Shmei seic sto mˆjhma Analutik GewmetrÐa

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

A' MEROS

3 Eisagwg Suntetagmènwn H perðptwsh tou epipèdou (E) E epðpedo thc EukleÐdiac Gewmètriac me thn eureða ènnoia (ta shmeða miˆc eujeðac kai oi metaxô touc sqèseic odhgoôn se ta twn meletˆme tic idiìthtec), E 1 1 R dhlad M E (x, ψ) R. epð O R èqei dom dianusmatikoô q rou me diˆstash dimr =. Sto epðpedo E den orðzetai prìsjesh shmeðwn me profan trìpo. Autì pou sundèei ton R me ton E eðnai h ènnoia tou dianôsmatoc. H ènnoia efarmostì diˆnusma = diatetagmèno zeôgoc (A, B),ìpou A, B E. JewroÔme mða sqèsh metaxô twn zeug n (A, B) (Γ, ) ta eujôgramma tm mata AB kai Γ eðnai Ðsou m kouc parˆllhla kai omìrropa me ˆkra sto Ðdio hmiepðpedo() Isqurismìc. H parapˆnw sqèsh metaxô twn zeug n (A, B) (Γ, ) eðnai sqèsh isodunamðac, dhlad eðnai anaklastik (A, B) (A, B) summetrik (A, B) (Γ, ) (Γ, ) (A B) } (A, B) (Γ, ) metabatik (Γ, ) (A, B) 'Askhsh: Na apodeiqjeð o isqurismìc. EleÔjero Diˆnusma. [(A, B)] = {(Γ, ) (A, B) (Γ, )} gia ìla ta (Γ, ) pou eðnai Ðdiou mètrou, parˆllhla, omìrropa me to (A, B). [(A, B)] [(K, Λ)] [(A, B)] = [(K, Λ)] 'Estw ω [(A, B)] kai ω = (P, Σ) tìte ω = (P, Σ) (P, Σ) (A, B) (P, Σ) (K, Λ)

4 Apì thn opoða sunepˆgetai, (K, Λ) (A, B) [(K, Λ)] = [(A, B)] StoiqeÐa Ðdiac klˆshc ω SumbolÐzoume ìla ta stoiqeða Ðdiac klˆshc me èna diˆnusma. SÔnolo klˆsewn: D sônolo phlðkou Ta eleôjera dianôsmata den èqoun shmeðo efarmog c se antðjesh me ta efarmostˆ dianôsmata pou èqoun shmeðo efarmog c. Apì kˆje shmeðo xekinˆei èna mìno eleôjero diˆnusma pou an kei se miˆ sugkekrimènh klˆsh. Dhlad, an dojeð shmeðo O tou epipèdou tìte: ω D!(akrib c èna) shmeðo M E ètsi ste [(O, M)] = ω. paðrnw antipros pouc me koin arq kai orðzoume, D 0 D 0 : ìla ta dianôsmata thc klˆshc ω pou xekinoôn apì to O. Apeikìnish φ 0 : D D 0 ìpou φ 0 : 1 1 kai epð φ 0 (M) = OM = ω D 0. R : èqei dom dianusmatikoô q rou diˆstashc, dimr = SÔndesh D 0 me ton R Sto D 0 orðzontai oi prˆxeic: (+) : D 0 D 0 D 0 prìsjesh, ( ) : R D 0 D 0 shmeiakìc pollaplasiasmìc. Prìsjesh Kanìnac Parallhlogrˆmmou Pollaplasiasmìc (prˆxh pol/smou sto D) [(O, K)] = z D 0 ω + σ = z prˆxh prìsjeshc sto D (OΛ) = λ(om) λ > 0, omìrropo λ < 0, antðrropo Prìtash: To D 0 eðnai dianusmatikìc q roc Apìdeixh: (deðqnoume ìti oi prˆxeic thc prìsjeshc (+) kai tou pollaplasiasmoô ( ) èqoun ìlec tic idiìthtec tou dianusmatikoô q rou)

5 p.q. a + ( β + γ) = ( a + β) + γ kataskeuastikˆ me kanìna parallhlogrˆmmou. 'Eqoume touc dianusmatikoôc q rouc: D 0, R kai jèloume na touc susqetðsoume, dhlad na broôme mða apeikìnish metaxô twn sunìlwn R = R R pragmatikoð arijmoð, kai D 0 gewmetrða. H epituq c sôndesh ja eisagˆgei arijmoôc sthn GewmetrÐa. UpenjÔmish apì GewmetrÐa H gewmetrik eujeða eðnai se 1 1 antistoiqða me to R. Oi eujeðec eðnai ˆxonec me prosanatolismì. M λ : (OM) = λ(oa), λ R èqei prìshmo, λ, N : λ (OA). 'Eqoume ta E, O, D 0. Epilègoume dôo temnìmenec eujeðec tou epipèdou pou dièrqontai apì to O kai tic tic kajist 'AXONES stouc opoðouc orðzw tic monˆdec i, j. {O, i, j} Autì mou epitrèpei na kataskeuˆsw apeikìnish Ψ : D 0 R. 'Estw ω D 0 kai M : [(OM)] = ω OM = ω. parallhlogrˆmmou. BrÐskw Γ ϵ 1 kai ϵ ètsi ste: OM = OΓ + O OΓ = x i me x, ψ R prˆxeic sto D 0 O = ψ j Dhlad OM = x i + ψ j D 0 ω ψ (x, ψ) R AnalÔw to OM me bˆsh ton kanìna tou H diadikasða Ψ eðnai 1 1 kai epð". H epilog thc arq c kai h epilog twn axìnwn eðnai aparaðthtec gia thn eisagwg suntetagmènwn (ìtan allˆzoun oi ˆxonec allˆzoun kai oi suntetagmènec). D 0 : Dianusmatikìc q roc (prˆxeic dianusmˆtwn) R : Dianusmatikìc q roc (prˆxeic zeugari n) ψ(0, i, j) = ψ : D 0 R H apeikìnish eðnai grammikìc isomorfismìc EÐnai 1 1 kai epð kai GRAMMIKH.

6 GRAMMIKH { Prìtash Oi suntetagmènec tou ajroðsmatoc (dianusmˆtwn) eðnai to ˆjroisma twn suntetagmènwn (zeugari n) D 0 D 0 ψ ψ R R D 0 + ψ R ìpou ekfrˆzei ˆjroisma dianusmˆtwn kai + ekfrˆzei ˆjroisma suntetagmènwn. GRAMMIKH: ψ( u w) = ψ( u) + ψ( w) APODEIXH: me kanìna parallhlogrˆmmou. AfoÔ D 0 R kai D 0 D E tìte prokôptei E R. E : EPIPEDO ENNOIA EFARMOSTOU DIANUSMATOS - KLASEIS DIANUSMATWN SUNOLO D ELEUJERWN DIANUSMATWN SUNOLO D 0 KLASEWN DIANUSMATWN ME KOINH ARQH APEIKONISH φ 0 : D D 0 me φ 0 : 1 1 kai epð To D 0 eðnai DIANUSMATIKOS QWROS APEIKONISH ψ : D 0 R me ψ : GRAMMIKOS ISOMORFISMOS E ènnoia dianôsmatoc D φ 0 D 0 ψ R

7 ApodeÐxame ìti D 0 = R apì ìpou prokôptei dimd 0 = GEWMETRIKH ERMHNEIA (Orismìc Grammik n Anexart twn Dianusmˆtwn) 'Estw u 0 tìte u grammikì anèxˆrthto. Prìtash 'Estw u, v grammikˆ exarthmèna dianôsmata, tìte ta u, v eðnai suneujeiakˆ. Apìdeixh: Upˆrqoun λ, µ R ìqi kai ta dôo mhdèn tètoia ste λ u + µ v = 0. 'Estw λ 0 tìte u = ( µ ) v u = ρ v, ρ R. λ Apì ton orismì tou pollaplasiasmoô prokôptei ìti ta dianôsmata u, v èqoun koinì forèa. AfoÔ èqoun koinì forèa kai koin arq eðnai sthn Ðdia eujeða. Prìtash: TrÐa dianôsmata sto epðpedo eðnai pˆntote grammikˆ exarthmèna Apìdeixh. (ErmhneÐa-Dikaiolìghsh) 'Estw ta dianôsmata u, v, w D 0 D. 1. An kˆpoio apì autˆ eðnai mhdèn, p.q. to u tìte 1 u + 0 v + 0 w = 0 kai epeid 1 0 èpetai ìti ta u, v, w eðnai exarthmèna.. An kanèna apì autˆ den eðnai mhdèn, tìte: w = w 1 + w, w 1 = λ u, w = µ v. 'Ara ta u, v, w eðnai grammikˆ exarthmèna. Perigraf Gewmetrik n Antikeimènwn mèsw thc taôtishc D 0 dianusmatik morf = R analutik morf (suntetagmènec) Perigraf twn Gewmetrik n Antikeimènwn shmeðou kai eujeðac sta D 0, D. Epilog thc KOINHS ARQHS To shmeðo M perigrˆfetai mèsw tou dianôsmatoc OM M E OM

8 To eujôgrammo tm ma twn shmeðwn M, N E perigrˆfetai M N : prosanatolismèno tm ma (eujôgrammo) Perigraf sto R D MN = OM + ON D 0 To shmeðo M : OM = i x M + j ψ M ìpou x M,ψ M monadikˆ M (x M, ψ M ) To eujôgrammo tm ma MN : MN (x N x M, ψ N ψ M ) 'Askhsh: EÔresh mèsou (Perigraf tou mèsou gewmetrikˆ kai analutikˆ). 'Estw ta shmeða M, N kai to MN eujôgrammo tm ma pou orðzetai apì ta shmeða, kai èstw K to mèson Apìdeixh: Apì thn idiìthta tou mèsou,, EujeÐa sto EpÐpedo x K = x M + x N, ψ K = ψ M + ψ N OK = OM+ ON, u K = u M + u N MK = KN tìte MO + OK = KO + OK = OM + ON OK = ON OM + ON Oi eujeðec den eðnai pˆnta UPOQWROI. Gewmetrikˆ h eujeða kajorðzetai ( ): Klˆsh parallhlðac: DieÔjunsh r M = r A + t u, t R Dianusmatik exðswsh thc eujeðac. Analutik exðswsh thc eujeðac 0, i, j eðte apì dôo shmeða thc eðte apì èna shmeðo kai ˆllh mða eujeða parˆllhlh proc aut n (dhlad thn ( ) klˆsh parallhlðac). OM = OA + AM eujeða AM = t AB t u, t R OM = OA + t u. DÐnontai A OA = r A diˆnusma jèshc, M OM = r M

9 A (x A, ψ A ) u = (α, β) M = (x, ψ) (x, ψ) = (x A, ψ A ) + t(α, β) (x, ψ) = (x A + tα, ψ A + tβ) { SÔsthma exis sewn thc eujeðac me x = x A + tα 'Ara ψ = ψ A + tβ me t R PARAMETRO t (Parametrikèc exis seic) APALOIFH thc PARAMETROU } 1 (x A x) + tα = 0 omogenèc grammikì sôsthma me (1, t) (0, 0) 1 (ψ A ψ) + tβ = 0 èqei ˆpeirec lôseic kai orðzousa mhdèn. x A x α 'Agnwstoc eðnai to t, ψ A ψ β = 0 (x x A)β (ψ ψ A )α = 0 βx αψ (βx A αψ A ) = 0 Morf : Ax + Bψ + Γ = 0, A + B = 0. PROBLHMA: EUJEIES SUNTREQOUSES (ε 1 ), (ε ), (ε 3 ) sunj kh ste na dièrqontai apì to IDIO SHMEIO (ε 1 ) :A 1 x + B 1 ψ + Γ 1 = 0 (ε ) :A x + B ψ + Γ = 0 (ε 3 ) :A 3 x + B 3 ψ + Γ 3 = 0 An dièrqontai apì to prèpei na eðnai lôsh twn dôo pr twn kai na ikanopoieð to trðto. Pìte lème ìti to SUSTHMA eðnai SUMBIBASTO? SUNJHKH: A 1 B 1 Γ 1 A B Γ = 0 A 3 B 3 Γ 3 'Askhsh: Na apodeiqjeð ìti oi Diˆmesoi enìc trig nou dièrqontai apì to Ðdio shmeðo (ME DIANUSMATA: GEWMETRIKA) ANALUTIKA: sto R LUSH: SumbolÐzoume me µ α, µ β, µ γ tic diamèsouc apì tic korufèc Â, ˆB, ˆΓ antðstoiqa. 'Eqoume, AM 1 = ( x B + x Γ x A, ψ B + ψ Γ µ α r A + t AM 1 r = (x A, ψ A ) + t ( x B + x Γ x A ψ A ) = (x B + x Γ x A, x B + x Γ x A, ψ B + ψ Γ ψ A ) )

10 Morf OrÐzousac: x A x µ α x B + x Γ x A OmoÐwc brðskoume tic orðzousec gia µ β, µ γ. ψ A ψ ψ B + ψ Γ ψ A = 0 ALLH LUSH: manteôoume to shmeðo tom c twn dôo diamèswn kai deðqnoume ìti apì ekeð pernˆei kai h trðth diˆmesoc.

11 GRAMMIKH ANEXARTHSIA DIANUSMATWN ORISMOS TO MONHRES DIANUSMA u EINAI GRAMMIKO ANEXARTHTA PROTASH 1: AN u, w GRAMMIKA EXARTHMENA TOTE u, w SUNEUJEIAKA PROTASH : TRIA DIANUSMATA STO EPIPEDO EINAI GRAMMIKA EXARTHMENA PERIGRAFH GEWMETRIKWN ANTIKEIMENWN GEWMETRIKA KAI ANALUTIKA SHMEIO TOU EPIPEDOU EUJUGRAMMO TMHMA MESON EUJUGRAMMOU TMHMATOS EUJEIA STO EPIPEDO SUNTREQOUSES EUEJEIES PROTASH: OI DIAMESOI ENOS TRIGWNOU DIERQONTAI APO TO IDIO SHMEIO

1 Plagiog nio sôsthma suntetagmènwn ψ{0, i, j} ALLAGH SUSTHMATWN SUNTETAGMENWN Pwc sqetðzontai dôo sust mata {0, i, j} {0, i, j } EQOUN IDIA ARQH {0, i, j} {0, i, j } DIAFEROUN OI ARQES SUSTHMATA ME IDIA ARQH Ekfrˆzw ta i, j wc proc i, j i = α i + β j j = γ i + δ j α β me α, β, γ, δ R kai γ δ 0 An αδ βγ = 0 kai α 0 tìte δ = βγ α. 'Ara j = γ i + βγ α j j = γ α (α i + β j) = λ i, dhlad ta j kai i ja tan grammikˆ suggramikˆ { } [ ] { } i α β i = j γ δ j M = [ α γ ] β ALLAZEI TH BASH. δ P c allˆzoun oi suntetagmènec 'Estw A(x A, ψ A ) wc proc { i, j} kai A(x A, ψ A ) wc proc { i, j }, tìte OA = x i + ψ j = x i + ψ j OA = x (α i + β j) + ψ (γ i + δ j) OA = (x α + ψ γ) i + (x β + ψ δ) j = x i + ψ j ProkÔptei ìti: x = x α + ψ γ ψ = x β + ψ δ [ ] x = ψ [ α β ] [ γ δ x ψ ]

13 [ ] x = ψ [ x ψ [ α β ] t [ γ δ x ψ ] ] [ ] = (M 1 ) t x ψ giatð detm 0 ˆra M 1. [ ] [ x = M t ψ x ψ ] Kˆje antistrèyimoc pðnakac orðzei mða allag (plagiog niwn) suntetagmènwn. PWS SUNDEONTAI DUO SUSTHMATA me DIAFORETIKH ARQH i = OO + α i + β j j = OO + γ i + δ j { } [ i = j x 0 ψ 0 ] x = x 0 + αx + γψ ψ = ψ 0 + βx + δψ + [ α β γ δ ] { } i j PROBLHMA: Na anagnwrisjeð h gramm pou perigrˆfetai apì thn exðswsh κx + λψ + µ = 0, κ + λ = 0. (Ta x, ψ eðnai suntetagmènec wc proc èna sôsthma). Prìkeitai gia eujeða parˆllhlh sto diˆnusma ( λ, κ) sto sôsthma (O, x, ψ). An allˆxoume sôsthma h MORFH [ ] exðswshc den allˆzei, [ ] allˆzoun [ ] ìmwc oi suntelestèc. Allˆzoume [ sôsthma, ] tìte: x x x x (κ, λ) + µ = 0 ìpou = T ψ ψ ψ me T pðnaka. 'Ara (κ, λ)t = 0 ψ [ ] [(κ, λ)t ] x ψ + µ = 0 (1) Gia na ( ) eðnai h (1) eujeða prèpei (κ, λ)t (0, 0). AfoÔ T ANTISTREYIMOS eðnai diˆforoc 0 tou kai epeid κ + λ = 0 èpetai ìti kai (κ, λ) (0, 0). 'Ara (κ, λ)t (0, 0). An eðqame 0 kai ALLAGH ARQHS tìte [ ] [ ] [ ] x x 0 x = + T ψ ψ opìte [(κ, λ)t ] ìpou jètontac (κ, λ ) = (κ, λ)t kai µ = [ [ x ψ ] [ ψ 0 + [ x 0 ψ 0 ] [ x 0 ψ 0 ] (κ, λ) + µ] (κ, λ) + µ] paðrnoume (κ, λ ) Oi GEWMETRIKES ENNOIES eðnai ANEXARTHTES apì to SUSTHMA SUNTETAGMENWN [ x ψ ] +µ. p.q. MESON EUJUGRAMMOU TMHMATOS

14 AM = MB M : mèson AB x M = x A + x B, ψ M = ψ A + ψ B (x A, ψ A ) (x A, ψ A ) (x M, ψ M ) (x M, ψ M ) p.q. PARALLHLES EUJEIES To akìloujo sôsthma eðnai ADUNATO { κx + λψ + µ = 0 κ x + λ ψ + µ = 0 κ λ Prèpei det κ λ = 0 AX = B A A deta = 0 A T A ( ) ( ) ( ) κ λ x µ = κ λ ψ µ H ENNOIA TRIGWNO A, B, Γ mh suneujeiakˆ shmeða { AB, AΓ} GRAMMIKA ANEXARTHTA {0, i, j}, A(x A, ψ A ), B(x B, ψ B ), Γ(x Γ, ψ Γ ) AB (x B x A, ψ B ψ A ) AΓ (x Γ x A, ψ Γ ψ A ) { AB AΓ} x B x A ψ B ψ A x Γ x A ψ Γ ψ A 0 x A ψ A 1 x B ψ B 1 0 x Γ ψ Γ 1 To trðgwno ABΓ odhgeð sto { AB, AΓ} {A, AB, AΓ} Se autì to sôsthma to AB eðnai to (1, 0) kai to AΓ eðnai to (0, 1) {A, AB, AΓ} {0, i, j} ASKHSH: OI DIAMESOI enìc TRIGWNOU SUNTREQOUN LUSH: qwrðc blˆbh thc genikìthtac mpor na jèsw A(0, 0), B(1, 0), Γ(0, 1)

15 µ α µ β µ γ K(x K, ψ K ), Λ(x Λ, ψ Λ ) x x K x Λ x K = 0 ψ ψ K ψ Λ ψ K 1 x 0 µ α 0 ψ 0 1 0 = 0 x ψ = 0 x 1 0 1 µ β 1 ψ 0 = 0 ψ = 1 x + 1 1 x 0 µ γ 0 ψ 0 0 1 = 0 LUSH tou SUSTHMATOS: x = ψ. PROBLHMA ArqÐzw me èna trðgwno ABΓ. Fèrw parˆl- lhlh proc th bˆsh thn eujeða ε. Jewr M ε to shmeðo tom c twn diagwnðwn tou TRAPE- ZIOU pou prokôptei. ZhteÐtai o Gewmetrikìc Tìtpoc tou shmeðou M ε. 'Estw ABΓ orjwg nio kai isoskelèc trðgwno, tìte o gewmetrikìc tìpoc eðnai Ôyoc, diˆmesoc, diqotìmoc. Allˆzw sôsthma apoorjogwnio se PLAGIOGWNIO. {A, AB, AΓ} {0, i, j} plagiog nio diathroôntai ìlec oi ènnoiec: EUJ. TMHMA, MESON, TRIGWNO. 'Ara o G.T. eðnai EUJEIA. Ja eðnai to Ôyoc h diˆmesoc h diqotìmoc. Upoqrewtikˆ eðnai h DIAMESOS. PARATHRHSH: sto plagiog nio sôsthma den èqoume MHKH kai GWNIES OdhgoÔmaste se èidikˆ sust mata suntetagmènwn: ta ORJOKANONIKA. ORJOKANONIKO SUSTHMA SUNTETAGMENWN z = x + ψ z = x + ψ (AB) = (OA) + (OB) Jewr ta sust mata {0, i, j} ìpou i j me i = j = 1.

16 M OM = x i + ψ j, OM = x + ψ A(x A, ψ A ) kai B(x B, ψ B ), AB = (x B x A ) + (ψ B ψ A ). SUSQETISMOS ORJOKANONIKWN SUSTHMATWN SUNTETAGMENWN ARQH Anˆlush probl matoc {0, i, j}, {0, i, j } tìte { } i = T j { } i j, T R me det(t ) 0. AfoÔ ta sust mata eðnai eidikoô tôpou to T DESMEUETAI kai den mporeð na eðnai tuqaðo. Ta pˆnta kajorðzontai apì th gwnða φ.

17 ALLAGH SUNTETAGMENWN sta PLAGIOGWNIA SUSTHMATA SUSTHMATA me KOINH ARQH p.q. {0, i, j} kai {0, i, j } SUSTHMATA me DIAFORETIKH ARQH p.q. {0, i, j} kai {0, i, j } PROTASH: OI GEWMETRIKES ENNOIES eðnai ANEXARTHTES tou SUSTHMATOS SUNTETAGMENWN EUJEIA MESON EUJUGRAMMOU TMHMATOS PARALLHLES EUJEIES TRIGWNO EFARMOGH: OI DIAMESOI TRIGWNOU SUNTREQOUN ORJOKANONIKO SUSTHMA POLIKES SUNTETAGMENES ALLAGH SUNTETSGMENWN sto ORJOKANONIKO SUSTHMA Sto PLAGIOGWNIO SUSTHMA h ALLAGH SUNTETAGMENWN DEN EPHERAZEI th GEWMETRIKH ENNOIA TA MHKH kai OI GWNIES OMWS DEN EINAI ANEXARTHTA.

18 ORJOKANONIKA SUSTHMATA EPIPEDOU {0, i, j} me i = j = 1, i j OM = x i + ψ j OM = x + ψ Pwc sqetðzontai dôo orjokanonikˆ sust mata me thn IDIA ARQH {0, i, j} {0, i, j } OM = x i + ψ j OM = x i + ψ j [ ] [ ] [ ] x x α β = P ψ ìpou P = ANTISTREYIMOS detp 0 ψ γ δ MORFH TOU PINAKA P OM = eðnai koinì kai sta dôo dianôsmata, 1 oc trìpoc: OM = x + ψ = (x ) + (ψ ) ( ) ( ) t ( ) x + ψ x x x = (x, ψ) = ψ ψ ψ ( ) t ( ) (x ) + (ψ ) = x ψ x ψ 'Omwc ProkÔptei: ( x ψ ) t ( x ψ ( ) ( ) ( ) t ( ) ( ) t x x x x = P = P t x ψ ψ ψ ψ ψ ) ( ) t ( ) ( ) t ( ) ( ) t ( ) x = P t x x P kai P t x x x P = ψ ψ ψ ψ ψ ψ Gia na isqôei h isìthta prèpei P t P = I. 'Enac tètoioc pðnakac onomˆzetai ORJOGWNIOS [ ] [ ] [ ] α γ α β 1 0 = β δ γ δ 0 1 oc trìpoc Prèpei x = αx + βψ ψ = γx + δψ α + γ = 1 β + δ = 1 αβ + γδ = 0 αδ βγ 0 IDIOTHTES TOU P } (x ) + (ψ ) = (αx + βψ) + (γx + δψ) = x + ψ

19 P t P = I det(p t P ) = 1 kai det(p t ) = det(p ) 'Ara (det(p )) = 1 det(p ) = ±1 P t = P, P 1 P = I, P P 1 = I PARADEIGMA 1 { i, j}, { i = i, j = j} i = 1 i + 0 j j = 0 i + 1 j ( ) ( ) 1 0 Tìte P = kai P t 1 0 = kai P = I, detp = 1 0 1 0 1 j i PARADEIGMA j i i = (cos φ) i + (sin φ) j j = cos(φ + π ) i + sin(φ + π ) j detp = 1 me P 1 = 'Ara P P t = I. [ cos φ sin φ ] sin φ cos φ = sin φ i + cos φ j [ ] [ ] [ ] x = P t x cos φ sin φ ψ ìpou P eðnai o pðnakac strof P = ψ sin φ cos φ TUPOS ALLAGHS SUNTELESTWN wc proc thn STROFH: PARADEIGMA 3: An T = [ cos φ sin φ x = cos θx + sin θψ ψ = sin θx + cos θψ ] sin φ tìte T t = cos φ [ cos φ sin φ ] [ ] sin φ, T T t 1 0 = cos φ 0 1

0 'Ara o PINAKAS T eðnai ORJOGWNIOS (AA t = I). 'Omwc èqei mða basik DIAFORA me ton P : det(t ) = 1. 'Ara DEN eðnai STROFH (ìloi oi orjog nioi pðnakec den eðnai STRO- FES). [ det(t ) = 1, T 1 = T. cos φ sin φ ] [ ] sin φ 1 0 = cos φ 0 1 [ cos φ sin φ strof,katoptrismìc ] sin φ = T cos φ ALLA SUSTHMATA [ ALLOI PINAKES ] cos θ sin θ PARATHRHSH: P (θ) = sin θ cos θ {P (θ) : θ R} = PINAKES STROFHS AFHRHMENH MELETH APODEIXH: {{P (θ) : θ R}, pollaplasiasmìc pinˆkwn} P (0) = I P ( θ) = ( P (θ) ) t = ( P (θ) ) 1 P ( θ) = P (θ) 1 P (θ 1 ) P (θ ) = P (θ 1 + θ ) ApoteleÐ mða PROSJETIKH } DOMH {{ OMADA }. TELIKO ERWTHMA: PoioÐ eðnai OLOI oi pðnakec Θ : Θ Θ t = I Prèpei na lôsoume to sôsthma α + γ = 1 β + δ = 1 (Σ) αβ + γδ = 0 αδ βγ 0 α, β, γ, δ R UPODEIXH: Xekinˆme apì thn 3 h kai lônoume me ˆgnwsto to β. PaÐrnoume peript seic gia ton suntelest α (α = 0, α 0) kai qwrðc blˆbh thc genikìthtac jewroôme α 0 opìte β = γ α δ. Jètoume sthn h ìpou β to Ðson tou. β + δ = 1 ( γ α δ) + δ = 1 α = δ ASKHSH 1 Sto R kai wc proc èna ORJOKANONIKO SUSTHMA Oxψ dðnetai to uposônolo C = {(x, ψ) R xψ = 1 } E = 1

1 Na brejeð h morf tou C wc proc to orjokanonikì sôsthma pou prokôptei apì to Oxψ me strof π 4. Zhtˆme C = {(x, ψ ) R?} x = cos π 4 x sin π 4 ψ = x ψ ψ = sin π 4 x + cos π 4 ψ = x + ψ Opìte 1 = xψ 1 = (x ψ )(x + ψ ) ( = (x ) (ψ ) ) (E ) : (x ) (ψ ) = 1 ISOSKELHS UPERBOLH C = {(x, ψ ) R (x ) (ψ ) = 1} PARATHRHSH: oi (E) kai (E ) eðnai ou BAJMOU. DIATHRHSH GWNIAS sta DIAFORA ORJOKANIKA SUSTHMATA. ASKHSH ( ψ = λ 1 x + µ ψ = λ x + µ ) Oxψ me lezˆnta Π = tan ω Π = λ λ 1 1 + λ 1 λ ìpou λ 1 λ 1. N.d.o. h Π DEN ALLZEI sta diˆfora orjokanonikˆ sust mata. GEWMETRIA EPIPEDOU 1. Anafèretai se orjokanonikˆ sust mata. Meletˆei idiìthtec pou isqôoun se OLA ta orjokanonikˆ sust mata LUKEIO/GEWMETRIA meletˆei m kh, gwnðec ANALUTIKH GEWMETRIA twn ORJOKANONIKWN SUSTHMATWN GEWMETRIA sto QWRO q roc (shmeða, eujeðec, epðpeda), (StereometrÐa: BiblÐo B LUKEIOU)

BASIKA AXIWMATA - 'Ola ta Axi mata tou Epipèdou isqôoun - Kˆje trða shmeða MH SUNEUJEIAKA orðzoun èna epðpedo - DÔo epðpeda pou èqoun èna koinì shmeðo èqoun kai mia koin eujeða - An mia EUJEIA èqei dôo koinˆ shmeða me èna epðpedo ìla thc ta shmeða eðnai to epðpedo - Sto q ro upˆrqoun 4 SHMEIA MH SUNEUJEIAKA

3 BASIKES ENNOIES sto QWRO SHMEIO-EUJEIA-EPIPEDO kai oi METAXU TOUS SQESEIS upˆrqoun toulˆqiston 4 mh sunepðpeda shmeða TrÐa MH SUNEUJEIAKA shmeða orðzoun akrib c èna epðpedo PORISMA Mia eujeða kai èna shmeðo ektìc aut c orðzoun èna epðpedo DÔo temnìmenec eujeðec orðzoun èna epipedo DÔo epðpeda me trða koinˆ shmeða tautðzontai. SQETIKH JESH DUO EPIPEDWN 1. Na mhn èqoun kanèna koinì shmeðo.. Na èqoun èna koinì shmeðo opìte èqoun mða koin eujeða () 3. Na sumpðptoun SQETIKH JESH DUO EUJEIWN STO QWRO 1. Na eðnai parˆllhlec, opìte brðskontai sto IDIO EPIPEDO. Na èqoun èna koinì shmeðo opìte orðzoun èna epðpedo () 3. AsÔmbatec eujeðec (oôte parˆllhlec, oôte temnìmenec, den upˆrqei epðpedo pou na tic perièqei). APODEIXH thc UPARXHS ASUMBATWN EUJEIWN 'Estw trða shmeða A, B, Γ Π kai / Π (Ta A, B, Γ eðnai MH SUNEJEIAKA) Ta B, Γ orðzoun mða (ε 1 ) Ta, A orðzoun mða (ε ). Oi (ε 1 ) kai (ε ) eðnai ASUMBATES giatð ta A, B, Γ, eðnai MH SUNEPIPEDA. KAJETOTHTA EUJEIAS kai EPIPEDOU (ε) (Π) ORISMOS A: Ðqnoc thc (ε) sto (Π) ( o pouc thc (ε) sto (Π)) Apì to A dièrqontai 1,, 3 eujeðec tou (Π) Lème tìte ìti (ε) (Π) (ε) (δ) (δ) eujeða tou (Π) pou dièrqetai apì to Ðqnoc A

4 (H (ε) me thn 1 dhmiourgoôn èna nèo epðpedo (Π 1 )). ArkeÐ na elegqjoôn mìno dôo eujeðec tou (Π) pou pernˆne apì to Ðqnoc thc (ε), dhlad to shmeðo A. JEWRHMA: An (Π) (ε) = A kai δ 1, δ (Π) me δ 1 δ. (ε) (ε) (Π). A (δ 1 ) (δ ) kai δ 1, δ APODEIXH Jewr shmeðo M (ε) kai M (ε): AM = AM Upìdeixh (δ) tuqaða eujeða tou (Π) AB = AΓ (A BΓ) orðzetai shmeðo (δ). MΓM isoskelèc MBM isoskelèc ArkeÐ na deðxoume ìti to trðgwno M M eðnai isoskelèc. Efarmìzoume () eujèwc kai antistrìfwc (h diˆmesoc eðnai diqotìmoc kai Ôyoc) TRISORJOGWNIO SUSTHMA. JEWRHMA TRIWN KAJETWN (Π), (ε) (Π) (ε) = A (δ) (Π) AB (δ), AB (Π) M (ε) kai fèrw th MB. 1. An (ε) (Π) MB (δ). An (ε) (Π) kai MB (δ) AB (δ) 3. An MB (δ) kai AB (δ) (ε) (Π) N (δ) kai fèrw thn AN. (ε) (Π). Ta trðgwna (M AN), (M AB) eðnai orjog nia. To trðgwno (ABN) eðnai orjog nio. Prèpei na deðxoume ìti to trðgwno (MBN) eðnai orjog nio sth ˆB. Efarmìzoume PUJAGOREIO JEWRHMA. EISAGWGH OROLOGIA

5 ASUMBATWS KAJETES (Π), (ε) (Π) (δ) ( (Π) ) δ δ (ε) (δ) A δ (ε), (δ) (en gènh) ORJOGWNIES (ε 1 ), (ε ) orjog niec (ε 1 ) (ε 1 ) (ε ) (ε ) } (ε 1 ) (ε ) DIATUPWSH An (ε) (Π) tìte h (ε) eðnai orjog nia me kˆje eujeða tou epipèdou. An h (ε) eðnai ORJOGWNIA me dôo eujeðec enìc epipèdou (Π) (ε) (Π). ASKHSH: DÐnetai epðpedo (Π) kai shmeðo A / (Π). Na aqjeð apì to A eujeða (ε) (Π) Efarmìzoume to JEWRHMA twn TRIWN KAJETWN (δ) (Π) H (δ) kai to A orðzoun epðpedo Fèrnw AB (δ). Pˆnw sto (Π) fèrw (x) (δ) opìte h (x) kai h AB orðzoun èna epðpedo. Sto epðpedo pou orðzoun oi (x) AB fèrw AO (x). PARALLAGH To shmeðo A (Π) (µ) (Π) µ, A orðzoun epðpedo fèrw sto (ε) (µ) µ apì to A. PORISMA: Upˆrqoun TRISORJOGWNIA SUSTHMATA Kataskeu : ArqÐzw me èna epðpedo (Π) kai 0 (Π). Fèrw (ε 3 ) (Π) me 0 (ε 3. Jewr (ε 1 ) (Π) kai na dièrqetai apì to O. Fèrw (ε ) (ε 1 ) me (ε (Π). TRISORJOGWNIO SUSTHMA AXONWN ASKHSEIS-EFARMOGES

6 ASKHSH1 'Olec oi eujeðec tou q rou pou dièrqontai apì to O kai eðnai kˆjetec sthn eujeða (ε) brðskontai se èna epðpedo (Π) pou onomˆzetai kai eðnai KAJETO EPIPEDO sthn (ε) PORISMA: DÔo epðpeda kˆjeta sthn Ðdia eujeða den èqoun koinˆ shmeða 'Estw ìti èqoun koinì shmeðo K. jewr thn KO kai KO. To KOO ja eðnai trðgwno me orjèc gwnðec. ATOPO! KOINH KAJETOS duo ASUMBATWN p.q. o kôboc èqei anˆ dôo asômbatec me mða koin kˆjeto 'Estw x (ε ) kai fèrw (ε 1 ) (ε 1 ). Jewr O (ε 1 ) kai fèrw OO (Π). Fèrw (ε 1 ) (ε 1 ) ìpou (ε 1 ) an kei sto epðpedo pou perièqei thn (ε 1 ) kai eðnai kˆjeto sthn (ε ).

7 GEWMETRIA sto QWRO T sun jhc q roc (perièqei shmeða, eujeðec, epðpeda, tic metaxô touc sqèseic) T = R 3 H TAUTISH gðnetai mèsw tou TRISORJOGWNIOU SUSTHMATOS Epilègoume arq O {O, i, j, κ} Epilègoume AXONES Tìte to tuqaðo M T OM = x M i + ψ M j + z M κ ìpou M(x M, ψ M, z M ). JEWRHMA: Oi suntetagmènec tou ajroðsmatoc eðnai to ˆjroisma twn suntetagmènwn. i = j = κ = 1 Sto R 3 ja bˆloume DOMES 1. Eswterikì ginìmeno. Exwterikì ginìmeno Efìson R 3 = T ja prokôyei eswterikì exwterikì ginìmeno dianusmˆtwn. ESWTERIKO GINOMENO prˆxh metaxô dianusmˆtwn ìpou to apotèlesma eðnai pragmatikìc arijmìc. EÐnai mia APEIKONISH: <, >: R 3 R TUPOS thc <, > < (x 1, x, x 3 ), (ψ 1, ψ, ψ 3 ) >= (x 1 ψ 1 + x ψ + x 3 ψ 3 ) Onomˆzetai kai ANALUTIKH EKFRASH IDIOTHTES(ALGEBRIKES, GEWMETRIKES) thc APEIKONISHS ALGEBRIKES IDIOTHTES < α, β > R 1. < α, β >=< β, α > SUMMETRIKOTHTA. < α + γ, β >=< α, β > + < γ, β } > 3. < λ α, β >= λ < α, β DIGRAMMIKOTHTA > λ R 4. < α, α > 0 kai < α, α >= 0 α = 0 JETIKA ORISMENO APODEIXH: α(x, ψ, z) = x + ψ + z 0 kai x + ψ + z = 0 α = 0

8 APEIKONISH NORMA Me th bo jeia tou ESWTERIKOU GINOMENOU, orðzw mia APEIKONISH : R 3 R me tôpo α = < α, α > Thn APEIKONISH aut thn onomˆzw NORMA. afoô to < α, α > 0 α R 3 mpor na bˆlw rðza. IDIOTHTES 1. α 0, α = 0 α = 0. λ α = λ α, λ R APODEIXH: λ α =< λ α, λ α >= λ λ < α, α >= λ < α, α >= λ α 3. α + β} α + β (trigwnik anisìthta) metrˆei m kh dianusmˆtwn APODEIXH: jewr th sunˆrthsh φ(λ) = α + λβ IsqÔei φ(λ) 0 R AnaptÔssoume: φ(λ) =< α + λβ, α + λβ >=< α, α > +λ < α, β > +λ < β, β > φ(λ) = ( β )λ + ( < α, β >)λ + α. DiakrÐnoume peript seic: (i) β = 0 β = 0 ˆra h IDIOTHTA (3) isqôei (ii) β = 0 φ(λ) tri numo kai gðnetai pˆnta omìshmo tou ( β ) 4 0 dhl. 4(< α, β >) 4 α β 0 4 < α, β > α β majhmatikˆ sumpðptei me thn IDIOTHTA (3) IDIOT.(3): α + β ( α + β ) < α + β, α + β > α + β + α β < α, α > + < α, β > + < β, β > α + β + α β < α, β > α β < α, β > < α, β > α β < α, β > α β An α, β 0 tìte < α, β > α β 1 SumperaÐnoume : NORMA MHKOS DIANUSMATOS APOSTASH DUO SHMEIWN ston R 3 = T

9 'Estw M, N T tìte anazhtoôme d(m, N) Epilègoume to sôsthma {0, i, j, κ} M OM = α(x 1, ψ 1, z 1 ) N ON = β(x, ψ, z ) MN ON OM = (x x 1, ψ ψ 1, z z 1 ) d(m, N) = β α = (x x 1 ) + (ψ ψ 1 ) + (z z 1 ) H d(m, N) eðnai METRIKH 1. d(m, N) 0, d(m, N) = 0 M N. d(m, N) = d(n, M) 3. d(m, N) d(m, K) + d(k, N) APODEIXH: d(m, N) = β α kai èstw OK = (x 3, ψ 3, z 3 ) = γ d(m, N) = β α = β γ + γ α β γ + γ α = d(m, K) + d(k, N) 'Ara d(m, N) d(m, K) + d(k, N). ASKHSH 1. α + β + α β = [ α + β ]. < α, β >= 1 4 [ α + β α β ] LUSH 1 NOMOS PARALLHLOGRAMMOU α + β =< α + β, α + β >= < α, α > + < α, β > + < β, β >= α + < α, β > + β α β = α < α, β > + β TI SHMAINEI < α, β >= 0 < α, β >= 0 PUJAGOREIO JEWRHMA {}}{ α + β = α + β APODEIXH: α + β = α + β apì thn prohgoômenh ˆskhsh. 'Ara α, β ORJOGWNIA. GEWMETRIKH EKFRASH ESWTERIKOU GINOMENOU

30 Qreiazìmaste thn ènnoia gwnða PUJAGOREIO (AB) = (OA) +(OB) (OA)(OB) cos φ ìpou (AB) APOSTASH (AB) (d(a, B)) = (d(o, A)) + (d(o, B)) (d(o, A))(d(O, B)) cos φ 'Omwc (d(a, B)) = AB = OB OA (d(o, A)) = OA (d(o, B)) = OB ProkÔptei: OB OA }{{} = OA + OB OA OB cos φ OA + OB < OA, OB >= OA + OB OA OB cos φ < OA, OB >= OA OB cos φ GEWMETRIKH EKFRASH < OA, OB > OA, OB 0 cos φ = OA OB Sqìlio: < α, β > α β < α, β > α β 1 θ (0, π) : cos θ = < α, β > α β ìpou θ h gwnða twn dianôsmatwn α, β H seirˆ twn α, β den paðzei rìlo giatð cos θ = cos( θ).

31 (R 3, < >) α(x 1, ψ 1, z 1 ), β(x, ψ, z ) < α, β >= (x 1 x + ψ 1 ψ + z 1 z ) ORJES PROBOLES DIANUSMATOS se AXONA PROBLHMA: DÐnetai èna α R 3 kai β R 3. ZhteÐtai na grafeð to β sth morf β = β 1 + β ìpou β 1 α kai β eðnai ORJOGWNIO me to α. Anˆlush: (I) Gia thn kataskeu tou toc qrhsimopoðhsa to epðpedo pou orðzoun ta α, β (II) Gia α, β me α 0 β = β 1 + β β 1 = λ α < β, α >= 0 Gl ssa thc analutik c gewmetrðac α (I) α = α 0 dianusmatik monˆda pou orðzei to α α 1 = α = 1, NORMA MHKOS DIANUSMATOS ( METRO DIANUSMATOS) α α β 1 = β cos φ β 1 = β α cos φ (1), ìpou α dianusmatik monˆda pou orðzei to α α. GnwrÐzoume ìti: < α, β >= α β cos φ. 'Ara h sqèsh (1) grˆfetai β 1 = < α, β > α (II) eðnai ALGEBRIKO SUSTHMA sth DOMH (R 3, <, >). α, β α 0 β = λ α + β β = β λ α β = β 1 + β opìte β 1 = λ α, λ R < β, α >=< β λ α, α >= 0 < β, α >= 0 < α, β > < α, λ α >= 0 < α, β >= λ < α, α > Epeid α 0 < α, α > 0. 'Ara λ = < α, β > < α, α > λ = < α, β > α α

3 Tìte β 1 = < α, β > < α, α > α kai β = β < α, β > < α, α > α Exetˆzoume an upˆrqoun ˆllec lôseic An β = β 1 + β kai β = β 1 + β ìpou { β1, β 1 α β, β α Tìte β 1 β 1 }{{} w 1 α = β β }{{} w α UpologÐzoume to eswterikì ginìmeno: < w 1, w > < w 1, w >=< β 1 β 1, β β >= 0 β 1 β 1 = 0 β 1 = β 1 TO EXWTERIKO GINOMENO sto QWRO R 3 PROBLHMA: DÐnontai α, β mh suggrammikˆ stoiqeða tou R 3. ZhteÐtai c, c 0 kai na eðnai kˆjeto sto epðpedo twn α, β. EPIPEDO POU ORIZOUN duo MH SUGGRAMIKA DIANUSMATA (Π) = L({ α, β}) = {ρ α + σβ, ρ, σ R} eðnai GRAMMIKOS SUNDUAsq ma SMOS mh SUGGRAMIKWN DIANUSMATWN DIANUSMA se EPIPEDO (Π) an < c, α >=< c, β >= 0 c (Π). Prˆgmati tìte < c, ρ α + σ β >= ρ < c, α > +σ < c, β >= 0. ArkeÐ loipìn na eðnai ORJOGWNIO sta α, β. EpilÔoume to SUSTHMA < c, α >=< c, β >= 0. 'Estw α = (x 1, ψ 1, z 1 ) kai β = (x, ψ, z ) kai c = (x, ψ, z). } { < c, α >= 0 < c, β >= 0 α β x 1 x + ψ 1 ψ + z 1 z = 0 x, ψ, z =? x x + ψ ψ + z z = 0 [ ] x [ ] x 1 ψ 1 z 1 0 ψ = AX = B x ψ z 0 z ranka = giatð α β α, β grammikˆ anexˆrthta. UPOORIZOUSA D 0 x 1 ψ 1 x 1 ψ 0

33 Tìte dhmiourgoôme sôsthma me agn stouc. ( ) x 1 x + ψ 1 ψ = z 1 z ProkÔptei to diˆnusma c x x + ψ ψ = z z z 1 z ψ 1 D x = z z ψ x = D x D, ψ = D ψ D D ψ = x 1 x z 1 z z z z 1 ψ 1 x 1 z 1 D x = z D ψ = z z ψ x z D = x 1 ψ 1 x ψ ψ 1 z 1 x 1 z 1 ψ z x z x = x 1 ψ z ψ = 1 x 1 ψ z, z R 1 x ψ x ψ c = (x, ψ, z) c = ( D x D, D ψ D, z) ( ψ 1 z 1 c = κ ψ z, x 1 z 1 x z, x 1 ψ ) 1 x ψ Sumpèrasma: to diˆnusma c α, β eðnai to pollaplˆsio tou (D x, D ψ, D) ORISMOS Sto R 3 orðzetai mia EXWTERIKH PRAXH X : R 3 R 3 R 3 wc ex c ( α, β) ( α β) pou onomˆzetai exwterikì ginìmeno twn α, β kai an jewr sw ORJOKANONIKO SUSTHMA SUNTETAGMENWN { i, j, κ} ìpou α = a 1 i + a j + a 3 κ β = β 1 i + β j + β 3 κ tìte to α β = γ 1 i+γ j+γ 3 κ ìpou ta γ 1, γ, γ 3 prokôptoun apì to anˆptugma thc ORIZOUSAS i j κ α 1 α α 3 wc proc thn pr th gramm β 1 β β 3 α β = α α 3 β β 3 α 1 α 3 i β 1 β 3 j + α 1 α β 1 β κ γ 1 γ γ 3

34 i j = κ i j ìpou i(1, 0, 0) j = (0, 1, 0). 'Ara j κ = i κ i = j 'Ara α β = c kai sth jèsh pollaplasðou h monˆda. EXWTERIKO GINOMENO R 3 {0, i, j, κ} ORJOKANONIKO α = α 1 i + α j + α κ β = β 1 i + β j + β 3 κ tìte α β = i j κ α 1 α α 3 β 1 β β 3 α β = ( (α β 3 β α 3 ) i (α 3 β 1 α 1 β 3 ) j + (α 1 β α β 1 ) κ ) α β = α α 3 β β 3 α 1 α 3 i β 1 β 3 j + α 1 α β 1 β κ IDIOTHTES EXWTERIKOU GINOMENOU 1 h LISTA: ALGEBRIKES IDIOTHTES 1. α α = 0, α 0 = 0 α R. α β = β α (ANTISUMMETRIKH IDIOTHTA) 3. α ( β + c) = ( α β) + ( α c) 4. λ( α β) = (λ α β) = α (λ β) APODEIXEIS: α β i j κ = α 1 α α 3 β 1 β β 3 1. ìpou β to α orðzousa me Ðdiec grammèc=mhdenik. antimetˆjesh gramm n stic orðzousec 3. kai 4. ˆmmesa apì thn grammikìthta thc orðzousac PROSOQH: α β = ( β α) h seirˆ tou ginomènou paðzei rìlo....ja sundejeð me th forˆ diagraf c. h LISTA IDIOTHTWN: 1. < α β, c >= det{ α, β, c}. ( α β) c =< α, c > β < β, c > α

35 3. α β = α β ( α, β >) D 1 D D 3 1) α β α α 3 = β β 3 i α 1 α 3 β 1 β 3 α 1 α j + β 1 β κ, c = γ 1i + γ j + γ 3 κ < α β, c >= γ 1 D 1 i γ D j + γ 3 D 3 κ < α β, γ 1 γ γ 3 c >= α 1 α α 3 = det{ α, β, c} det{ α, β, c} β 1 β β 3 < α β, α 1 α α 3 c >= β 1 β β 3, opìte < α β, c >= < β α, c > γ 1 γ γ 3 p.q. < β c, α >= det{ β, c, α} = det{ β, α, c} = det{ α, β, c} ) ( α β) c =< α, c > β < β, c > α α = α 1 i + α j + α 3 κ ìpou β = β 1 i + β j + β 3 κ kˆnoume prˆxeic...kai pistopoðhsh. α = γ 1 i + γ j + γ 3 κ Prèpei na broôme se poio epðpedo brðsketai to ( α β) c to opoðo ja eðnai kˆjeto sto α β kai sto c. 3) α β =< α β, α β >=< α, β ( α β) >= < α, ( α β) β >= < α, < α, β > β < β, β > α >= < α, < α, β > β > + < α, < β, β > α >= < α, β >< α, β > + < α, α >< β, β > afoô < u v, w >=< u, v w >. 'Ara α β = α β (< α, β >) α β = α β α β cos φ = (1 cos φ) α β = sin φ α β, ìpou φ h gwnða pou sqhmatðzoun ta α, β. 'Ara, α β = α β sin φ (giatð φ [0, π]) GEWMETRIKH EKFRASH EXWTERIKOU GINOMENOU α, β tìte α β: 1. α β = α β sin φ (METRO). α β sto epðpedo twn α, β (DIEUJUNSH) 3. FORA: jewroômc ta sust mata { i, j, κ} 1 h BASH kai { α, β, α β} h BASH

36 PINAKAS ALLAGHS BASHS α 1 α α 3 det β 1 β β 3 > 0 γ 1 γ γ 3 OrÐzei ton Ðdio prosanatolismì me to { i, j, κ}. =< α β, α β > α β = α β (< α, β >) An α = α 1 i + α j + α 3 κ, β = β 1 i + β j + β 3 κ tìte α = α 1 + α + α 3 = 3 i=1 α i opìte EpÐshc EpÐshc α β = α β α α 3 = β β 3 β = < α, β >= 3 i=1 α i 3 i=1 β i 3 α i β i i=1 3 βi ( i=1 α 1 α 3 + β 1 β 3 3 α i β i ) i=1 α 1 α + β 1 β = i<j D ij i<j D ij = 3 3 i=1 α i i=1 β i ( 3 i=1 α iβ i ) 0 TAUTOTHTA Lagrange ProkÔptei ( α i )( β i ) ( α i β i ) α i β i α i β i ANISOTHTA CAUCHY < x, ψ > x ψ gia eswterikì ginìmeno EFARMOGH 1: GEWMETRIKH ERMHNEIA METROU EXWTERIKOU GINOMENOU, EMBADON PARALLHLOGRAMMOU

37 E = bˆsh Ôyoc ìpou bˆsh= α kai Ôyoc= b sin φ Tìte E = α b sin φ, φ [0, π] Dhlad E = α b EFARMOGH : GEWMETRIKH ERMHNEIA METROU MIKTOU GINOMENOU < α b, c >= det{ α, b, c} OGKOS PARALLHLEPIPEDOU MIKTO GINOMENO ENNOIA thc PROBOLHS prob u v = < u, v > < u, u > u v = u 1 + u u 1 u, u u ORJH PROBOLH ENNOIA OGKOS PARALLHLEPIPEDOU ParallhlepÐpedo: { a, b, c} ìpou a, b, c mh suggramikˆ V ( a, b, c) = (embadìn bˆshc) (Ôyoc) ìgkoc tou P. Ôyoc = m koc (prob a b c) embadìn bˆshc = a b AnazhtoÔme prob a b c = or. < c, a b > < a b, a b > ( a b) < c, a b > a a b = < c, a b > b a b 'Ara Ôyoc = < c, a b > a b 'Ara 'Ara V ( a, b, c) = embadìn bˆshc Ôyoc = a b < c, a b > a b V ( a, b, c) = < c, a b > APOLUTH TIMH MIKTOU GINOMENOU SUGGRAMMIKOTHTA TRIWN DIANUSMATWN SUNJHKH SUGGRAMMIKOTHTAS TRIWN DIANUSMATWN 3 dianôsmata eðnai sunepðpeda ìtan to miktì ginìmeno eðnai mhdèn < a b, c >= 0 det{ a, b, c} = 0

38 Ti deðqnei h orðzousa an eðnai mhdèn: ta a, b, c eðnai sunepðpeda Ti deðqnei h orðzousa an den eðnai mhdèn: ton ìgko V ( a, b, c) H orðzousa sundèetai me ton ìgko. EMBADON TRIGWNOU (apì to EXWTERIKO GINOMENO) 'Estw ta shmeða A(x A, ψ A, z A ), B(x B, ψ B, z B ), Γ(x Γ, ψ Γ, z Γ ) emb(abγ) EMB(ABΓ) = 1 AB AΓ i j κ x B x A ψ B ψ A z B z A EMB(ABΓ) = 1 (Dψz ) + (D xz ) + (D xψ ) x Γ x A ψ B ψ Γ z Γ z A