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ΕΥΣΤΑΒΙΟΥ ΒΑΣΙΛΕΙΟΥ
ΣΗΜΕΙΏΣΕΙΣ ΓΡΑΜΜΙΚΗΣ ΓΕΟΜΕΤΡΙΑΣ
(ΤΕΥΧΟΣ 1)
ΑΒΗΝΑ 1986
I Σ Υ Σ Χ Ε Τ Ι Σ Μ Ε Ν Ο Ι X q p q j
Ο. £/ΖΑΓΰΓΗ
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(< /d (/a i Oat C°v {/t//7 (Po υ /rc a o jo /7oo S (* O f 'jo r t o u t ac/*oz{\7o</y la 4 a (/d a i' (Co/.
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cm ^ (cou ty it 'a i aoc/is ^Srirau. ju l cn 4o/tdt/<x af^o/uun/ ct£iUt^*dxtttv, av<xu£(«· fca
o/roi'a 4f><iatccu Uou co S i af'un/a. Cy cucncta ) cou i t /u J c /J n .
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a /rou ta ' cn S o o n fyJi ^yaouru/uou 0 S ta rtstjuactaou ) acjoov, M t c/jjue^oSo cu /co ,
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z n z a s t /ro* j/fa*ufzcu 6 to ZtJtaccuo e te i itt/v rt-xvtrt/icnct/acUarY 6/roi/ cJy.
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Ctau 6t/u/T^rTpeud/?u( 6zn f y 'y o v n (/to x jj αηό τον 2>. # t f 6 e r { ' /S ) y i rr7
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yyayyrtuo czaiyo f Jtuy/S fa. ja/ityto/to/oJat ααϊοσ/Λ. aujQcysf zn oyaxrp.
ffui Q ' <χα*>λ*υ9η«ουο( zy yiSoSo JB) acu ya^/tza. Θα f/?fazadoJy/£ 61 yt«x
η/ό apjywytfvn t η 0/7<3/Λ 9a /Ty>t('xu zn fto/xou/j'n y/tuytty»t’a 6otv nayocr-
Styaq Cf/f/ari /rey/rrztoe n ). 2 nyiuu/roi/ut on ^rta ftivtoan yrurp/a/a. af cn a (So
ot) y/vtzcu (to ucidnua tns φ>ο/ο2/α*)Λ /fayteytaS.
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€ o u y t t C ( / y > o ί λ η α α ζ α y / ( z y n e n s . O r a * a y a y f t o < j a i a v c r j t r y $ c / y a t o r n r c * , ( t t f t v o
η ο υ U n o y o o i t t * x Z t d v o c y t f e t n t t o / A U u j S n y t a j y t t y j < ' a . { / r a u 0 / ϊ / α ρ y o f S y t i l a z o -
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1.2
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VoS Caffine"} ucu avz/S/tvfceJLJf zat ar/r® w £ t/u\*/£t/t> jtu)yoo f /tot/ e/rau cuyo/ftoi
rf/ai (OiXiz/tUfroi Jctyoos j 6 to* ono/or trneyoxu n St/rtxzoznttx ens f i t Z /ontn s (f/S-
( u) £(ωζγ>/αοο j//voa*roo ).
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IAN KtNHTPO ΤΗΣ ΜΕΛΕΤΗΣ TS.N I Y Σ X £ Τ/ΣΜ £N SN XOPSAf.
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tni*ty>nua’zm* jueis.
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u r n C t i t 0 n u > l Ζ Α a i / c / J c u t J a v t i f s e t f c C U f S ' a t i n p v f / u > j / y > e y y t a e / c u > r i7 **c
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f x x u c i c t o ( X r i ^ / a . ) . O / r to s j t f o v u r , 6 r « £ o f i j t z a a n t i / r o / e r n J {trcyaa'JJn ? n s ) f t f r t t z o -
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POL (Jrou CO ltsd<yy>eutfto zjtri·
Oct tiov oj>tjovv Svo ί/·*ρ<y>( ~
ztuai f no do P aai Q zoo£f
ΐό ζ ( η Οτ^&ί'Λληλη) α(ζία.ζό.
n u n 7 o a r e i Ρ Λ o ^ t f ( z o a
u * zoir t jf f i y > 0 n o : o r A
e/rau o/to/oSrin°Zt tnftf/o Cot/£
zo'tt y (/aava T (A ) ζου A (Sn>. zo t πακ /ο πον /ra/joyot/at < /(tatZantj/nrzas ro A aau*' y i -
aa 7} λο Λα/ /#o /Jtiab to P Q ) Qa t/y.-u to * 'la t/o to a n j r i i (</eva «Ιλν r?i) t u dti-
aS £j t /Του $/y*xtzcu από ι o A u-u. t/s -u /τ^α '^ Λ η λη /y^>j to /°c3 f u< ( f ,m m
ItSYaS/urj) (t/dt/a /toe/ £/(jpxtzau οπό to <3 i/au t/rnu παγ>αΆΑη)η ry>os to Cu»u<t /*A .
M 'a iJ J * ‘λ γ '/a. Τ/Ά) iirau n c tto jz η u o fV fri f j y /T^yxx.JJ/7 }ojy,au/jc/ou /roc/ (y^/jfeou.
από vex enf/eiac P, CL uau A . frrajutytoi^ (pajia0J*vzaii tr>v /S/a S/a S/uat/o //au y//a
ono/o%n/7o t ( d j j o 6nut/o £ , S/an/t cairoijut o r/ acu co ct/dJypoutuo tariaa AB o f -/ ‘ / S ' / / / / /
taco/7/J( ecu 6eo Cjuriaa a t cujoa TfA ) acu TCB).
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6ior eaufio got/. £ oa oX a S /a /r /itu /r te c u o r / r> 7" (/y o u α π (/α ο ν / 4 η / - J { : au /p tuot/o .
f r i a a v r n ) Hou (/7 / , A y ty o a 6υα£α X /to o u t a t t ( £ ) το turo^o ό λ ω ν zu>r S i/i/a ttvr
ju t la t o />/ 6turv ίο υ £ f u /tsjovut ya o j / t o u u t a/<* /y > a jn / y i o e d t t f r j y ( ,au/io a t ζν
Τ.3
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/Tfapuai/uot/S cyo/lUtovs , /y>(nit r * oj>/toc/jut aou to //routro Jϊ . ( ρ α ) i \ t/R t S/ΐλ. &»
jr/ ro juva f yoj /y>ay/uai/ttov ayo/t)aou j u t i r a ti/d<yy>cLu uo CurjctA. £Su> ouu/i ^ ( π α
zo e/f//fffo y<\ tyoS/ar i t u at t v Suy n tJ> t r?t a. ya. uayei/uf a t tfi/rtn aeu y*. of/JovuC
to a riu o i ( t o i z a n a a re i. ( l^u SnX rj (/ta^t*>j >i isv a iy oa v ( : a a ra ia j c/haoi/S)
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lo o O'TO/OV on/7(Kou r} tx/ roe ta tn Suo drjue/u/v a t t » r /7r>oo/radttn i,vt/a .* Ais o/ ft / ’ / / r
Ju^os tuy><* (X U aou So a » a u /v /u o J a^o^ot/ (a t a a t*a })„ )k n cexJtstn cou £ CoY
P 2 ) , £ i t / t a v y o o ^ t r ( r<x O ja/1οι/ u t Co λ (Ρ Ο , ) e a r t o Cufijsci f Cou o />o /JV Zo f * A -
tio± (/ ra t Λ pyotS u> uriujoi c<jv f id . i A * 4d4aua. a^onitao/)v/riiouuf rnv Ccttjrj-
(? ) £ wt f i t /too t/i/eu^/XtJH/t tou ut tnr (α ^ γ *i <ryci/v^ u //■yaoua e m foJut
^ . f fl v />°Ct)t6t4a m i /tj*en j/oisatvni /yiajsrs fti/ « « ejoy>cdt<x4 α,ηά zns tuA>y*i
Ζ 4
zns ay>xns 1 lortnwi t &r T = T-~ c/rau n e/tc<xt0/i/6n ααΐα. /3θ;
ft u t t a t o n t f n T m A· Tpg ^/'rtccu at cn β κ ί ιη
fc ta n tfttv v o o u f ct/αολα o t t Co tthfo>o tC £ ) f fj>oS>a*<t vo u ( r/J Soo /yx>njsouf/f-
V(S tyzaj(/S + oat · (f/ri /yoaj/uaztuotf cy>*dji0) HVcu f 'vai fyatM utuoi jcuy>oS. £ n ) .
£ (u ir u h l* J Λπο zfy £ t o S n j/ a o a a tc t fcu tt ■uaca.iatc"; coo ffau^s/uoc/ 3tuy>oi/ Y fe ).
Ot Svo autpot £ acu * C £ ) 6UY Stdvccu C e V 6 x. € τ< g , o V z a t ) rav t p v -
6/oJ^OJr/U0 ψ>0/?ο ;
U a d f ^ (u j/o i in ud urr C & 4 .) zoo £ (oovo*r>aatrc<x.) cn
U (Ca.z0n/tn 7 ^ ( Uaza P Q ) oca f uotTci tavthfiAv,
n cx n ffU ov ttn
i : £ x £ ---- ( P , Q )> - + 7 ^ .
£ras Sro/ae/a>5ni (J t ^ X o i aai /r f/d f/ o r/ 9 / f x f / r/J sS /otnceJ .·
4) fia aaidf P m £ , n a n t /u o w tn : £ -----* t { £ ) t /t o o o f/jtca u f t f
zn e x i t n
i p ( Q ) ; . Tr a , ν α . ε
e/vau / - / Oou ( n t .
//J //a o/r*/airi/7or( 6 nut /a /? Q. # zoo £ f n f-x€tn
T* 5 + T5 k * T* f i ” 0 >
onou o f 0 6uuio "Xtjooi/f cn i/n$t*/ur> ftfza. ζάπ/* η $η>· τη *·* ζ*κζ*>» n n 'too oy,jr.
vet a*'«2?<3/coc& ·α ά ϊ ( 6nue/o zoo £ .
Α Σ Κ Η Σ Ε ΙΣ .
4. /Vo. anoSf/XZf/ o tt ono/alri rroz l (x n f/O o r /S n CnJ p o p p h i n ’roa
α/τ(/αόν/6η £ - i ζου £ € n i του tauzoo cου.
1.5
Ζ . Δ υ ό ono/eeSnnoz£ afcarfeyoc's Τ uou Τ ' C etty>*€t*zy>ct : Τα T ^ ctau
7“' · 7^j ) pczctct dcrzau SnA. To Τ ' — τ 'ο T.
S . Λ/a e x n o S f/x t ft ozt co C tC £ ), ° ) , όηου « /7cya/6za cn e o r d f fn
a n f/U o v i6 (ω ν , a n o z iJ ii α $ (3 /α ν ή oOez’Scr f α,υζό S ta cu o^orti uou Co ίυοβο-* / / /
0<t*po tns / y ic r jn j + y nou cyo/j.(zeu utzjugu ca t* tzo/ xa ’tov zoo e C s i /
4. /id yota ono/orSn no it 6 na (/<x. A, 3 , C cocr £ iSxOu η βχέ«η
T - o T ~ = T— .BC AS AC
XuJS 60vJ(£tcu au/cri & £ xnv S/dzneac 4e) cn J / .
5 . A no S f/ ζ ZL n'XripcvS ozt o C C £ ) c /va t J / f a p t t t ao i auyooi ctau 0zt n trn -
S fy /cth utt<xt0n /6n C/3\. I S/ό ζη ζa. i t ) ) (/you r> zaujtoz/urj a n e /a oy /tn zoo £.
6. Met et/roSe/xzroav ot Jb toznza / ) uou ΰ ) /too 6o6x (z/Jouy coy £ p i zov
Z (£ ). J fei u7ono/ftzca n p f za ζό/r/tn 0 « e r e ' ttriaoi cr*i fopatcyj/ufvot/·
iu d u yp a a u cv cu ripa zoJ .
1.6
2. Η ΓΕΝΙΚΗ £ΝΝΟ!Α ΤΟΥ Σ Υ Σ Χ £ Π Σ Μ Ε Ν Ο Ύ ΧβΡΟΥ
Ο zpon°S ρ ( ro ν ο/το/ο so fS tta u f f tznv /rp onp a jp frn /7 < ^ > α ^ β ρ ο ^ zov £ p i co
p p cp p tU o A a p o t ( £ ) f olnpt·/ fen p f v /α ό ζ γ η dtu ipm n cat? evfXfz/epzi/urr SnX
afyanufi'W Y 6σγόλα/τ^ nou «um et/^Oveau α α ζαλληλ*. atpyoapp/aoJj Au>^>ouSt fr t / cuerf
rat CJvat furacri n a v a n z u jn p /a j / 7 /0 J'tis/urps (at/e* lu c /a ys ) p ( c o p ( yataS f aaZ<\
Co /y>0Tu/ro tffi 6zo/XC/utSoui ( cupt(7p/a&.
/ f ανά ητυζη ( η °υ Οαρου6/α.^ ( cat eSuif ( /vau p /a . a/?J> r/J /TaAAei OtoSOt/apes)
a/ropets now p /ta p f i ra 3p776/porto/riff/ zeaiff/S.
Θ (ω ρ ο ύ ρ ( fra lu xov β ύ /ολον £ , zoo 0 /70/ου car 6ro/ Λ (/a. 9 a Uct-
λ ο upe 6n o s i * acu <y« j/p a p p /u o g a p a V . 0 V ( / vcu (p /a fuu* λ /*.) p/t&pp/uo\
OiujpoS to* /tp°S /R I d a ρ/Ηγ>ούίαρ( r& opnt/po/ja /titoupe ttt λ ΛΑ<λ. eu /pata f (uxo's ro t/
f i / o'nwi ( /r/'fttS -Ctau S/apt z/ctouS Sou U»)> tOVS f atvci β ο α ρ ρ /tt an/ Xcopurv. Kau fc / j n t -
pz/rza ief/i oeozei 4tatei^7ipoupf « ( napdposct ίυρ /> (ρafepaCA. f a Jau /r*' /}ou 9 a S o d p f
6Tn Senrfxaa- Ορωί^ fr/J p rv sa o rrp fs owret n^s//?zu/*tti f ez/rcuzf/rau (itas auMtnpo-
z p o i (%fpX*»S t όπου tppav/J 't z*x* y Z a p a u r jo /fc /a ή tot/ ίίύραζ& ΐ y /yothf/ ra not-
paiicppdoay <&/όζηκ j /rpa'^f urv n o j $a/ /m Joiry tzn pt<//uo Tt^n /T(p//r zu>tn J ,
<?./ 0Ρ/ΖΜΟΣ. fra/S €OfZ€ Z / t U CroS CCwpoS ( atff/ne Spo'ce) t/rau a Sot cp/α
ία. (£ , Y , n t όπου £ uott V t/vau on cos rponpouua/tus t aau i t/i/aut pi<x anC/uoy/tn
zns p op f/is £ : £ * £ *--- -V " f ( Z6/ aiecf r& **χύουγ o/ gvYdriud ·*
(Z.X i ) Γ,Ο/ aaide P c £ n a n (/a o v / (n
6p : B - V : A * +tp C Q ) : . t ( p a )
f/rau i - i acu €n i .\
fax 2) /id o/to/a in froze frrpf/a. /f zoo £ t /6xi/C/ <7 cxetn
ί (ρ < 2 ) + / ία * > + t c * , p > - o
I ΧΟΛ/A . /) Τψν tturi έ (Ρ & ) e»i / *zat (no (/A /> a 6t/u&o>‘jou u ( ft/x rat'oat
lt( / ( P Q ) m PCL. M* ro rto i) ixftn tyi (t/vdriuni fz.x 2) j/faiytzoa acu
ji t znuapfri: M + 37l y RP - Ο. N extitn ecotfj t/rau j'ruHtri ctau eav rptyajyturj
(Stotnza. p gag'd rarS T°u CAas&l C/X. arr/iro/xp A/ocnca ctor S/avutadctov cou JR3
2 ) l e t r r / E t a ( x t t n z n i C j .X S ) z o 0 z o o S t i / ζ γ ο ο t K ^ e u i ( t i t a n 6 y > o y * r e 0 1 ) Co f n -
Stvtud itairi/iact cou V.
3 ) /(a id* <xnnud\tt6r> ip tr>ty>t'rtu ra vatoz/6ouo( zov £ o f r o ^-tutf/tuA auufo Y
(atttos cuykfuti tavltJoUU ( to :lajoo cm tza/xt/utSoui f t toojt(y>ta l a t cov 4fJ y co (/7/trtfo
1.7
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i ( P , Q ) + t * (Λ , P ) = 0
acu u jf t 'v a n α /'τό ίζ/ξη . ·
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tou £ .
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j o u * i ( ( t n £ ( y / u o x t^ n ( yro/a coo ( υ i X ( c / 6 a t you ν/τ&Αιήρόυ.
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I. 22
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Ay>a 5 » S (A t U ) - S (A , Ds ) .
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5 . { T ( A ) - Α + υ / v e U j
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4 5 STOP/Σ Μ Α . S(cou 5® S (A t ( / ) ί'κσί cuextc/tat'yol υ/roxuyoos cou £. Τόζί,
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( 4 4 ) / ( A , P ) * ( ( P Q ) + t ( Q , A > β 0 .
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