Shmei seic sto mˆjhma Analutik GewmetrÐa

Shmei seic sto mˆjhma Analutik GewmetrÐa Didˆskwn: Lˆppac D. Ejnikì Kapodistriakì Panepist mio Ajhn n

B' MEROS

3 EPIFANEIES sto QWRO Epifˆneia gia thn perigraf thc qreiˆzontai dôo parˆmetroi mia eidik epifˆneia EpÐpedo HremoÔn ugrì Efarmog eujeðac (t, s) x 0 + t w + s u = x (PARAMETRIKH E- XISWSH) epðpedo pou dièrqetai apì èna shmeðo to x 0 kai eðnai kˆjeto sto ( w u). ( w u) grammikˆ anexˆrthto. < AM, l = 0 (E) : Ax + Bψ + Γz + = 0 (ANALUTIKH E- XISWSH) EpÐpedo kˆjeto sto l = (A, B, Γ). x x 0 = R prokôptei me thn peristrof thc hmiperifèreiac. Perigraf thc epifˆneiac thc SfaÐrac me dôo paramètrouc SfaÐra (x, ψ) + R x ψ (giatð x + ψ 1 = 0, x = ± 1 x ψ ) (x, ψ, R x ψ ) EmpÐptei ston ORISMO tmhmatikˆ Polikèc suntetagmènec dôo parˆmetroi x + ψ + z = 1 (POLUWNUMIKH ou BAJMOU TRIWN ME- TABLHTWN) KWNIKES EPIFANEIES (C): kôkloc K: ShmeÐo tou q rou (ektìc epipèdou tou kôklou) AK: eujeða (GENETEIRA) Af noume to A na trèqei ston kôklo tìte h (KA) parˆgei mða epifˆneia (S) S: k noc me koruf to K kai odhgì thn kampôlh (C) oi eujeðec eðnai olìklhrec K(x 0, ψ 0, z 0 ) A (C) ìpou A(x 1, ψ 1, z 1 ) (C)

4 (C) f 1 (x, ψ, z) = 0 f (x, ψ, z) = 0 TOMH EPIFANEIWN (SFAIRA-EPIPEDO) P : TUQAIO SHMEIO thc EPIFANEIAS (S) (S) P (x, ψ, z) (S) = P A (C) : P, K, A SUNEUJEIAKA} P, K, A suneujeiakˆ KP = t KA, t R KP = (x x 0, ψ ψ 0, z z 0 ) t KA = (x 1 x 0, ψ 1 ψ 0, z 1 z 0 ) x x 0 = t(x 1 x 0 ) ψ ψ 0 = t(ψ 1 ψ 0 ) (S) z z 0 = t(z 1 z 0 ) f 1 (x 1, ψ 1, z 1 ) = 0 f (x, ψ, z ) = 0 AnazhtoÔme sqèsh anˆmesa sta x, ψ, z (Prèpei na apaleifjoôn ta t, x 1, ψ 1, z 1 ) PARADEIGMA K(1, 3, ) 3x + ψ = 1 (C) z = 0 èlleiyh sto epðpedo xoψ. ZhteÐtai h kwnik epifˆneia me koruf K kai odhgì kampôlhc (C) 'Estw P (x, ψ, z) shmeðo thc epifˆneiac (S). Tìte A(x 1, ψ 1, z 1 ) (C) ste KP = t KA, t R. x 1 = t(x 1 1) ψ + 3 = t(ψ 1 + 3) z = t(z 1 ) } 3x 1 + ψ1 = 1 z 1 = 0 Epeid z 1 = 0 tìte z = t( ) t = z x 1 = z (x 1 1) x 1 = ψ + 3 = z (ψ 1 + 3) ψ 1 = 1 h PERIPTWSH

5 z 0 x 1 1 = ψ + 3 = (x 1) z (ψ + 3) z x 1 = ψ 1 = Apì thn antikatˆstash sthn 3x 1 + ψ 1 = 1 prokôptei h PERIPTWSH 3 [ (x 1) + ( z) z (x 1) + ( z) z (ψ + 3) 3( z) z ] [(ψ + 3) 3( z) ] + + 1 = 0 z (S) : 3[(x 1) + ( z)] + [(ψ + 3) 3( z)] = ( z) z = 0 z = prokôptei to K(1, 3, ) (S). MAJHMATIKH MELETH An kˆnoume prˆxeic ja prokôyei exðswsh thc morf c: αx + βψ + γz + δxψ + εψz + ζxz + (κx + λψ + µz) + ν = 0 ExÐswsh poluwnumik, ou bajmoô, tri n metablht n. Jètw X = x 1, Ψ = (ψ + 3), Z = z (dhlad arq to K). Tìte h exðswsh thc S paðrnei th morf : 3[X Z] + [Ψ + 3Z] = Z ExÐswsh thc S sto K, XΨZ}. H exðswsh thc S sto K, XΨZ} eðnai OMOGENHS ou BAJMOU. F (X, Ψ, Z)(= 0) x λx OMOGENHS: ψ λψ Z λz F (λx, λψ, λz) = λ F (X, Ψ, Z) (X, Ψ, Z) (λx, λψ, λz) eðnai kai pˆli shmeða thc epifˆneiac. SUMPERASMA: An mða kwnik epifˆneia grˆfei se sôsthma anaforˆc me arq thn koruf tìte eðnai OMOGENHS EXISWSH. Mia exðswsh ou bajmoô perigrˆfei k no eˆn me katˆllhlh metaforˆ eðnai OMOGENHS f(x, ψ, z) x x + x 0 ψ ψ + ψ 0 z z + z 0?(x 0, ψ 0, z 0 ) : f(x, ψ, z ) OMOGENHS. APLOUSTERES MORFES KWNWN x + ψ + z = 0 to kèntro.

6 x + ψ z = 0 x ψ z = 0 x + ψ + z = 0 αx + βψ + γz = 0 K(0, 0, 0) (C) : z = 1 x + ψ = 1 k noc me koruf to K kai odhgì thn kampôlh (C) ProkÔptei z 1 = 1 z = t x = zx 1 x 1 = x z ψ = zψ 1 ψ 1 = ψ z KP = t KA x = tx 1 ψ = tψ 1 z = tz 1 x + ψ = 1 z 1 = 1 (S) : x + ψ z = 0 KWNIKES TOMES TOMH KWNOU kai EPIPEDOU x + ψ z = 0 : (S) αx + βψ + γz + δ = 0 : (Π) (S) (Π) : z = 1 x + ψ = 1 KUKLOS

(Kwnik epifˆneia) Kèntro(K) odhgìc kampôlhc(c) } 7 (S) = p A (C) : KP = t KA, t R} An jewr sw sôsthma me arq to K, tìte h exðswsh thc S sto nèo sôsthma eðnai OMOGENHS ou BAJMOU. POTE MIA EXISWSH B' BAJMOU sto QWRO eðnai KWNIKH EPIFANEIA g(x, ψ, z) = 0 anazht (x 0, ψ 0, z 0 ) x x + x 0 ψ ψ + ψ 0 z z + z 0 g(x + x 0, ψ + ψ 0, z + z 0 ) OMOGENHS wc proc x, ψ, z. An isqôei tìte (x 0, ψ 0, z 0 ) KENTRO tou nèou sust matoc. PARADEIGMA 1 Na exetasteð an paristˆnei kwnik epifˆneia Prosjètoume katˆ mèlh b' bˆjmio x 4ψ z + 4ψz + 4z 4 = 0 x = x + x 0 + x x 0 4ψ = 4ψ 4ψ 0 8ψ ψ 0 z = z z 0 4z z 0 4ψz = 4ψ z + 4ψ 0 z 0 + 4ψ z 0 + 4z ψ 0 4z = 4z + 4z 0 x 4ψ z + 4ψ z + x x }} 0 + ψ ( 8ψ 0 + 4z 0 ) + z (4ψ 0 4z 0 + 4) }} EÐnai omogenèc (Σ): sumbibastì a' bˆjmio + x 0 4ψ 0 z 0 + 4ψ 0 z 0 + 4z 0 4 }} stajeroð ìroi (1) x 0 = 0 () 8ψ 0 + 4z 0 = 0 (Σ) (3) 4ψ 0 4z 0 + 4 = 0 (4) x 0 4ψ0 z0 + 4z 0 ψ 0 + 4z 0 4 = 0 Prèpei to (Σ) na eðnai sumbibastì x 0 = 0 ψ 0 = 1 z 0 = epalhjeôoun thn (4) = 0 suntelestèc prwtobˆjmiwn stajerìc

8 Tìte prokôptei h exðswsh: x 6 4ψ z + 4ψ z = 0 OMOGENHS K(0, 1, ) Br kame to kèntro K(0, 1, ). AnazhtoÔme thn odhgì kampôlh ProkÔptei me tom thc epifˆneiac kai enìc epipèdou. jewroôme to epipèdo z = 0 x 4ψ = 4 x 4ψ = 4 (C) : uperbol sto xoψ. z = 0 z = 0 K(0, 1, ) Koruf 'Ara o k noc eðnai (C) odhgìc kampôlh PARADEIGMA JewroÔme to shmeðo K(1, 1, 1) kai thn eujeða x 1 1 tou q rou me thn idiìthta: = ψ 1 1 = z 1 1 (ε) kai ìla ta shmeða M ( KM, (ε) = STAJERH = π 4. Na exetasteð eˆn ta M sqhmatðzoun stajer epifˆneia. Jewr to diˆnusma w(1, 1, 1) (ε). ( KM, w) = π 4 cos( KM, w) = cos( KM, w) = < KM, w > KM w = < (x 1, ψ 1, z 1), (1, 1, 1) > (x 1) + (ψ 1) + (z 1) 1 + 1 + 1 = = [(x 1) + (ψ 1) + (z 1)] [(x 1) + (ψ 1) + (z 1) ] 3 = 1 paristˆnei k no me koruf K(1, 1, 1). Jètoume x 1 = X ψ 1 = Ψ z 1 = Z Tìte h exðswsh gðnetai: (X + Ψ + Z) = 3(X + Ψ + Z ). F (X, Ψ, Z) = 0 F (X, Ψ, Z) = 3(X + Ψ + Z ) (X + Ψ + Z) X +Ψ +Z 4XΨ 4XZ 4ΨZ = 0 OMOGENHS b' BAJMOU orjìc kuklikìc k noc φ : gwnða thc genèteirac me ton ˆxona φ : gwnða tou k nou

9 KULINDRIKES EPIFANEIES odhgìc kampôlh (C). genèteira: kineðtai pˆnw sthn odhgì kampôlh paramènontac parˆllhlh se miˆ stajer dieôjunsh (δ) u dieôjunsh (S) = p A (G) : 'Estw P (x, ψ, z), A 1 (x 1, ψ 1, z 1 ) C, C AP u AP = t u t R ProkÔptei to SUSTHMA (C) : kampôlh AP u} KULINDRIKH EPIFANEIA x x 1 = tα ψ ψ 1 = tβ z z 1 = tγ f 1 (x, ψ, z) = 0 f (x, ψ, z) = 0 f 1 (x ψ, z) f (x, ψ, z) Apaloif twn t, x 1, ψ 1, z 1. f 1 (x 1, ψ 1, z 1 ) = 0 f (x, ψ, z ) = 0 Me thn apaloif twn t, x 1, ψ 1, z 1 prokôptei F (x, ψ, z) = 0 h exðswsh epifˆneiac. PARADEIGMA: ORJOS PARABOLIKOS KULINDROS ψ = px (C) : PARABOLH sto Oxψ z = 0 u(0, 0, 1) sto Oxψ A(x 1, ψ 1, z 1 ) C S = p A (G) : AP u, AP = t u} ìpou P (x, ψ, z) tuqaðo shmeðo. H epifˆneia (S) pou ja prokôyei eðnai ORJOS PARABOLIKOS KULINDROS. x x 1 = 0t ψ ψ 1 = 0t z z 1 = 1t } ψ1 = px 1 z 1 = 0 x x 1 = 0 ψ ψ 1 = 0 AfoÔ z 1 = 0 z = t tìte: ψ = px z R Sqìlio H exðswsh ψ = px sto q ro eðnai KULINDROS. H exðswsh pou prokôptei eðnai b' bajmoô MH OMOGENHS.

10 PARADEIGMA: PARABOLIKOS KULINDROS ψ = px C = PARABOLH sto Oxψ z = 0 w = (1,, 3) A(x 1, ψ 1, z 1 ) C x x 1 = t ψ ψ 1 = t z z 1 = 3t ψ 1 = px 1 z 1 = 0 z = 3t t = z 3 Tìte x 1 = x z 3 ψ 1 = ψ z 3 Jètoume ta x 1, ψ 1 sthn ψ 1 = px 1 kai prokôptei (ψ z 3 ) = p(x z 3 ) 3(ψ z 3 ) = 6p(3x z) p = 1 6 (E) : 9ψ + 4z 1ψz 3x + z = 0 An exetˆsoume thn (E) wc proc thn OMOGENEIA prokôptei ìti KENTRO. SFARIKOS KULINDROS ASKHSH: JewroÔme th sfaðra x + ψ + z = 1 kai to mègisto kôklo pou prokôptei wc tom thc (S) kai tou epipèdou x + ψ + z = 0. ZhteÐtai h orj kulindrik epifˆneia me odhgì kampôlh thn (C) Gia sfaðra (S) kai kampôlh (C) ìpwc prin zhteðtai o G.T. twn efaptomènwn sth sfaðra kai sthn kampôlh P (x, ψ, z) : shmeðo thc epifˆneiac A(x 1, psi 1, z 1 ): shmeðo thc (C) u(1, 1, 1) sto epðpedo thc (C). x x 1 = t 1 (1) ψ ψ 1 = t () z z 1 = t (3) x 1 + ψ 1 + z 1 = 0 (4) x 1 + ψ1 + z1 = 1 (5)

11 t 0 tìte x x 1 t = ψ ψ 1 t = z z 1 t opìte t = x + ψ + z 3 ψ z + x (1) x 1 =, () ψ 1 = 3 AntikajistoÔme ta x 1, ψ 1, z 1 sthn (5) = x + ψ + z (x 1 + ψ 1 + z 1 ) 0 3t x + ψ z, (3) z 1 = 3 x ψ + z 3 (ψ+z x) +(x ψ+z) +(x+ψ z) = 9 KULINDROS pou efˆptetai sth sfaðra. ja tan OMOGENHS. An eðqe sto deôtero mèloc 0 ja tan k noc giatð ELLEIPTIKOS KULINDROS ASKHSH: Na brejeð h exðswsh tou elleiptikoô kulðndrou me odhgì thn kampôlh (C) : 3x + ψ 1 = 0, z = 0 kai dieôjunsh orismènh apì to u = (1, 3, ) 3x + ψ = 1 (C) : ELLEIYH sto Oxψ: ODHGOS KAMPULH z = 0 u(1, 3, ): STAJERH DIEUJUNSH A(x 1, ψ 1, z 1 ) C P (x, ψ, z) tuqaðo shmeðo thc epifˆneiac S = p A (G) : AP u} AP u dhlad AP = t u. AP = (x x 1, ψ ψ 1, z z 1 ) x x 1 = 1 t ψ ψ 1 = 3t z z 1 = t 3x 1 + ψ 1 = 1 z 1 = 0 ProkÔptei ìti z 1 = 0 z = t t = z. Tìte x 1 = x z, ψ 1 = sthn 3x 1 + ψ1 = 1 kai prokôptei ψ + 3z. AntikajistoÔme 3( x z ) ψ + 3z + ( ) = 1 3(x z) + (ψ + 3z) 4 = 0 'Ara (S) : 3(x z) + (ψ + 3z) 4 = 0 exðswsh elleiptikoô kulðndrou.

1 EPIFANEIES EK PERISTROFHS Dedomèna: C: KAMPULH ξ: EUJEIA pou leitourgei wc ˆxonac me peristrof thc (C) gôrw apì thn ξ parˆgetai mia epifˆneia (S). Kˆje shmeðo thc epifˆneiac brðsketai pˆnw se èna KUKLO. Autìc lègetai KUKLOS ANAFORAS kai èqei akrib c èna koinì shmeðo A me thn kampôlh MAJHMATIKH PERIGRAFH (S) : P!(K (ξ) kai A (C))} ( (KAP ) (ξ) KA = KP ξ: exðswsh eujeðac (C): exðswsh tom epifanei n (Σ): sunj kec: K (ξ) kai A (C). KWNOS (DIQWNO) to epðpedo tou kôklou eðnai kˆjeto ston ˆxona ξ kai ton tèmnei se èna shmeðo K: kèntro tou kôklou. Dhlad (KAP )(epðpedo) ξ PARADEIGMA: Na upologisteð h epifˆneia pou prokôptei apì peristrof thc (C) gôrw apì ton ˆxona x x ìpou x x = (x, 0, 0) x R. M (Σ) M(x, ψ, z) A (C), A(x 1, ψ 1, z 1 ) kai K ξ, K(x 0, 0, 0) x 1 ψ 1 = 0 z 1 = 0 (KAM) epðpedo (ξ) x = x 0 = x 1 KA = KM (Σ) k noc sto q ro (x 1 x 0 ) + ψ1 + z1 0 = (x x 0 ) + (ψ ψ 0 ) + (z z 0 ) x 1 =x ψ1 = (ψ ψ 0 ) + (z z 0 ) ψ 0=0, z 0 =0 = ψ1 = ψ + z ψ 1=x x ψ = 0 z = 0 }

13 x = ψ + z x = ψ + z } x = ψ + z x + ψ + z = 0 KWNOS H koruf den eðnai omalì shmeðo. PARADEIGMA: ELLEIYOEIDES ek PERISTROFHS (C) x α + ψ β = 1 z = 0 (ξ) x = z = 0 ˆxonac ψ ψ M(x, ψ, z) A(x 1, ψ 1, z 1 ) x 1 α + ψ 1 β = 1, z 1 = 0 K(0, ψ 0, 0) KM = KA x + (ψ ψ 0 ) + z = x 1 + (ψ 1 ψ 0 ) + z 1 (1) (KAM) epðpedo (ξ) ψ 1 = ψ = ψ 0 Kˆnoume apaloif twn x 0, x 1, ψ 1, z 1 x 1 α + ψ 1 β = 1 ψ 1=ψ = x 1 α + ψ β = 1 (1) x + z = z 1 x + z = x 1 α α = 1 ψ β x α + ψ β + z α = 1 An α = β = R (KUKLOS) x + ψ + z = R sfaðra ek peristrof c. ShmeÐwsh x α + ψ β + z = 1. ParathroÔme ìti oi paranomastèc tou pr tou kai trðtou klˆsmatoc α eðnai ìmoioi kai sunep c prìkeitai gia epifˆneia ek peristrof c. α + ψ β + z = 1 γ oi paranomastèc tou pr tou kai trðtou klˆsmatoc den eðnai ìmoioi kai sunep c h epifˆneia den einai ek peristrof c. An x UPERBOLOEIDH EK PERISTROFHS (α) KampÔlh (C) ψ b z α = 1 x = 0 (β) ˆxonac peristrof c: O ˆxonac z z KA = KM

14 x 1 = 0 ψ1 b z α = 1 x + ψ = ψ 1 x + ψ b = ψ 1 b K(0, 0, z 0 ) (KAM) z z z 0 = z 1 = z to A èqei x 1 = 0 giatð A (C) A(0, ψ 1, z 1 ) = 1 + z α ProkÔptei: x + ψ b z α = 1 MONOQWNO UPERBOLOEIDES SHMEIWSH: x + ψ z = 1 DEN eðnai ek peristrof c 3 x ψ z = 1 DEN eðnai MONOQWNO. PARADEIGMA: MH SUNEKTIKO UPERBOLOEIDES/DIQWNO ψ C b z c = 1 UPERBOLH sto ψoz x = 0 ˆxonac peristrof c ψ ψ A(x 1, ψ 1, z 1 ) K(0, ψ 0, 0) P (x, ψ, z) ψ = ψ 0 = ψ 1 x + z c + ψ b = 1 x c z c + ψ b = 1 PARABOLOEIDH ek PERISTROFHS KAMPULH: ψ = α cz, x = 0 PARABOLH ston ψoz AXONAS: z z Peristrèfoume th C gôrw apì ton z z ˆxona. IsqÔei (KAP ) z z, KA = KP A(x 1, ψ 1, z 1 ) K(0, 0, z 0 ) P (x, ψ, z) KP = x + ψ KA = ψ 1 ψ 1 = (α c)z x +ψ (α c)z = 0, ìpou z prwtobˆjmioc ìroc.

15 EPIFANEIES EK PERISTROFHS KWNOI: αx + βψ + γz = 0 ELLEIYOEIDH: x α ± ψ β ± z β = 1 UPERBOLOEIDH anhgmènec morfèc b' bˆjmia ta ELLEIYOEIDH-UPERBOLOEIDH èqoun KENTRO SUMMETRIAS kai POLLA EPIPEDA SUMMETRIAS (x, psi, z) ( x, ψ, z) PARABOLOEIDH: x + ψ (α c)z = 0 DEN èqei KENTRO SUMMETRIAS 'Eqei EPIPEDO SUMMETRIAS.

16 H DEUTEROBAJMIA EXISWSH sto QWRO (E) : α 11 x + α ψ + α 33 z + α 1 xψ + α 13 xz + α 3 ψz + α }} 14 x + α 4 z + α 34 z + α } 44 = 0 } b-bˆjmio a-bˆjmio 3 i, j=1 α ijx ij x 1 x x ψ x 3 z α ij = α ji [ + 3 i=1 α i4x i ] + α 44 = 0 b'bˆjmio a'bˆjmio Ekfrˆzoume thn (E) wc ginìmeno pinˆkwn A = α ij, A = A t α 11 α 1 α 13 x x (x, ψ, z) α 1 α α 3 ψ + (α 14, α 4, α 34 ) ψ +α 44 = 0 α 31 α 3 α 33 z z }}}} B = (α 14, α 4, α 34 ) x X = ψ z Γ = α 44 (E) : X t AX + BX + Γ b'bˆjmio a'bˆjmio SUSTHMA SUNTETAGMENWN pou APLOPOIEI thn (E) H (E) anafèretai se èna sôsthma suntetagmènwn Oxψz M(x, ψ, z) (E) = 0}. AnazhtoÔme sôsthma suntetagmènwn O x ψ z } ste h (E) na paðrnei thn aploôsterh morf. ALLAGH SUNTETAGMENWN sto QWRO x Oxψz ψ O x ψ z X = P ψ + H ìpou P : pðnakac allag c bˆshc pou diathreð ta m kh, STROFH (GENIKEUSH) kai H: METAFORA P : GENIKEUSH thc STROFHS ìpou P P t = I dhlad o P eðnai ORJOGWNIOS. APODEIXH x, ψ, z, x, ψ, z x + ψ + z = (x ) + (ψ ) + (z )

x (x, ψ, z)i ψ = X t IX z (x, ψ, z )I x ψ z = Ψ t IΨ 17 Prèpei X t IX = Ψ t IΨ X = P Ψ IsqÔei X t = Ψ t P t Tìte Ψ t P t P Ψ = Ψ t IΨ ψ F ν ν ProkÔptei ìti P P t = I b 11 b 1 b 13 b 11 b 1 b 31 P = b 1 b b 3 b 1 b b 3 = I b 31 b 3 b 33 b 13 b 3 b 33 Oi grammèc eðnai metaxô touc ORJOGWNIA DIANUSMATA Oi st lec eðnai metaxô touc ORJOFWNIA DIANUSMATA Kˆje gramm, kˆje st lh èqoun m koc 1. EpÐshc isqôei det(p ) = 1 det(p ) = 1 det(p ) = 1 Allˆzoume suntetagmènec: X = P Ψ + H Sthn exðswsh (E): X t AX + BX + Γ = 0 antikajistoôme to X kai prokôptei: b'bˆjmio a'bˆjmio P 1 = P t Ψ t (P AP t )Ψ + [(HA + B)P t ]Ψ + (HAH t + BH t + Γ) = 0 }}}}}} P t stajerìc ìroc SHMEIWSH A P AP t B (HA + B)P t Γ (HAH t + BH t + Γ) ERWTHMA: Upˆrqei mða metaforˆ ste sthn exðswsh pou prokôptei na mhn upˆrqei prwtobˆjmioc ìroc } (HA + B)P t = 0 Mìno an upˆrqei H : HA + B = 0 P t ANTISTREYIMOS det(a) 0 èqei lôsh Dhlad prèpei HA = B det(a) = 0 den èqei lôsh

18 An det(a) 0 H = BA 1 PROTASH H E eðnai epifˆneia pou èqei KENTRO SUMMETRIAS an kai mìno an det(a) 0. Tìte to KENTRO eðnai h lôsh tou sust matoc H = BA 1. MELETH twn EPIFANEIWN 1 h PERIPTWSH: det(a) 0 dhlad upˆrqei KENTRO se mia metaforˆ. α 11x + α ψ + α 33ψ + α 1x ψ + α 13x z + α 3ψ z + α 44 = 0 Z t BZ + = 0 ìpou B: summetrikìc kai B = A me det(b) 0 kai Z t BZ : b'bˆjmioc ìroc, : stajerìc ìroc. Prèpei na gðnei ALLAGH BASHS gia na fôgoun oi ìroi x ψ, ψ z, x z Upˆrqei allag bˆshc Ω = P Z ste Ω t (P BP t ) Ω + = 0 }} diag nioc ( ) Dhlad exafanðzontai tautìqrona ta ginìmena UpenjÔmish apì GRAMMIKH II: DIAGWNIOPOIHSH 'Estw M summetrikìc pðnakac pragmatik n suntelest n tìte o M eðnai ìmoioc me ènan diag nio pðnaka pou sthn diag nio upˆrqoun oi idiotimèc pou eðnai PRAGMATIKES λ 1 0 0 0 K 1 0 λ 0 0 MK = λ i R, λ i 0 0 0 0 λ n Epiplèon o K mporeð na epilegeð ste na eðnai ORJOGWNIOS SUMPERASMA An det(a) 0 upˆrqei mða allag suntetagmènwn X = P Ψ + H pou sto nèo sôsthma h (E) ja paðrnei th morf αx + βψ + γz + δ = 0, α, β, γ 0 αx + βψ + γz = δ

19 δ = 0 αx + βψ + γz = 0, α, β, γ 0 KWNOS (pragmatikìc fantastikìc) δ 0 x ( δ ) + ψ ( δ ) + z ( δ ) = 1 α β γ ELLEIYOEIDES UPERBOLOEIDES monìqwno, dðqwno pragmatikì fantastikì anˆloga me ta prìshma twn δ, δ, δ α β γ h PERIPTWSH: det(a) = 0 den upˆrqei KENTRO SUMMETRIAS Upˆrqei prwtobˆjmioc ìroc (Den mporeð na gðnei metaforˆ) MporeÐ na gðnei STROFH (o pðnakac diagwniopoieðtai allˆ upˆrqoun mhdenikˆ sthn diag nio). αx + βψ + γz + δx + εψ + jz + n = 0 kˆpoio apì ta α, β, γ eðnai mhdèn. 'Estw ìti upˆrqei èna mhdenikì p.q. γ = 0 αx + βψ + γz + δx + εψ + jz + n = 0 α, β 0 me katˆllhlh metaforˆ: kai me ˆllh metaforˆ α x + β ψ + jz + n }} = 0 α x + β ψ + j z = 0 ELLEIPTIKOI KULINDROI UPERBOLOKOI KULINDROI an j = 0. ELLEIPTIKO UPERBOLIKO PARABOLOEIDES an j 0. PARADEIGMA x 3 + ψ 4 = 1 z = 0 u(0, 0, 1) x 3 + ψ 4 = 1 KULINDROS.

0 JEMA 1 α) An a, b 0 kai isqôei a + b = a + b na deðxete ìti a, b eðnai suggrammikˆ. APODEIXH: a + b = a + b + a b < a + b, a + b >= a + b + a b < a, b >= a b < a, a > + < b, b > + < a, b >= a + b + a b 1 oc TROPOS: < a, b >= a b cos( a, b) 0 a, b cos( a, b) = 1 a, b suggramikˆ oc TROPOS: ANALUTIKH EKFRASH a = (a 1, a ) b = (b1, b ) a 1 b 1 + a b = (a 1 b a b 1 ) = 0 tìte < a, b >= a 1 b 1 + a b a 1 + a b 1 + b = a 1b 1 + a b + a 1 b 1 a b = ( a 1b 1 + a 1b + a b 1 + a b ) a 1 b = a b 1 0 a, b p.q. b 1 0 a 1 = a = λ a 1 = λb 1 (a 1, a ) = λ(b 1, b ) b 1 b a = λb < a, b > a b ANISOTHTA CAUCHY-SWARTZ a + λ b φ(λ) = a + λ b 0 φ(λ) = 0 a + λ b = 0 a, b suggrammikˆ. β) a, b, c R 3 kai isqôei < a b, c >= 0 tìte ta a, b, c eðnai sunepðpeda. SWSTO LA- JOS. APANTHSH: eðnai SWSTO. APODEIXH (A' TROPOS): < a b, c > = V ( a, b, c) = 0 a, b, c sunepðpeda ˆra den sqhmatðzetai parallhlepðpedo. APODEIXH (B' TROPOS): det a, b, c} = 0 a, b, c grammikˆ exarthmèna. An a, b, c 0 kai λ 1 a + λ b + λ3 c = 0

1 (λ 1, λ, λ 3 ) (0, 0, 0) tìte gia λ 1 0 eðnai a = ( λ λ 1 ) b λ 3 λ 1 ) c a = ρ b + κ c a, b, c sunepðpeda. An a = 0 tìte a = 0 b + 0 c. γ) Sto sun jh q ro jewroôme to epðpedo (Π) : x ψ + z = 0 kai to shmeðo M : OM(1, 1, 1). Na brejoôn shmeða N tou epipèdou (Π) tètoia ste OM ON kai ON = 1. LUSH: 'Estw ON(x, ψ, z) tìte N (Π) kai isqôei (a) x ψ + z = 0 (b) x + ψ + z = 0 (a) x ψ + z = 0, (b) < OM, ON >= x + ψ + z = 0, (c) ON = 1 x + ψ + z = 1. } ( ) ψ = 0 z = x x = z giatð N (Π); (giatð OM ON); tìte (x, ψ, z) = λ(1, 0, 1). () 'Ara N = (x, ψ, z) R 3 (x, ψ, z) = λ(1, 0, 1)}. (c) ON = 1 λ(1, 0, 1) = 1 λ = ±. 'Ara (, 0, ), (, 0, ) Dhlad upˆrqoun shmeða tou epipèdou pou ikanopoioôn tic sunj kec tou erwt matoc. δ) Sto R 3 kai sto Oxψz} jewroôme thn apeikìnish f : (x, ψ) f(x, ψ) = ( ( x + κψ), kai κλ = 3. Na upologistoôn ta κ, λ ste: 4 f(x, ψ) = (x, ψ) x, ψ R (λx + ψ) ), κ, λ R LUSH: f(x, ψ) = 1 4 ( x + κψ) + 1 4 (λx + ψ) = 1 4 [x + 4κ ψ + 4λ + ψ 4κxψ + ψ + 4λxψ] = = (1 + 4λ ) 4 x + (1 + 4κ ) ψ + (λ κ)xψ x, ψ 4

(x, ψ) = x + ψ Prèpei LÔnontac to sôsthma prokôptei ìti (1 + 4λ ) = 1 4 (1 + 4κ ) = 1 4 λ = κ κ = λ = 3 3 κ = λ = giatð κλ = 3 4. JEMA DÐnetai h exðswsh tou R 3 sto sôsthma Oxψz} (Σ) : x + ψ + z 8x + 6z 11 = 0 α) Na apodeiqjeð ìti h (Σ) paristˆnei sfaðra kai na prosdioristeð to kèntro kai h aktðna Kèntro: (4, 0, 3) R = 6. (x 4) 16 + (ψ ) + (z + 3) 9 11 = 0 (x 4) + ψ + (z + 3) 36 = 0 β) DÐnetai to epðpedo (Π) tou q rou (Π) : x ψ + z 5 = 0. (i) Na apodeiqjeð ìti (Π) (Σ) = kôkloc kai na prosdioristeð to kèntro kai h aktðna tou. ArkeÐ na deiqjeð ìti upˆrqei A (Π) : M [(Π) (Σ)] AM = stajerì Tìte sumperaðnoume ìti [(Π) (Σ)] = (C) : kôkloc. An up rqe autì to shmeðo A tìte KA = l (Π) l(, 1, ) (ε) : (eujeða apì K) (Π) x 4 (ε) : = ψ 0 1 = z + 3 'Ara A = (Π) (E). (1) Jètoume t touc Ðsouc lìgouc sthn (1) dhlad x 4 = ψ 0 1 = z + 3 = t opìte: x = t + 4 ψ = t z = t + 3

3 AntikajistoÔme sthn exðswsh tou (Π) : x + ψ + z + 5 = 0 kai prokôptei: (t + 4) ( t) + (t + 3) + 5 = 0 9t = 7 t = 7 9 opìte x = 9, ψ = 7 9, z = 41 9 A(x, ψ, z) = ( 9, 7 9, 41 9 ) KENTRO tou KUKLOU KA = d(k, Π) 1 = ( 4) ψ(0) + ( 3) 5 + 1 + KA = 3 'Ara R = (aktðna sfaðrac) KA R = 36 9 R = 7 'Ara h tom (Π) (Σ) : C kôkloc me K( ), R = 7 (ii) (Σ ) omìkentrh thc (Σ) efaptìmenh sto (Π) K(4, 0, 3) kai R = d(k, Π) = 3. γ) A( 1, 7, 1), B(9, 7, 5). Na brejeð o G.T. twn shmeðwn M tou q rou pou blèpoun to eujôgrammo tm ma me ˆkra ta shmeða A kai B me orj gwnða (dhlad ÂMB = 90 o ). 1 oc TROPOS EPILUSHS: GEWMETRIKOS K MESON tou AB K(4, 0, ) R = 1 AB = 1 1 100 + 8 + 36 = 164 (x 4) + ψ + (z ) = 41 oc TROPOS < MA, MB >= 0. 1 d(p, Π) = Ax 0 + Bψ 0 + Γz 0 + A + B + Γ