l
nonepxHOCTb BToporo nopH,li.K3. B ,li.ClHHo~r c.riy'Iae paneHCTBa (j) noc.r1e co
Kpaw.e HHi1 npHMYT BH,ll.:
(j ' ) ·
Jl.H c}lc}l ep e HIJ.HpyH CW.C pa3 3TH p3Be HCTB3 B,li.OJ!b ,li.3HHO I~i KpHBOii, llOJl Y'lHI\1
( Wti 1 + wsfs) / 1 + Aw12I" = 0, ro12I2l8 = 0;
OTCIO,ll.a
WI7:II + WI2.Ai-; = 0, IJ3 = 0.
BHOCH p3BCHCTB3 Wai = wa2 = 0 B QlOPMYJibi (3), !lOJlY'lHM, B qacTHOCTH ,
WI2 = XIWI.
0 0,li.CT3BJI 5l5l 3TO 3 H34CI-IHe WI2 B rr e pBOe H3 npC,ll.bi,li.YliJ.HX p3BeHCTB H COKp3-
IU35l 1!3 WI, npHBC,li.CM 3TH paBCHCTBa K BH,li.Y
(k) ·
320
EcJJH 3a KoopJJ.HHaTHbie ocH npHH51Tb pe6pa noJJ.BIDKHoro HopMa.rJbHoro
"TpexrpaHJ-JHKa, TO paBCHCTBa (i), (j')) lJ (k) npHMYT BH,ll.:
Coo = 0, Coa = 0, C33 = 0, Co 1 = 0, C 1 ~ = 0, C11 + X1Coz = 0, C23 = 0.
YpaBHCime conpHKacmomerocH U.H.'II1HJJ.pa noJIYliHM, BI-ICC51 3TH Ko3<1Jqm
U.HCHTbl B ypaBHCHHC (h),
- x1X2 + c22 Y 2 + 2c 12XY + 2Y = 0. ( 13)
3TO - lJ.CJ1351 CB513K3 U.HJIHHJJ.pOB C napaMCTp3MH C22 I1 C!2·
2) ,U.TJ51 KpHBOH, onpCJJ.CJ151C~lOH paBCHCTB3MH -WI = WJI = 0, ypaBHCHHK
(j) I1pH1-1HM310T BHJ].:
0 = 0, Alz = 0, l zl ., = 0. (l)
BJJ.O.'Ib 9Toi1 KpHBOi-i dA = w 3l 3 , CJICJJ.OB<HCJJbHO, 3Ta KpHBa51 om6aeTC51 JIY
<raMH KOMnJJeKca. Mbl nmeM, TaKHM o6pa3oM, conpHKacaJOLUYIOCH nosepx
HOCTb BTOporo nOp51JJ.K3, npOXOJl.51lli.YIO 4Cpe3 TpH 6CCKOHCl!HO 6JJH3KHX KaCa
TCJib!-lbiX K npoCTpaHCTBCHI-IOi-i KpHsoii. OtleBHJJ.HO, T3I<351 nosepxHOCTb 6yJJ.eT
IIC e,ll.H HCTBeJUIOH . ,Uncp<lJepeHU.HpyH p3BCHCTB3 (!) B,ll.OJ!b )].31-IIIOH KpHBOH,
IlO.J I YliH~l
OTCI0Jl.3
(m)
ITpHHHMaH 33 KOOpJJ.HJ-raTJ-IbiH TpexrpaHHHK nOJJ.BH)KI-IOi-i 1-IOpMaJJbHbiH
TpexrpaHHHK KOMilJJCKCa, Mbl 113 paBCHCTB (i ), {J) H (m) IlOJJYliHM
Coo = 0, Co3 = 0, c .1 3 = 0, Coz = 0, C23 = 0, Co 1 = 0, - XaC13 + C22 = 0.
CJJeJJ.OB3Te.ribHO, ypaBHCHHe conpHKacaiomeikH nose pxHOCTH nprrMeT BHJJ.:
c 11X 2 + x 3Y 2 + 2c1 2X Y + 2XZ = 0. (14)
'3TO TaJOKC lJ.CJI351 CB513K3 IlOBCpXHOCTCi-i C I13paMCTp3MH C11 H Ci2·
3) Jlmi KpHBOH, onpCJJ.C.,151CMOH paBC!-ICTB3MH W I = W 32 = 0 ypaBHCHH51
(j) npm1HM310T BH,ll.:
(n)
Jln<lJ<iJep CH U.HPY51 3TH p3BCHCTB3 ClUC p3 3 B,ll.OJJb TOi1 )I<C C3MOI1 KpHBOH,
.IlOJJYl!Hl\1
'CTB3
(lo2I2 +cu5Is)f~ + Acu2 1~ =0,
(cu27;. +cu3fs)/; +A (w12.4 +cu13l, )+ dai;J;; +aro2 1l;E +a£v31 1211 =0, (p)
(t•lt2/~ +totls) la +rl)s1/1/1 =0 .
.nerKO BHJl.CTb, liTO Jl.Jl51 p3CCM3TpHB3CMOH KpHBOH cnpaBC,ll..TJHBbl p3BCH-
BI-IOCH 3TH 3H3l!CHH51 B paBCHCTB3 (p) rr npoH3BOJJ.H coKp3u~eHHH B CHJJY
npe,ll.biJl.YlUHX COOTHOWCHHH, IlOJJYliHM
aJ;I----; + (axz- x 5 )lzl3- x2Al1 = 0,
aY,Tz + x4l2T3 + a~fz = 0,
x2lzla + T1~ = o.
(q)
321
npH COBDa,ll.eHHH KOOp,li.HHaHIOfO TpexrpaHHHKa C DO,li.BH)KHbiM H3 YCJIO
BHI{ (i), (n) 11 (q) 6y,11.eM HMeTb COOTHOUleHHH ,li.JIH K03cpcpHU,HeHTOB ypaB
IIeHHH CODpHKaCaJOlll,eikH DOBCpXHOCTH BTOpOrO DOpH,li.Ka:
Coo = 0, Co3 = 0, C33 = 0, Co2 = 0, Cot + ac23 = 0, C13 = 0,
ac22 + (ax2- xs) C23- X2Co1 = 0, 2ac1 2 + X4C23 = 0, x2c23 + c11 = 0.
C'!e,ll.oBaTeJJhHO, ypaBHeHHe HCKOMOH noBepxHOCTH 6y.neT HMeTh BH,ll.:
ax2X 2 + (2ax2 - x5 ) P + x.1XY- 2aYZ + 2a2X = 0. (15)
l(acaTeJJbl-!aH D.TJOCKOCTb 3TOi1 DOBepXHOCTH B TO'!Ke M (0, 0, f) JJy'la HMeeT
ypaBHeHHe
a - tY = 0.
0TCIO,ll.a 3aKJIIO'laeM, 'ITO 31 a DJIOCKOCTb o6pa3yeT C BeKTOpOM ~~
( OCbiO X) yroJJ 81, onpe,ll.eJJHeMb!H paBe HCTBOM
a
·g !7: = t.
CpaBHHBaH .:ny ¢opMyJJy c ¢opMyJJ~ (6), 3aKJJIO'la eM, 'ITO ()1 =-e. CJJe
,li.OBaTeJihHO, KaCaTCJib~!aH DJIOCKOCTb ~~·pXHOCTH ( 15) B TO'IKC M CHM
MeTpii'IHa DJIOCKOCTII KpiiBOH S, KaCaiOlll,e HCSI JJy'la B TOH )I<C TO'!Ke, OTHOCH
TeJ1bHO r .riaBHOH HOpMa.TJII KOMDJICKCa.
U e HTP noBepxHOCTII (15) IIMeeT Koop,li.HHaTbi
a
X=- - ·
X!!'
Y=O,
CJJe,ll.oBaTeJJbHO, noBepxHoCTb ( 15) HBJJHeTCH O,li.HonoJJOCTHbiM rHne p6oJJOH
,li.OM. 3Ta DOBe pXHOCTb CTaHOBIITCH rHn ep60JJ H'ICCKH M napa60JJOH,li.OM ,li.JIH
I<OMDJICKCOB
X2 = 0.
UenTp noBepxHOCTH JJe)I{HT Ha r JiaBHOH HOpMaJJH KOMnJJeKca, eC-~lH
X 4 = 0. Y TaKHX KOMDJi e KCOB ,li.JIH KpHBOii Wt = w 32 = 0 cnpaBC,li.JiliBO TaK)Ke
paBeHCTBO da = 0, T. e. B,li.OJ!b paCCMaTpHBael\IOll KpHBOH a = COn st.
l(acaTe.TJbHaH D.TIOCKOCTb DOBe pXHOCTH (15) B U,e~ITpe JJy'la COBna,ll.aeT
c nJJoCKOCTbiO 121;. 4 e p e3 u.e HTp npoxo,li.HT ,li.Be o6pa3yiolll,He runep6oJJou.na:
O,li.Ha H3 ~fHX COBDa,ll.aeT C JJY'IOM (l 3 ) (x = y = 0), ,11.pyraH onpe,ll.eJJHeTCH
ypaBI-Iei-IHHMH
x = 0, (2ax2 - xs) Y- 2aZ = 0.
JlerKO BH,ll.eTb, 'ITO
r,ne if! - yroJJ, o6pa3yeMbiH BTOpOH npHMOJIHHeH HOH o6pa3yiOIU,eif C BeKTO
pOM f;. C ,11.pyroH CTOpOHbl, KaCaTeJibHaH K JIHHIIII WI = -W32 = 0 onpe,ll.e
JJHeTCH Be KTOpOM
Ho
322
CJie.D.oBaTeJihHO,
dA = ·W31[aT2 + (ax2- xs)I3].
0TCIO,ll,3 33KJIIOll3eM, liTO
r.D.e 9J - yroJl Me)K,LJ,y KacaTeJJhHOH H 6IIHOpMaJJhJO (I2) KOMrrJJeKca. Kaca
TeJJhtiaH I< KpHBOH ;WI = W32 = 0 COBITa,ll,aeT CO BTOpOH rrpHMOJJHHeHHOH o6pa -
3 YIOW:eif ITOBepXHOCTH ( 15), eCJJH qJ = ·if!. JlerKO BH,ll,eTb, liTO T3KOe COBIT3,ll,e
HHe HMeeT MeCTO J!Hlllb ,li,JIH KOMITJJeKCOB
Xs= 0.
TeM caMhiM Mbi orrpe.D.eJJHJJH reoMeTpHlleCKHH CMbiCJJ HHBapuaHTOB x 2 ,
X4, Xs.
4. KpHBhie WI = ·W3J = 0 raaK H33bJBaeT u.eHTpaJJhHh!MH KpHBh!MH. Bee
KaCaTeJJbHbJe K 3THM KpHBb!M 5JBJJHIOTCH JIYli3MH KOMITJJeKCa, rrpHlleM K3)K,li,3H
TOliK3 K3C31-IIIH rrpe,li,CT3BJJHeT C060H JJ.e!Hp COOTBeTCTBYJOW:ero JJylla.
E.D.HHHliHbiM BeKTopoM KacaTeJihHOH K u.ei-ITpaJJbHOH KpHBoif HBJJHeTCH
BeKTOP 1 3. B.D.OJJh KpHBOH HMeeM paBeHCTBO di3 = W 32lz- CJJe,li,OBaTeJihHO, .
rJJaBHOH 1-10p~13J!hl0 U.e HTpaJJbHOH KpHBOH HBJJfleTCH 6m-IOpM3Jlb KOMITJJeKca.
E.D.HHHliHhiH BeKTOP 6mwpMaJIH u.ei-npaJJhHOH Kpimoif 6y.D.eT rro3TOMy orrpe
,ll,eJJHTbCH paBeHCTBOM
T. e . 3TOT BeKTOp rrpOTHBOITOJJO:tl<eH e,li,HHHliHOMY BeKTOpy f 1 rJJaBHOH HOpMaJJH
KOMITJJeKca. KpylleHue u.eHTpaJJbHOH KpHBoi:f 7. orrpe.D.eJJ HM H3 paBeHCTBa
d{J dl; (!)121~ - 7
ds = - ds - ds -7. 2 •
CJJe,ll,oBaTeJJhHO,
113 paBeHCTBa dA = w 31; 33KJJIOllaeM, liTO ds = w 3 . CJJe,ll,oBaTeJJhHO,
(1) 12
x=- Wa 0
(r)
O e pBoe H rpeThe paBeHCTBa (3) .LI:JJH u.e nrpaJJbHOH KpHBOH npHHHM3IOT BH.LJ::
WJ2 = X3W32,
W3 = (aX.1 - X6)W32o
BnocH B (r) 3TH 3HalleHHH qJOpM, rroJJyl!HM ¢opMyJJy Kpyl!eHHH rrpocTpaH-
CTBei-IHOH KpHBOH (U.eHTpaJihHOH):
Y KOMITJJCKCa
X a
x=- axs-XG 0
X3=0
Bee u.e HTpaJJbHhie KpHBbie HBJJHJOTCH ITJJOCKHMHo TeM caMh!M orrpe.D.eJJeH reo
MeTpuqecKHH CMb!CJJ H HHBapHaHTa X3.
323'·
J1I1TEPATYPA
1. P l ii c k e r, Neue Geometrie des Raumes, gegrundet auf die Betrachtung der
geraden Linie als Raumelement, 1868-1869, Leipzig.
2. W. H a a k , Differentialgeometrie der Strahlenkorpplexe: I, Mathematische
Zeitschrift, B. 40, 1935, S . 560-561; II, ibid., S. 703-718; IV, B. 41, S . 252-260.
3. C. n. Cl> HIll! K 0 B, feO~I CTj) I!H KOMIIJICKCa npH~IbiX, YtiCI!biC 33 11HCKH M ocKOBCKO fO
;- opO!lCEOrO II C)I.. HHCTHT)'T3 I!M. !3. TI. TIOTCMKI!IHl , 13blll. I, T. I, 1940.
4. c. n. ci> H H H K 0 13, M eTO,u: LlHCWHHX ljlOpM K apTa na, M. - - J1., 1948.
5. n. K. p a w e B c K H ii , Kypc )l.li'<!JijJepenuua JiuHOii I'CO MCTj)HII, f>"\.-J1., 1938.
TIOJJY'I e Ha 12 HHBapll 1953 r.
3anopO)!{be.
0312
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0315
0316
0317
0318
0319
0320
0321
0322
0323
0324
|
| id | umjimathkievua-article-7737 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus |
| last_indexed | 2026-03-24T03:33:19Z |
| publishDate | 1953 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/f6/d933881c4a8728de09ba6d5a6eb009f6.pdf |
| spelling | umjimathkievua-article-77372023-08-15T10:12:02Z Кривые линейчастого комплекса Кривые линейчастого комплекса Кованцов, Н. И. Кованцов, Н. И. - Institute of Mathematics, NAS of Ukraine 1953-08-04 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/7737 Ukrains’kyi Matematychnyi Zhurnal; Vol. 5 No. 3 (1953); 312-324 Український математичний журнал; Том 5 № 3 (1953); 312-324 1027-3190 rus https://umj.imath.kiev.ua/index.php/umj/article/view/7737/9434 |
| spellingShingle | Кованцов, Н. И. Кованцов, Н. И. Кривые линейчастого комплекса |
| title | Кривые линейчастого комплекса |
| title_alt | Кривые линейчастого комплекса |
| title_full | Кривые линейчастого комплекса |
| title_fullStr | Кривые линейчастого комплекса |
| title_full_unstemmed | Кривые линейчастого комплекса |
| title_short | Кривые линейчастого комплекса |
| title_sort | кривые линейчастого комплекса |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/7737 |
| work_keys_str_mv | AT kovancovni krivyelinejčastogokompleksa AT kovancovni krivyelinejčastogokompleksa |