SofÐa ZafeirÐdou: GewmetrÐec
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- ÍΕρρίκος Καλαμογδάρτης
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1 Tm ma Majhmatik n Panepist mio Patr n Bohjhtikèc Shmei seic gia to mˆjhma GewmetrÐec SofÐa ZafeirÐdou Anaplhr tria Kajhg tria Pˆtra 2018
2 Oi shmei seic autèc grˆfthkan gia tic anˆgkec tou maj matoc GewmetrÐa. Me qarˆ ja deqt opoiesd pote parathr seic epð tou keimènou diorj seic. SofÐa ZafeirÐdou (
3 Perieqìmena 1 Basikèc ènnoiec Basikˆ stoiqeða thc jewrðac omˆdwn Metrikìc q roc R n Aplìc lìgoc tri n suneujeiak n shmeðwn tou R Diplìc lìgoc tessˆrwn suneujeiak n shmeðwn tou R Summetrikˆ sq mata sto R Orjog nioi pðnakec Ask seic Ti eðnai GewmetrÐa; Axi matik mejodoc Ta <<StoiqeÐa>> tou EukleÐdh Axiwmatik JemeleÐwsh thc EukleÐdeiac GewmetrÐac apì ton Hilbert Mh EukleÐdeiec GewmetrÐec Orismìc thc GewmetrÐac katˆ Klein Omˆdec metasqhmatism n tou R n kai oi antðstoiqec gewmetrðec Omˆda isometri n tou R n H antimetajetik omˆda metafor n tou R n Omˆda afinik n metasqhmatism n tou R n Ask seic Afinik GewmetrÐa tou R Jemelei dec Je rhma thc Afinik c GewmetrÐac AnaloÐwtec afinik n metasqhmatism n tou R Parˆllhlec probolèc Ask seic EukleÐdeia GewmetrÐa tou R Orismìc thc EukleÐdeiac GewmetrÐac tou R Omˆda peristrof n tou R 2 gôrw apì èna shmeðo Anaklˆseic tou R 2 wc proc mia eujeða ε IsometrÐec tou R AnalloÐwtec eukleðdeiwn metasqhmatism n Merikˆ Jewr mata EukleÐdeiac epipedometrðac Ask seic
4 2 5 Sfairik GewmetrÐa Orismìc thc sfairik c gewmetrðac Basikèc idiìthtec thc sfaðrac Basikˆ stoiqeða thc sfairik c gewmetrðac IsqÔc twn protˆsewn anˆlogwn twn EukleÐdeiwn axiwmˆtwn sth sfairik gewmetrða Sfairikèc isometrðec Sfairikèc eujeðec wc elˆqistec diadromèc Embada sqhmˆtwn sth sfairik gewmetrða Embadìn dig nou Embadìn sfairikoô trig nou Sfairik trigwnometrða Isìthta sfairik n trig nwn Stereografik probol Didiˆstath elleiptik gewmetrða Ask seic Probolik GewmetrÐa Probolikì epðpedo RP OmogeneÐc suntetagmènec probolik n shmeðwn Probolikèc eujeðec tou RP EpÐpeda probol c (enallaktikì montèlo probolikoô epipèdou) tou RP Probolikèc apeikonðseic tou RP Jemeli dec Je rhma thc Probolik c GewmetrÐac Duïsmìc sthn probolik gewmetrða Diplìc lìgoc tessˆrwn suneujeiak n ShmeÐwn tou RP AnalloÐwtec thc probolik c gewmetrðac (RP 2, P (2)) Ask seic GewmetrÐa thc antistrof c Antistrof wc proc kôklo H eikìna miac eujeðac kai enìc kôklou dia mèsou thc antistrof c wc proc ton kôklo Orismìc thc gewmetrðac thc antistrof c tou R 2 { } Merikèc idiìthtec thc gewmetrðac thc antistrof c H sqèsh thc gewmetrðac thc antistrof c me ˆllec gewmetrðec Antistrof kai orjog nioi kôkloi Efarmogèc thc antistrof c wc proc ton kôklo gia thn apìdeixh jewrhmˆtwn thc EukleÐdeiac GewmetrÐac Ask seic MetasqhmatismoÐ tou migadikoô epipèdou MetasqhmatismoÐ Möbius tou Ĉ Idiìthtec metasqhmatism n Möbius
5 3 8.2 Genikeumènoi metasqhmatismoð Möbius Ask seic Uperbolik GewmetrÐa Montèlo tou ˆnw hmisfairðou Montèlo tou uperboloeidoôc Montèlo tou dðskou Beltrami-Klein Montèlo tou dðskou tou Poincaré Montèlo tou hmiepipèdou tou Poincaré ApeikonÐseic metaxô twn montèlwn tou uperbolikoô epipèdou H uperbolik apìstash 'Ajroisma gwni n trig nou sto uperbolikì epðpedo Ask seic H ierˆrqhsh twn gewmetri n UpogewmetrÐec thc probolik c gewmetrðac
6 4 SumbolismoÐ S X summetrik omˆda tou sunìlou X. S n summetrik omˆda sunìlou me n stoiqeða. Ro φ peristrof katˆ th gwnða φ sto R 2 gôrw apì thn arq. Re φ anˆklash sto R 2 apì thn eujeða pou dièrqetai apì thn arq kai èqei suntelest dieôjunshc tan φ. E(n) omˆda isometri n tou R n. GL(n, R) omˆda twn antistrèyimwn n n pinˆkwn me stoiqeða apì to R. SL(n, R) omˆda twn antistrèyimwn n n pinˆkwn me stoiqeða apì to R orðzouasac 1. O(n, R) omˆda twn orjog niwn n n pinˆkwn me stoiqeða apì to R SO(n, R) omˆda twn orjog niwn n n pinˆkwn me stoiqeða apì to R orðzouasac 1. A(n) omˆda ìlwn twn afinik n metasqhmatism n tou R n. RP 2 probolikì epðpedo (2-diastatoc probolikìc q roc) pˆnw sto R.
7 Kefˆlaio 1 Basikèc ènnoiec. 1.1 Basikˆ stoiqeða thc jewrðac omˆdwn. Omˆda. 'Ena mh kenì sônolo G efodiasmèno me mða prˆxh, dhlad mia apeikìnsh : G G G kaleðtai omˆda, an ikanopoioôntai oi ex c idiìthtec 1. Prosetairistik idiìthta: gia opoiad pote f, g, h G isqôei: (f g) h = f (g h) 2. 'Uparxh oudetèrou stoiqeðou: upˆrqei e G ste gia kˆje g G na isqôei e g = g e = g 3. 'Uparxh summetrikoô stoiqeðou: gia kˆje g G upˆrqei g 1 G tètoio ste g 1 g = g g 1 = e. An epiplèon g h = h g gia opoiad pote g, h G, tìte h omˆda G me thn prˆxh kaleðtai antimetajetik abelian. Upoomˆdec. 'Estw (G, ) mia omˆda kai H G, H. Ac upojèsoume ìti (i) gia kˆje h H to h 1 eðnai stoiqeðo tou H (ii) gia opoiad pote h 1, h 2 H to h 1 h 2 eðnai stoiqeðo tou H. Gia h H apì tic parapˆnw idiìthtec (i) kai (ii) sunepˆgetai ìti h h 1 = e H. 'Ara, to oudètero stoiqeðo e thc omˆdac G ˆn kei sto H. Epomènwc (H, ) eðnai omˆda, h opoða kaleðtai upoomˆda thc (G, ). Lème aplˆ ìti H eðnai upoomˆda thc omˆdac G. H upoomˆda H kaleðtai gn sia an H G kai H {e}. ApodeiknÔetai ìti èna uposônolo H miac omˆdac G me prˆxh eðnai upoomˆda an kai mìnon an h 1 h 1 2 H gia kˆje h 1, h 2 H. 5
8 6 OmomorfismoÐ omˆdwn. Orismìc 'Estw (G, ) kai (H, ) dôo omˆdec. Miˆ apeikìnish i : G H kaleðtai omomorfismìc, ìtan 'Enac omomorfismìc i : G H kaleðtai i(g 1 g 2 ) = i(g 1 ) i(g 2 ), g 1, g 2 G. (i) monomorfismìc, ìtan eðnai 1-1 (i) epimorfismìc, ìtan eðnai epð (i) isomorfismìc, ìtan eðnai 1-1 kai epð Summetrik omˆda enìc sunìlou. 'Estw X èna mh kenì sônolo. SumbolÐzoume S X = {g : X X : g eðnai 1-1 kai epð}. To sônolo S X efodiasmèno me thn prˆxh sônjeshc eðnai omˆda. Prˆgmati, h tautotik apeikìnish τ X : X X eðnai to oudètero stoiqeðo thc S X kai gia kˆje g S X, to summetrikì stoiqeðo eðnai h apeikìnish g 1. Profan c, g 1 (g 2 g 3 ) = (g 1 g 2 ) g 3. H omˆda S X kaleðtai summetrik omˆda tou X. ParadeÐgmata An ta sônola X kai Y èqoun ton Ðdio plhjˆrijmo, tìte oi summetrikèc omˆdec S X kai S Y eðnai isìmorfikèc. 2. H summetrik omˆda tou {1,..., n} sumbolðzetai me S n. H S n apoteleðtai apì n! stoiqeða eðnai ìlec oi metajèseic tou {1,..., n}. 1.2 Metrikìc q roc R n. JewroÔme to sônolo R n ìlwn twn diatetagmènwn n-adwn x = (x 1, x 2..., x n ) pragmatik n arijm n. SumbolÐzoume 0 n = (0, 0,..., 0) R n. OrÐzoume thn prìsjesh sto R n, jètontac gia ā = (a 1, a 2,..., a n ), b = (b 1, b 2,..., b n ) R n : ā + b = (a 1 + b 1, a 2 + b 2,..., a n + b n )
9 7 OrÐzoume ton pollaplasiasmì stoiqeðou tou R n epð pragmatikì arijmì, jètontac gia ā = (a 1, a 2,..., a n ) R n kai λ R λ ā = λā = (λa 1, λa 2,..., λa n ) H triˆda (R n, +, ) eðnai dianusmatikìc q roc pˆnw sto R. Ta grammik c anexˆrthta shmeða ē 1 = (1, 0,..., 0), ē 2 = (0, 1,..., 0),..., ē n = (0, 0,..., 1) apoteloun mða bˆsh tou R n. Gia kˆje x = (x 1, x 2..., x n ) R n isqôei: x = x 1 ē 1 + x 2 ē x n ē n. An x, y R n, x = (x 1, x 2..., x n ) kai y = (y 1, y 2..., y n ), tìte h apìstash d n ( x, ȳ) metaxô twn x kai y upologðzetai apì ton tôpo d n ( x, ȳ) = (x 1 y 1 ) 2 + (x 2 y 2 ) (x n y n ) 2. H apeikìnish d n : R n R n R eðnai metrik. ParathroÔme ìti d n ( x, ȳ) = d n ( x ȳ, 0 n ). Gia kˆje x = (x 1, x 2,..., x n ) R n sumbolðzoume x = d( x, 0 n ). Tìte x = x x x 2 n. Gia opoiad pote x, ȳ R n èqoume ApodeiknÔetai eôkola ìti x ȳ = d n ( x, ȳ). x ȳ x + ȳ x + ȳ x + ȳ Prìtash Apì ìla ta diforðsima tìxa tou epipèdou R 2 me ˆkra ta shmeða A, B R 2 to eujôgrammo tm ma AB èqei to mikrìtero m koc. Apìdeixh. 'Estw r : [a, b] R 3 eðnai mia diafor simh parametrikopoihmènh kampôlh me A = r(a) kai B = r(b). Tìte to m koc thc r apì to A ewc to B eðnai l(a, B) = b a r (t) dt. 'Estw v èna tuqaðo monadiaðo diˆnusma kai f(t) = r(t) v, t [a, b]. Tìte Sunep c f (t) = r (t) v = ( B A ) v = b a b a ( r (t) v)dt = ( r (t) v)dt = f(t) b a = [ r(t) v ] b a = ( B A ) v b a r (t) v) cos( r (t), v)dt b a r (t) = l(a, B). Gia v = B A B A, eðnai ( B A ) v = B A = AB. 'Ara, AB l(a, B).
10 Aplìc lìgoc tri n suneujeiak n shmeðwn tou R 3. Orismìc Aplìc lìgoc (ABM) tri n suneujeiak n shmeðwn A, B, M tou R 3 eðnai o arijmìc λ gia ton opoðo AM = λ MB. AM O aplìc lìgoc twn A, B, M sumbolðzetai epðshc me MB. Gia kˆje N R 3 sumbolðzoume me N to diˆnusma jèshc ON tou N. Je rhma 'Estw A, B, M trða suneujeiakˆ shmeða tou R 3. AM = λ MB M = 1 λ A + B 1 + λ 1 + λ Je rhma Gia kˆje shmeðo M miac eujeðac (AB) upˆrqei monadikì zeôgoc pragmatik n arijm n (α, β) tètoio ste M = α A + β B kai α + β = 1. Pìrisma 'Estw A, B, M trða suneujeiakˆ shmeða tou R 3. M = α A + β B kai α + β = 1 = (ABM) = β/α Diplìc lìgoc tessˆrwn suneujeiak n shmeðwn tou R 3. Orismìc Diplìc lìgoc (ABMN) tessˆrwn suneujeiak n shmeðwn A, B, M, N tou R 3 eðnai o arijmìc pou orðzetai wc ex c (ABMN) = (ABM) (ABN). Prìtash 'Estw A, B, M, N tèssera suneujeiakˆ shmeða tou R 3. M = α A + β B kai N = γ A + δ B, ìpou α + β = 1 kai γ + δ = 1 = (ABMN) = β α/ δ γ Summetrikˆ sq mata sto R 3. 'Ena sq ma Σ tou R 3 kaleðtai summetrikì wc proc to shmeðo K R 3, ìtan gia kˆje A Σ to shmeðo A K summetrikì tou A wc proc to K an kei epðshc sto Σ. 'Ena sq ma Σ tou R 3 kaleðtai summetrikì wc proc thn eujeða ε tou R 3, ìtan gia kˆje A Σ to shmeðo A ε summetrikì tou A wc proc K an kei epðshc sto Σ. 'Ena sq ma Σ tou R 3 kaleðtai summetrikì wc proc to epðpedo Π tou R 3, ìtan gia kˆje A Σ to shmeðo A Π summetrikì tou A wc proc to Π an kei epðshc sto Σ.
11 9 1.3 Orjog nioi pðnakec. c c 1n c c 2n Orismìc O pðnakac..... c n1... c nn kaleðtai orjog nioc an: (i) c 2 1k + c2 2k c2 nk = 1 gia kˆje k = 1,..., n (ii) c 1k c 1j + c 2k c 2j c nk c nj = 0, an k j, k, j = 1,..., n. To sônolo ìlwn twn orjog niwn n n pinˆkwn me stoiqeða apì to sônolo R twn pragmatik n arijm n sumbolðzetai me O(n, R). Je rhma Oi akìloujec sunj kec eðnai isodônamec: 1. C eðnai orjog nioc 2. C T C = E 3. C T = C 1 4. C T eðnai orjog nioc Je rhma H orðzousa enìc orjog niou pðnaka isoôtai me ±1. Je rhma To ginìmeno dôo orjog niwn pinˆkwn eðnai orjog nioc pðnakac kai o antðstrofoc enìc orjog niou pðnaka eðnai orjog nioc pðnakac. ( ) c11 c Je rhma Gia kˆje orjog nio pðnaka C = 12 upˆrqei monadikì φ ( π, π] c 21 c 22 gia to opoðo o C èqei mða apì tic morfèc: ( ) ( ) cos φ sin φ cos φ sin φ. sin φ cos φ sin φ cos φ Apìdeixh. O C eðnai orjog nioc, epomènwc c c 2 21 = 1 (1.1) c c 2 22 = 1 (1.2) c 11 c 12 + c 21 c 22 = 0 (1.3) Apì thn (1.1) èpetai ìti upˆrqei monadikì φ ( π, π] tètoio ste c 11 = cos φ, c 21 = sin φ. H (1.3) grˆfetai c 12 cos φ = c 22 sin φ (1.4) Uy nontac thn (1.4) sto tetrˆgwno kai prosjètontac kai sta dôo mèlh c 2 12 sin 2 φ paðrnoume diadoqikˆ: c 2 12 cos 2 φ + c 2 12 sin 2 φ = c 2 22 sin 2 φ + c 2 12 sin 2 φ = c 2 12 = (c c 2 12) sin 2 φ. Apì thn (1.2) kai thn teleutaða isìthta paðrnoume: c 2 12 = sin 2 φ. Sunep c c 12 = sin φ c 12 = sin φ. Tìte apì thn (1.4): c 22 = cos φ c 22 = cos φ, antðstoiqa.
12 Ask seic Na apodeiqjeð ìti to sônolo GL(n, R) ìlwn twn antistrèyimwn n n pinˆkwn me stoiqeða apì to R me prˆxh pollaplasiasmoô twn pinˆkwn ( eðnai mh ) antimetajetik omˆda. a b Na apodeiqjeð ìti to sônolo twn pinˆkwn tic morf c me a b a 2 +b 2 = 1 me prˆxh pollaplasiasmoô twn pinˆkwn eðnai antimetajetik omˆda Na apodeiqjeð ìti to sônolo O(n, R) me prˆxh pollaplasiasmoô pinˆkwn eðnai omˆda, h opoða gia n > 3 eðnai mh antimetajetik. Upìdeixh: deðxte ìti AB BA gia A = kai B = Na apodeiqjeð ìti an A = (x, y, z), tìte summetrikì tou A = (x, y, z) wc proc thn arq O = (0, 0, 0) eðnai to A O = ( x, y, z). summetrikì tou A wc proc ton Ox eðnai to A Ox = (x, y, z). summetrikì tou A wc proc to Oxy eðnai to A Oxy = (x, y, z) Na brejoôn ta shmeða summetrikˆ tou A = (1, 4, 2) wc proc thn arq twn axìnwn, wc proc ton ˆxona Oz kai wc proc to epðpedo Oxz Na apodeiqjeð ìti an èna sq ma Σ tou R 3 eðnai summetrikì wc proc kˆje epðpedo suntetagmènwn, tìte Σ eðnai summetrikì wc proc kˆje ˆxona suntetagmènwn.
13 Kefˆlaio 2 Ti eðnai GewmetrÐa; 2.1 Axi matik mejodoc. Mia apì thc mejìdouc oikodìmhshc thc gewmetrðac eðnai h Axiwmatik Mèjodoc, pou eðnai kai h basikìterh mèjodoc twn Majhmatik n. H mèjodoc aut sunðstatai sthn eisagwg enìc axiwmatikoô sust matoc, dhlad diatôpwsh merik n basik n ennoi n kai ˆxiwmˆtwn - protˆsewn pou apodèqontai wc alhjeðc. Apìdeixh miac prìtashc se èna axiwmatikì sôsthma jewreðtai mia alusðda apì protˆseic pou eðte eðnai axi mata eðte kˆpoia prìtash h al jeia thc opoðac sunepˆgetai apì tic prohgoômenec protˆseic sômfwna me touc kanìnec thc logik c pou èqoun sumfwnhjeð Ta <<StoiqeÐa>> tou EukleÐdh. Klasikì parˆdeigma thc axi matik c mejìdou oikodìmhshc miac gewmetrðac eðnai h EukleÐdeia GewmetrÐa. O EukleÐdhc sta <<StoiqeÐa>> tou diatup nei ta pènte ait mata-axi mata: 1. Gia opoiad pote dôo diaforetikˆ shmeða upˆrqei monadikì eujôgrammo tm ma me ˆkra ta shmeða autˆ. 2. 'Ena eujôgrammo tm ma mporeð na proektajeð aperiìrista kai stic dôo kateujônshc se mia eujeða. 3. Upˆrqei kôkloc me kèntro opoiod pote shmeðo kai me opoiad pote aktðna. 4. 'Olec oi orjèc gwnðec eðnai Ðsec (mia gwnða kaleðtai orj, ìtan eðnai Ðsoi me thn paraplhrwmatik thc) 5. An mia eujeða ε tèmnei ˆllec dôo eujeðec ε 1 kai ε 2 kai to ˆjroisma kˆpoion entìc kai epð ta autˆ gwni n eðnai < 180, tìte ε 1 kai ε 2 tèmnontai se kˆpoio shmeðo, to opoðo brðsketai apì thn meria thc ε pou perièqei tic gwnðec autèc. To 1795 o skotsèzoc majhmatikìc kai gewgrˆfoc John Playfair apèdeixe ìti o 5 o axðwma eðnai isodônamo me to akìloujo gnwstì wc axðwma parallhlðac: 5. Apì kˆje shmeðo A èxw apì mia eujeða ε dièrqetai monadik eujeða ε parˆllhlh sthn ε. 11
14 Axiwmatik JemeleÐwsh thc EukleÐdeiac GewmetrÐac apì ton Hilbert. O Hilbert to 1899 basizìmenoc sta <<StoiqeÐa>> tou EukleÐdh diatup nei nèo axiwmatikì sôsthma sthn ergasða tou <<Grundlagen der Geometrie>>. O Hilbert taxinìmhse ta axi mata thc EukleÐdeiac GewmetrÐac se 5 kathgorðec: Axi mata prosptwshc (jèshc). Axi mata diˆtaxhc (endiamesìthtac). Axi mata kðnhshc (isìthtac). Axi mata thc sunèqeiac (tou Dedekind). AxÐwma thc parallhlðac (tou EukleÐdh) Mh EukleÐdeiec GewmetrÐec. Apì thn epoq tou EukleÐdh polloð majhmatikoð parat rhsan ìti h diatôpwsh tou 5ou axi matoc den tan tìso apl ìso oi diatôpwsh twn ˆllwn axiwmˆtwn kai anarwt jhkan m pwc to 5o axðwma apodeiknôetai apì ta ˆlla tèssera. ApodeÐqjhke telikˆ ìti to 5o aðthma den sunepˆgetai apì ta pr ta 4 ait mata. Oi prospˆjeiec na apodeiqjeð to 5 o axðwma upojètontac ìti h ˆrnhsh tou odhgeð se ˆtopo èqoun apobeð ˆkarpec, wc ìtou apì ta tèlh tou 18ou èwc ta mèsa tou 19ou ai na oi majhmatikoð parat rhsan ìti h ˆrnhsh tou 5ou axi matoc mporeð na od ghsei se nèa Jewr mata miac ˆllhc gewmetrðac. 'Etsi genn jhkan oi mh EukleÐdeiec gewmetrðec. MerikoÐ apì touc jemeleiwtèc twn mh EukleÐdeiwn gewmetri n eðnai: Gauss ( ), Lobachevsky ( ), Bolyai ( ), Beltrami ( ), Riemann ( ). H ˆrnhsh tou 5ou axi matoc shmaðnei ìti: Apì kˆpoio shmeðo A èxw apì mia eujeða ε mporeð (i) na mhn dièrqetai kamiˆ eujeða parˆllhlh sthn ε, (ii) na dièrqontai toulˆqiston dôo eujeðec parˆllhlec sthn ε. Allˆzontac apaloðfontac pl rwc to 5o axðwma tou EukleÐdh kai diathr ntac ta pr ta tèssera odhgoômaste se nèec gewmetrðec: Elleiptik GewmetrÐa jemelei netai diathr ntac ta pr ta tèssera ait mata tou EukleÐdh kai antikajist ntac to 5o me to ex c: (5 e ) Kˆje eujeða dierqìmenh apì èna shmeðo A èxw apì mia eujeða ε tèmnei thn ε. (Den upˆrqoun parˆllhlec eujeðec.) Uperbolik GewmetrÐa jemelei netai diathr ntac ta pr ta tèssera ait mata tou EukleÐdh kai antikajist ntac to 5o me to ex c: (5 u ) Apì kˆje shmeðo A èxw apì mia eujeða ε dièrqontai toulˆqiston dôo eujeðec pou den tèmnoun thn ε.(upˆrqoun toulˆqiston dôo eujeðec pou dièrqontai apì to A kai eðnai parˆllhlec sthn ε.) Oudèterh ( Apìluth) GewmetrÐa jemelei netai diathr ntac ta pr ta tèssera ait mata tou EukleÐdh kai katarg ntac pl rwc to 5o aðthma thc parallhlðac.
15 Orismìc thc GewmetrÐac katˆ Klein. 'Otan dôo trðgwna T 1 kai T 2 enìc epipèdou Π eðnai Ðsa, tìte sunduˆzontac thn peristrof tou epipèdou gôrw apì kˆpoia koruf tou enìc trig nou me mia metatìpish ìlwn twn shmeðwn tou epipèdou katˆ to Ðdio diˆnusma, to èna trðgwno ja sumpèsei me to ˆllo. 'Ara, upˆrqei mia 1-1 kai epð apeikìnish ϕ : Π Π, h opoða diathreð tic apostˆseic metaxô twn shmeðwn, tètoia ste ϕ(t 1 ) = T 2. Prìgramma tou Erlangen. To prìgramma tou Erlangen (Erlangen program) eðnai h prìtash tou F. Klein gia ton trìpo taxinìmhshc twn gewmetri n, to opoðo èqei diatupwjeð sthn omilða tou to 1872 sto Panepist mio tou Erlangen. SÔmfwna me ton Klein gia na orðsoume mia gewmetrða se èna sônolo X arkeð na epilèxoume mia omˆda metasqhmatism n G tou X (G apoteleðtai apì 1-1 kai epð autoapeikonðseic tou X). O kôrioc stìqoc thc melèthc thc gewmetrðac pou ja orðsteð eðnai h eôresh twn idiìthtwn twn sqhmˆtwn (uposunìlwn) tou X pou paramènoun analloðwtec (den allˆzoun) stouc metasqhmatismoôc g G. Orismìc thc GewmetrÐa katˆ Klein. KaloÔme gewmetrða kˆje zeôgoc (X, G), ìpou X èna mh kenì sônolo kai G èna mh kenì sônolo 1-1 kai epð autoapeikonðsewn tou X me tic ex c idiìthtec: (i) gia kˆje g, h G h sônjesh g h eðnai stoiqeðo thc G, (ii) gia kˆje g G h antðstrofh apeikìnish g 1 eðnai stoiqeðo thc G. Apì tic idiìthtec (i) kai (ii) prokôptei ìti h tautotik apeikìnish τ X = g g 1 eðnai stoiqeðo thc G. Epeid h sônjesh twn apeikonðsewn èqei prosaiteristik idiìthta, prokôptoun oi akìloujoi isodônamoi orismoð thc gewmetrðac 1. Kˆje zeôgoc (X, G), ìpou X eðnai èna mh kenì sônolo kai G eðnai mia omˆda 1-1 kai epð autoapeikonðsewn tou X me prˆxh sônjeshc twn apeikonðsewn, kaleðtai gewmetrða. 2. Kˆje zeôgoc (X, G), ìpou X eðnai èna sônolo kai G mia upoomˆda thc summetrik c omˆdac S X tou X, kaleðtai gewmetrða. ParadeÐgmata 'Estw X èna mh kenì sônolo, τ X tautotik apeikìnish tou X. H mikrìterh gewmetrða tou X eðnai h (X, {τ X }) kai h megalôterh (X, S X ). 2. 'Estw X ènac topologikìc q roc kai H h omˆda ìlwn twn omoiomorfism n tou X epð tou X. Tìte (X, H) eðnai gewmetrða. 3. 'Estw S X = ABC èna isìpleuro trðgwno tou epipèdou kai X = {A, B, C}. Tìte {( A B C A B C ) ( A B C, B C A ) ( A B C, C A B ) ( A B C, A C B ) ( A B C, C B A ) ( A B C, B A C )}
16 14 ( ) ( ) A B C A B C parathroôme ìti eðnai h tautotik, prokôptei apì thn peristrof A B C B C A ( g 1 tou trg nou ) gôrw apì to kèntro tou trig nou katˆ jetik forˆ katˆ th gwnða 240, A B C prokôptei apì thn peristrof g C A B 2 tou trig nou gôrw apì to kèntro tou trig nou katˆ jetik forˆ katˆ th gwnða 120. ParathroÔme ìti g 2 = g1 1. Ta upìloipa stoiqeða thc S X antistoiqoôn stic anaklˆseic g 3, g 4, g 5 tou trig nou apì touc ˆxonec summetrðac tou trig nou pou dièrqontai apì ta shmeða A, B, C, antðstoiqa. ParathroÔme ìti g i = g 1 i gia i = 3, 4, 'Estw G = {τ X, g 1, g 2, g 3, g 4, g 5 }, ìpou oi metasqhmatismoð g i eðnai tou prohgoômenou paradeðgmatoc. Tìte ( ABC, G) eðnai gewmetrða. ParathroÔme ìti h G apoteleðtai apì isometrðec tou ABC. 'Estw (X, G) mia gewmetrða. Ta stoiqeða thc omˆdac G kaloôntai metasqhmatismoð. Kˆje uposônolo A tou X kaleðtai sq ma thc (X, G). To sq ma A thc (X, G) kaleðtai isodônamo (Ðso, ìmoio) me to sq ma B thc (X, G), ìtan upˆrqei metasqhmatismìc g G tètoioc ste g(a) = B. Grˆfoume tìte A B. Profan c eðnai sqèsh isodunamðac. Opìte ta sq mata thc (X, G) qwrðzontai se klˆseic isodunamðac isodônamwn ( Ðswn ) sqhmˆtwn. ParadeÐgmata Sth gewmetrða (X, S X ) opoiad pote sq mata Σ 1 kai Σ 2 gia ta opoða upˆrqei mia 1-1 kai epð apekìnish f : Σ 1 Σ 2 eðnai isodônama. Prˆgmati, epeid Σ 1 = Σ 2 upˆrqei 1-1 kai epð apeikìnish g : Σ 2 \ Σ 1 Σ 1 \ Σ 2. OrÐzoume F : X X me g(x), x Σ 2 \ Σ 1 F (x) = g 1 (x), x Σ 1 \ Σ 2 x, x X \ (Σ 2 \ Σ 1 ) \ (Σ 1 \ Σ 2 ). 2. ApodeiknÔetai ìti gia kˆje trðgwno T kai gia kˆje kôklo C tou R 2 upˆrqei omoiomorfismìc h : R 2 R 2 gia ton opoðon h(t ) = C. Epomènwc to trðgwno kai o kôkloc eðnai isodônama sq mata sth gewmetrða (R 2, H), ìpou H eðnai h omˆda twn autoomoiomorfism n tou R 2.
17 AnaloÐwtec miac gewmetrðac. O kôrioc stìqoc thc melèthc miac gewmetrðac (X, G) eðnai o prosdiorismìc ekeðnwn twn idiot twn twn sqhmˆtwn thc pou diathroôntai apì tic apeikonðseic g : X X gia kˆje g G. Dhlad, an kˆpoio sq ma Σ èqei mia idiìthta P, tìte kai to sq ma g(σ) èqei thn diìthta P. 'Enac genikìc orismìc thc ènnoiac thc analloðwthc eðnai o parakˆtw. 15 Orismìc 'Estw ìti (X, G) eðnai mia gewmetrða, S èna mh kenì sônolo sqhmˆtwn thc (X, G). 1. To sônolo S kaleðtai analloðwto thc (X, G), ìtan gia kˆje Σ S kai gia kˆje g G isqôei ìti g(σ) S. 2. Mia apeikìnish α : S Y, ìpou Y èna mh kenì sônolo kaleðtai analloðwth tou sônolou S wc proc thn (X, G), ìtan g(σ) S kai α(σ) = α(g(σ)), gia kˆje Σ S kai gia kˆje g G. 2.3 Omˆdec metasqhmatism n tou R n kai oi antðstoiqec gewmetrðec Omˆda isometri n tou R n. Orismìc Mia apeikìnish F : R n R n kaleðtai isometrða ìtan F (ā) F ( b) = ā b, (a, b) R n. To sônolo ìlwn twn isometri n tou R n sumbolðzetai me E(n). Ta stoiqeða tou E(n) kaloôntai epðshc EukleÐdeioi metasqhmtismoð. Je rhma Kˆje isometrða tou R n eðnai 1-1 kai epð. Apìdeixh. An ā b, tìte ā b > 0. Epeid F (ā) F ( b) = ā b, èpetai ìti F (ā) F ( b) > 0. 'Ara, F (ā) F ( b). ApodeiknÔetai ìti gia kˆje isometrða F : R n R n upˆrqei orjog nioc n n pðnakac C kai ā = ȳ = y 1. y n a 1. a n R n, tètoia ste F ( x) = C x + ā gia kˆje x = R n, tìte ȳ = F ( x), ìpou x = C 1 (ȳ ā). x 1. x n R n. 'Ara, an EÔkola apodeiknôetai to parakˆtw Je rhma. Je rhma To sônolo isometri n E(n) tou R n efodiasmèno me thn prˆxh sônjeshc apeikonðsewn eðnai omˆda. Orismìc To zeôgoc (R n, E(n)) kaleðtai EukleÐdia gewmetrða tou R n.
18 H antimetajetik omˆda metafor n tou R n. Gia kˆje ā R n, h apeikìnish Tā : R n R n pou orðzetai apì ton tôpo Tā( x) = x + ā kaleðtai metaforˆ katˆ diˆnusma ā. Je rhma To sônolo metafor n T (n) = {Tā : ā R n } efodiasmèno me thn prˆxh sônjeshc apeikonðsewn eðnai antimetajetik omˆda. Apìdeixh. H tautotik apeikìnish id X T 0 : X X eðnai to oudètero stoiqeðo thc T kai gia kˆje Tā, to antðstrofo stoiqeðo eðnai h metaforˆ T ā. Oi sônjesh dôo metafor n eðnai metaforˆ. Prˆgmati, (Tā T b)(p ) = Tā(T b(p )) = Tā(P + b) = P + b + ā = T b+ā. Epeid oi sônjesh apeikonðsewn eðnai prosaiteristik, (Tā T b) T c = Tā (T b T c ). EpÐshc Tā T b = T b+ā = Tā+ b = T b Tā. H antimetajetik omˆda (T (n), ) kaleðtai omˆda metafor n tou R n. Epeid kˆje metaforˆ eðnai 1-1 kai epð, apì ton orismo thc gemetrðac, to zeôgoc (R n, T (n)) eðnai gewmetrða. Prìtash Kˆje metaforˆ eðnai isometrða. Apìdeixh. 'Estw Tā : R n R n metaforˆ katˆ diˆnusma ā. Tìte gia ìpoiad pote x, ȳ R n : Tā( x) Tā(ȳ) = ( x + ā) (ȳ + ā) = x ȳ Omˆda afinik n metasqhmatism n tou R n. SumbolÐzoume me GL(n, R) thn omˆda antistrèyimwn n n pinˆkwn me stoiqeða apì to R me prˆxh pollaplasiamoô twn pinˆkwn. Orismìc Afinikìc metasqhmatismìc tou R n eðnai mia apeikìnish f : R n R n pou orðzetai wc ex c f(v) = Av + a, ìpou A GL(n, R) kai a R n. To sônolo ìlwn twn afinik n metasqhmatism n tou R n sumbolðzeati me A(n). Je rhma To zeôgoc (R n, A(n)) eðnai gewmetrða.
19 17 Apìdeixh. EÔkola apodeiknôetai ìti kˆje afinikìc metasqhmatismìc eðnai 1-1. Ja deðxoume ìti kˆje afinikìc metasqhmatismìc eðnai epð. 'Estw f(v) = Av + a A(n) kai v R n. v = Av + a = v = A 1 v A 1 a = v = f(a 1 v A 1 a). H sunjesh afinik n metasqhmatism n f 1 (v) = A 1 v +a 1 kai f 2 (v) = A 2 v +a 2 eðnai afinikìc metasqhmatismìc: (f 2 f 1 )( v) = f 2 (f 1 ( v)) = f 2 (A 1 v + a 1 ) = A 2 (A 1 v + a 1 ) + a 2 = (A 2 A 1 )v + (A 2 a 1 + a 2 ) jètontac A = A 2 A 1 kai a = A 2 a 1 + a 2, paðrnoume (f 2 f 1 )( v) = Av + a, ìpou A GL(n, R) kai a R n. Apì ta parapˆnw, an f(v) = Av + a A(n), tìte f 1 ( v) = A v + a, ìpou A = A 1 kai ā = A 1 a. 'Ara, f 1 A(n). Sunep c apì ton orismì thc gewmetrðac to zeôgoc (R n, A(n)) eðnai gewmetrða. Orismìc To zeôgoc (R n, A(n)) kaleðtai afinik gewmetrða. 2.4 Ask seic ( ) ( ) An A = kai ā =, na brejoôn oi eikìnec mèso tou afinikoô metasqhmatismoô f( v) = A v + ā twn shmeðwn: (0, 0), (1, 0), (0, 1) Na apodeiqjeð ìti oi parakˆtw metasqhmatismoð t 1, t 2 : R 2 R 2 eðnai afinikoð kai na brejeð o afinikìc metasqhmatismìc t pou eðnai h sônjesh t 1 t 2 : ( ) ( ) ( ) ( ) t 1 ( x) = x +, t ( x) = x ( ) ( ) ( ) 1 3 x DÐnontai metasqhmatismoð tou R 2 : f(x, y) = +, 1 2 y 2 ( ) ( ) ( ) ( ) ( ) ( 2 1 x x 2 g(x, y) = + kai h(x, y) = y y 1 ). (a') Na prosdioristeð poioð apì parapˆnw metasqhmatismoôc eðnai afinikoð. (b') Na prosdioristeð an o metasqhmatismìc f g eðnai afinikìc DÐnetai o afinikìc metasqhmatismìc t( v) = ( (a') Na brejeð o afinikìc metasqhmatismìc antðstrofoc tou t. ) ( v ) + ( 2 1 (b') Na brejeð h eikìna thc eujeðac l : 2x + 3y + 1 = 0 wc proc ton t. ).
20 18 Apˆnthsh: (a') Jètoume A = ( ) kai ā = ( 2 1 An v = t( v) = A v + ā, tìte v = A 1 v A 1 ā. BrÐskoume ìti ( ) ( ) ( ) 1/2 1/2 1/2 1/2 2 A 1 = kai A ā = = ( ) ( ) 1/2 1/2 3/2 'Ara, v = v ( ) ( ) 1/2 1/2 3/2 Sunep c t 1 ( v) = v (b') Gia v = (x, y) kai v = (x, y ), apì to upoer thma (α ) èqoume: ). x = 1 2 x 1 2 y 3 2 y = x + 2y + 4 ( 3/2 4 Antikajist ntac ta x kai y sthn exðswsh 2x + 3y + 1 = 0 paðrnoume thn exðswsh: 2x + 5y + 10 = 0. 'Ara, t(l) : 2x + 5y + 10 = Na apodeiqjeð ìti kˆje metaforˆ eðnai afinikìc metasqhmatismìc Na dojeð parˆdeigma enìc afinikoô metasqhmatismoô pou na mhn eðnai metaforˆ Na dojeð parˆdeigma enìc mh grammikoô afinikoô metasqhmatismoô Na dojeð parˆdeigma enìc afinikoô metasqhmatismoô pou den eðnai eðnai isometrða Na dojeð parˆdeigma miac isometrðac tou epipèdou pou den eðna grammik apeikìnish Na apodeiqjeð ìti h omˆda metafor n (T (n), ) eðnai isìmorfik me thn omˆda (R n, +). ).
21 Kefˆlaio 3 Afinik GewmetrÐa tou R 2. SumbolÐzoume me GL(2, R) thn omˆda antistrèyimwn 2 2 pinˆkwn me stoiqeða apì to R. Mia apeikìnish f : R 2 R 2 kaleðtai afinikìc metasqhmatismìc an upˆrqei A GL(2, R) kai ā R 2 tètoia ste f(v) = Av + a gia kˆje v R 2. To sônolo ìlwn twn afinik n metasqhmatism n tou R 2 sumbolðzeati me A(2). To zeôgoc (R 2, A(2)) kaleðtai afinik gewmetrða. 3.1 Jemelei dec Je rhma thc Afinik c GewmetrÐac. Je rhma Gia opoiad pote trða mh suneujeiakˆ shmeða P, Q kai R tou R 2 upˆrqei monadikìc afinikìc metasqhmatismìc f : R 2 R 2 pou apeikonðzei ta shmeða (0, 0), (1, 0) kai (0, 1) sta mh suneujeiakˆ shmeða P = (x P, y P ), Q = (x Q, y Q ) kai R = (x R, y R ), antðstoiqa. ( ) ( ) a b k Apìdeixh. 'Estw ìti f( v) = A v + ā, ìpou A = kai ā =. Tìte c d m ( xq y Q ( xr y R Sunep c ( xp y P ) ( ) ( ) ( ) ( ) a b 0 k k = f(0, 0) = + = = k = x c d 0 m m P, m = y P ) ( ) ( ) ( ) ( ) ( ) ( ) a b 1 k a k a = f(1, 0) = + = + = + c d 0 m c m c ) ( ) ( ) ( ) ( ) ( ) ( ) a b 0 k b k b = f(0, 1) = + = + = + c d 1 m d m d ( xq x A = P x R x P y Q y P y R y P ) kai ā = (x P, y P ). Epeid P, Q, R eðnai mh suneujeiakˆ det(a) 0, ˆra o A eðnai antistrèyimoc. ( xp y P ( xp y P ) ) ShmeÐwsh ApodeiknÔetai ìti gia oioiad pote tèssera mh sunepðpeda kai anˆ trða mh suneujeiakˆ shmeða P, Q, R kai S tou R 3 upˆrqei monadikìc afinikìc metasqhmatismìc f tou R 3 pou apeikonðzei ta shmeða (0, 0, 0), (1, 0, 0), (0, 1, 0) kai (0, 0, 1) sta shmeða P, Q, R kai S, antðstoiqa. 19
22 20 Je rhma (Jemelei dec Je rhma thc Afinik c GewmetrÐac.) An P, Q, R kai P, Q, R eðnai triˆdec mh suneujeiak n shmeðwn tou R 2, tìte upˆrqei monadikìc afinikìc metasqhmatismìc f : R 2 R 2 gia ton opoðon f(p ) = P, f(q) = Q kai f(r) = R. Apìdeixh. Apì to je rhma upˆrqoun monadikoð afinikoð metasqhmatismoð f 1 : R 2 R 2 kai f 2 : R 2 R 2 gia touc opoðouc isqôei ìti: f 1 apeikonðzei ta shmeða (0, 0), (1, 0) kai (0, 1) sta shmeða P, Q kai R, antðstoiqa. f 2 apeikonðzei ta shmeða (0, 0), (1, 0) kai (0, 1) sta shmeða P, Q kai R, antðstoiqa. O afinikìc metasqhmatismìc f = f 2 f1 1 tou R 2 apeikonðzei ta shmeða P, Q kai R sta shmeða P, Q kai R, antðstoiqa. Ac upojèsoume ìti q eðnai ènac afinikìc metasqhmatismìc tou R 2 pou apeikonðzei ta shmeða P, Q kai R sta shmeða P, Q kai R, antðstoiqa. Tìte o afinikìc metasqhmatismìc g f 1 apeikonðzei ta shmeða (0, 0), (1, 0) kai (0, 1) sta shmeða P, Q kai R, antðstoiqa. 'Ara, g f 1 = f 2, apì to Je rhma Sunep c g = f 2 f1 1 = f. ShmeÐwsh An P, Q, R, S kai P, Q, R, S eðnai tetrˆdec mh sunepðpedwn kai anˆ trða mh suneujeiak n shmeðwn tou R 3, tìte upˆrqei monadikìc afinikìc metasqhmatismìc f tou R 3 gia ton opoðon f(p ) = P, f(q) = Q, f(r) = R kai f(s) = S. 3.2 AnaloÐwtec afinik n metasqhmatism n tou R 2. Gia kˆje P = (x, y) R 2, sumbolðzoume me P to diˆnusma jèshc OP. To shmeðo P kai to diˆnusma P èqoun tic Ðdiec suntetagmènec. Epomènwc opoiad pote isìthta metaxô twn dianusmˆtwn jèshc shmeðwn mporeð na grafeð wc isìthta metaxô twn shmeðwn kai antðstrofa. Je rhma Kˆje afinikìc metasqhmatismìc tou R 2 apeikonðzei eujeðec se eujeðec. Apìdeixh. 'Estw f( v) = A v + ā ènac afinikìc metasqhmatismìc tou R 2 kai l mia eujeða pou dièrqetai apì to shmeðo P kai eðnai parˆllhlh sto diˆnusma q R 2. Tìte 'Ara, an M l, tìte M = P + λ q. Epomènwc l = {P + λ q : λ R} f(m) = A(P + λ q) + ā = (AP + ā) + λa q = f(p ) + λa q. 'Ara, f(l) eðnai eujeða, dierqìmenh apì to shmeðo f(p ) kai parˆllhlh sto diˆnusma A q. Pìrisma 'Olec oi eujeðec eðnai afinikˆ isodônamec. Apìdeixh. 'Estw ε kai ε dôo eujeðec. JewroÔme zeôgh diaforetik n shmeðwn P, Q ε kai P, Q ε. 'Estw S R 2 \ (ε ε ). Apì to Je rhma upˆrqei monadikìc f A(2) tètoioc ste f(p ) = P, f(q) = Q kai f(s) = S. To sônolo f(ε) eðnai eujeða apì to Je rhma EpÐshc P f(ε) kai Q inf f(ε). 'Ara, f(ε) = ε.
23 21 Je rhma Kˆje afinikìc metasqhmatismìc tou R 2 apeikonðzei parˆllhlec eujeðec se parˆllhlec eujeðec. Apìdeixh. 'Estw f( v) = A v + ā ènac afinikìc metasqhmatismìc tou R 2. JewroÔme zeôgoc parˆllhlwn eujei n Tìte l 1 = {P 1 + λ q : λ R} l 2 = {P 2 + λ q : λ R}. f(l 1 ) = {f(p 1 ) + λa q : λ R} f(l 2 ) = {f(p 2 ) + λa q : λ R}. Oi eujeðec f(l 1 ) kai f(l 2 ) eðnai parˆllhlec sto Ðdio diˆnusma A q. An f(p 2 ) f(l 1 ), tìte P 2 = f 1 (f(p 2 )) f 1 (f(l 1 )) = l 1, pou eðnai ˆtopo. 'Ara, f(p 2 ) f(l 1 ). Sunep c f(l 1 ) f(l 2 ). Je rhma Kˆje afinikìc metasqhmatismìc tou R 2 diathreð ton aplì lìgo tri n shmeðwn miac eujeðac. Apìdeixh. 'Estw f( v) = A v + c ènac afinikìc metasqhmatismìc tou R 2. JewroÔme trða shmeða P, Q, M miac eujeðac l. Tìte (P QM) = β/α, ìpou α + β = 1 kai M = αp + βq. Epomènwc f(m) = f(αp + βq) = αap + βaq + c = αap + βaq + (α + β) c = αf(p ) + βf(q). Pìrisma 'Estw ìti o afinikìc metasqhmatismìc f tou R 2 apeikonðzei ta suneujeiakˆ shmeða P, Q, R sta shmeða P, Q, R,antÐstoiqa. An Q eðnai metaxô twn P kai R, tìte Q eðnai metaxô twn P kai R. Apìdeixh. Epeid R eðnai metaxô P kai Q upˆrqoun λ P, λ Q [0, 1] me λ P + λ Q = 1 kai R = λ P P +λ Q Q. Epeid kˆje afinikìc metasqhmatismìc tou R 2 diathreð ton aplì lìgo tri n shmeðwn miac eujeðac, èpetai ìti f(r) = λ P f(p )+λ Q f(q). Sunep c R = λ P P +λ Q Q, ìpou λ P, λ Q [0, 1] me λ P + λ Q = 1. 'Ara, R eðnai metaxô P kai Q. Orismìc TrÐgwno ABC me korufèc ta mh suneujeiakˆ shmeða A, B kai C eðnai h ènwsh twn eujôgrammwn tmhmˆtwn AB, BC kai CA Pìrisma Opoiad pote dôo trðgwna enìc epipèdou eðnai afinikˆ isodônama.
24 Parˆllhlec probolèc. Orismìc 'Estw ìti Π 1 kai Π 2 eðnai dôo epðpeda tou q rou kai eðnai mia dèsmh parˆllhlwn eujei n pou tèmnoun kai ta dôo epðpeda Π 1 kai Π 2. Apì kˆje P Π 1 dièrqetai monadik eujeða l P. To shmeðo P = l P Π 2 eðnai h parˆllhlh probol tou P sto Π 2 pou orðzetai apì thn dèsmh. H apeikìnish p : Π 1 Π 2 pou gia thn opoða p(p ) = P kaleðtai parˆllhlh probol tou Π 1 sto Π 2. EÔkola apodeiknôetai ìti H parˆllhlh probol eðnai 1-1 kai epð. H antðstrofh apeikìnish miac parˆllhlhc probol c p : Π 1 Π 2 eðnai parˆllhlh probol p 1 : Π 2 Π 1. An Π 1 Π 2, tìte h parˆllhlh probol p : Π 1 Π 2 eðnai isìmetrða. Je rhma Kˆje parˆllhlh probol eðnai afinikìc metasqhmatismìc Apìdeixh. 'Estw ìti sto kajèna apì ta epðpeda Π 1 kai Π 2 dðnetai apì èna orjokanonikì sôsthma suntetagmènwn Oxy kai O x y. Tìte sth parˆllhlh probol p : Π 1 Π 2 antistoiqeð apeikìnish f p : R 2 R 2 pou orðzetai wc ex c: an (x, y) = P sto Oxy, p(p ) = P kai P = (x, y ) sto O x y, tìte f p (x, y) = (x, y ). An f p (O) = O, tìte f p eðnai grammik kai, epomènwc, upˆrqei 2 2 pðnakac A tètoioc ste f p ( v) = A v gia kˆje v R 2. Epeid p eðnai antistrèyimoc, o A antistrèyimoc. pðnakac. An f p (O) = ā, tìte upˆrqei antistrèyimoc 2 2 pðnakac A tètoioc ste f p ( v) = A v + ā gia kˆje v R 2. ShmeÐwsh 'Enac afinikìc metasqhmatismìc mporeð na mhn antistoiqeð se parˆllhlh probol. Gia parˆdeigma, f( v) = 2 v, v R ( 2, eðnai ) afhnikìc metasqhmatismìc, afoô mporeð na grafeð 2 0 se morf f( v) = A v + ā gia A = kai ā = (0, 0). f den mporeð na eðnai parˆllhlh 0 2 probol parˆllhlwn epipèdwn, pou eðnai isometrða, oôte epipèdwn pou tèmnontai katˆ eujeða oi apostaseic metaxô twn shmeðwn thc opoðac diathroôntai me parˆllhlh probol. Je rhma Kˆje afinikìc metasqhmatismìc eðnai sônjesh dôo parˆllhlwn probol n. Apìdeixh. 'Estw ìti o afinikìc metasqhmatismìc f : R 2 R 2 apeikonðzei ta shmeða (0, 0), (1, 0) kai (0, 1) sta mh suneujeiakˆ shmeða P = (x P, y P ), Q = (x Q, y Q ) kai R = (x R, y R ), antðstoiqa. 'Estw ìti Π 1 kai Π eðnai epðpeda me orjokanonika sust mata suntetagmènwn tètoia ste (0, 0) tou Π 1 sumpðptei me to (x P, y P ) tou Π. 'Estw p 1 : Π 1 Π h parˆllhlh probol gia thn opoða p 1 (1, 0) = (x Q, y Q ). Tìte p 1 (0, 1) eðnai kˆpoio shmeðo T = (x T, y T ) Π. 'Estw ìti Π 2 to epðpedo pou perièqei ta shmeða (x P, y P ), (x Q, y Q ) kai R = (x R, y R ) kai p 2 : Π Π 2 h parˆllhlh probol gia thn opoða p 1 (x T, y T ) = (x R, y R ). Tìte gia thn p = p 2 p 1 isqôei ìti p(0, 0) = (x P, y P ), p(1, 0) = (x Q, y Q ) kai p(0, 1) = (x R, y R ). 'Ara, p = f, dhlad f eðnai sônjesh dôo parˆllhlwn probol n.
25 Ask seic Na brejeð o afinikìc metasqhmatismìc f : R 2 R 2 pou apeikonðzei ta shmeða (0, 0), (1, 0) kai (0, 1) sta shmeða P = (1, 2), Q = (2, 1) kai R = ( 3, 5), antðstoiqa Na brejeð o afinikìc metasqhmatismìc g : R 2 R 2 pou apeikonðzei ta shmeða (0, 0), (1, 0) kai (0, 1) sta shmeða P = (2, 3), Q = (1, 6) kai R = (3, 1), antðstoiqa Na brejeð o afinikìc metasqhmatismìc h : R 2 R 2 pou apeikonðzei ta shmeða (2, 3), (1, 6) kai (3, 1) sta shmeða (1, 2), (2, 1) kai ( 3, 5), antðstoiqa. Upìdeixh: 'Estw g o afinikìc metasqhmatismìc pou apeikonðzei ta shmeða (0, 0), (1, 0) kai (0, 1) sta shmeða P = (1, 2), Q = (2, 1) kai R = ( 3, 5), antðstoiqa. 'Estw epðshc f o afinikìc metasqhmatismìc pou apeikonðzei ta shmeða (0, 0), (1, 0) kai (0, 1) sta shmeða P = (2, 3), Q = (1, 6) kai R = (3, 1), antðstoiqa. Tìte h = g f 1. An f( x) = A x + ā kai g( x) = B x + b, tìte f 1 ( x) = A 1 x A 1 ā kai h( x) = B(A 1 x A 1 ā) + b = BA 1 x BA 1 ā + b Na brejeð o afinikìc metasqhmatismìc pou apeikonðzei thn eujeða ε : 3x + 2y + 4 = 0 tou R 2 sthn eujeða x = Na brejeð o afinikìc metasqhmatismìc pou apeikonðzei thn uperbol x 2 y 2 = 1 sthn uperbol y = 1 x. ( ) ( ) 1 1 x Upìdeixh: f(x, y) =. 1 1 y Na apodeiqjeð ìti opoiesd pote dôo uperbolèc eðnai afinikˆ isodônamec. Upìdeixh: ArkeÐ na deðxoume ìti kˆje uperbol eðnai afinikˆ isodônamh me thn uperbol x 2 y 2 = 1. 'Estw γ mia uperbol tou R 2. Upˆrqei afinikìc metasqhmatismìc g (sônjesh peristrof c kai metaforˆc) pou apeikonðzei thn γ sthn uperbol γ pou èqei exðswsh thc morf c x2 a y2 = 2 1. JewroÔme ton afinikì metasqhmatismì b2 f(x, y) = ( 1/a 0 0 1/b ) ( x y ) = ( x a y b ) ( x = y ). O afinikìc metasqhmatismìc f g apeikonðzei thn uperbol γ sthn uperbol x 2 y 2 = Na apodeiqjeð ìti opoiesd pote dôo elleðyeic eðnai afinikˆ isodônamec.
26 Na apodeiqjeð ìti opoiesd pote dôo parabolèc eðnai afinikˆ isodônamec 'Estw f afinikìc metasqhmatismìc pou apeikonðzei ta suneujeiakˆ shmeða P, Q, R kai S sta suneujeiakˆ shmeða P, Q, R kai S, antðstoiqa. Na apodeiqjoôn ìti P Q RS = P Q R S Dhlad, kˆje afinikìc metasqhmatismìc diathreð ton lìgo twn mhk n twn eujeðgrammwn tmhmˆtwn katˆ m koc miac eujeðac 'Estw ìti o afinikìc metasqhmatismìc f : R 2 R 2 pou apeikonðzei ta mh suneujeiakˆ shmeða P, Q kai R sta shmeða P, Q kai R, antðstoiqa. Na apodeiqjeð ìti o f apeikonðzei to eswterikì tou trig nou P QR epð tou eswterikoô tou trig nou P Q R. Upìdeixh: 'Ena shmeðo M R 2 eðnai an kei sto eswterikì tou trig nou P QR an kai mìnon an kai mìnon an upˆrqoun λ P, λ Q, λ R (0, 1) me λ P + λ Q + λ R = 1 gia ta opoða M = λ P P + λ Q Q + λ R R.
27 Kefˆlaio 4 EukleÐdeia GewmetrÐa tou R Orismìc thc EukleÐdeiac GewmetrÐac tou R 2. To zeôgoc (R 2, E(2)), ìpou E(2) eðnai h omˆda twn isometri n tou R 2, kaleðtai EukleÐdeia gewmetrða tou R 2 (EukleÐdeio epðpedo) Omˆda peristrof n tou R 2 gôrw apì èna shmeðo. H peristrof enoc prosanatolismènou epipèdou Π katˆ gwnða θ gôrw apì to shmeðo P Π eðnai autoapeikìnish tou epipðdou pou se kˆje shmeðo M Π antistoiqeð M Π gia to opoðo P M = P M kai oi gwnða apì thn hmieujeða [P M) proc thn [P M ) eðnai θ. Prìtash To sônolo R(P ) ìlwn twn peristrof n enoc prosanatolismènou epipèdou Π gôrw apì opoiod pote shmeðo P efodiasmèno me thn prˆxh sônjeshc twn apeikonðsewn eðnai antimetajetik omˆda. Epomènwc (Π, R(P )) eðnai gewmetrða gia kˆje P Π. Apìdeixh. JewroÔme èna polikì sôsthma suntetagmènwn sto epðpedo Π me arq to shmeðo P. An Rθ P eðnai peristrof gôrw apì to P katˆ gwnða θ, tìte h RP θ apeikonðzei to M = (r, φ) sto Rθ P (M) = (r, φ + θ). 1. R P θ eðnai 1-1 kai epð. 2. H antðstrofh thc R P θ eðnai h RP θ 3. R P θ 1 R P θ 2 = R P θ 1 +θ Rθ P 1 Rθ P 2 = Rθ P 1 +θ 2 = Rθ P 2 Rθ P 1 Sunep c to sônolo ìlwn twn peristrof n eðnai omˆda. ShmeÐwsh Parathroume ìti gia kˆje ϕ R upˆrqei monadikì θ ( π, π] gia to opoðo ϕ = θ + 2nπ, ìpou n Z. Profan c Rϕ P = Rθ P. Sunep c to sônolo ìlwn twn peristrof n gôrw apì to P eðnai to sônolo R(P ) = {Rθ P : θ ( π, π]}. 25
28 26 Prìtash 'Estw Π èna epðpedo me èna orjokanonikì sôsthma suntetagmènwn Oxy. An Rθ O eðnai h peristrof tou Π gôrw apì thn arq twn axìnwn O = (0, 0) katˆ th gwnða θ ( π, π] kai Ro θ : R 2 R 2 eðnai apeikìnish pou orðzetai apì th sqèsh ( ) ( ) cos θ sin θ x Ro θ (x, y) =, θ ( π, π]. sin θ cos θ y tìte R O θ (M) = Ro θ(x, y) gia kˆje M = (x, y) Π kai gia kˆje θ ( π, π]. Apìdeixh. 'Estw M = (x, y) R 2. Tìte x = r cos θ M y = r sin θ M, ìpou r = OM kai θ M h gwnða apì ton Ox proc thn [OM)-hmieujeÐa. Tìte R O θ (M) = (x, y ), ìpou x = r cos(θ M + θ) = r cos θ M cos θ r sin θ M sin θ = x cos θ y sin θ y = r sin(θ M + θ) = r sin θ M cos θ + r cos θ M sin θ = x sin θ + y cos θ ( ) ( ) ( ) x 'Ara, Rθ O(M) = cos θ sin θ x y = = Ro sin θ cos θ y θ (x, y). Prìtash H omˆda R(O) twn peristrof n gôrw apì to O ( = (0, 0) ) katˆ gwnða θ a b ( π, π] eðnai isomorfik me thn omˆda SO(2, R) ìlwn twn pinˆkwn me a b a 2 + b 2 = 1. Apìdeixh. Epeid a 2 + b 2 = 1, upˆrqei monadikì θ A ( π, π], tètoia ste a = cos θ A kai b = sin θ A. OrÐzoume h : SO(2, R) R(O) me h(a) = Rθ O A. An A, B SO(2, R), tìte (( )( )) (( )) cos θa sin θ h(ab)=h A cos θb sin θ B cos(θa + θ = h B ) sin(θ A + θ B ) = sin θ A cos θ A sin θ B cos θ B sin(θ A + θ B ) cos(θ A + θ B ) = R O θ A +θ B = Ro θa Ro θb = h(a) h(b). 'Ara, h eðnai isomorfismìc. Prìtash Kˆje peristrof gôrw apì èna shmeðo eðnai isometrða. Apìdeixh. 'Estw Ro θ peristrof gôrw apì to O = (0, 0). Gia shmeða ū = (xū, yū) kai v = (x v, y v ) tou R 2 èqoume Epomènwc Ro θ (ū) = (xū cos θ yū sin θ, xū sin θ + yū cos θ) Ro θ ( v) = (x v cos θ y v sin θ, x v sin θ + y v cos θ) ((xū Ro θ (ū), Ro θ ( v) = x v) cos θ (yū y v) sin θ ) 2 ( + (xū x v) sin θ + (yū y v) cos θ ) 2 = (xū x v ) 2 + (yū y v ) 2 = ū v. 'Ara, R θ eðnai isometrða. 'Estw ìti P eðnai èna shmeðo tou epipèdou diforetikì apì thn arq O = (0, 0) kai T P eðnai h metaforˆ sto R 2 katˆ diˆnisma OP = P. Tìte h peristrof R θ (P ) gôrw apì to P katˆ gwnða θ paristˆnetai wc sônjesh isometri n: Rθ P = T P R θ T P.
29 Anaklˆseic tou R 2 wc proc mia eujeða ε. H anˆklash wc proc mia eujeða ε tou R 2 eðnai h apeikìnish a ε : R 2 R 2 pou kˆje shmeðo A R 2 antisoiqeð to summetrikì tou a ε (A) wc proc thn ε. Kˆje anˆklash a ε wc proc mia eujeða ε eðnai 1-1 kai epð metasqhmatismìc tou R 2 kai a 1 ε = a ε. Oi anaklˆseic wc proc mia eujeða ε den apoteloôn omˆda, afoô h sônjesh dôo anaklˆsewn eðnai tautotik apeikìnish, h opoða den eðnai den eðnai anˆklash. SumbolÐzoume me ε θ thn eujeða pou dièrqetai apì thn arq twn axìnwn kai sqhmatðzei gwnða θ ( π, π] 2 2 me ton jetikì hmiˆxona Ox. Me Re θ sumbolðzoume thn anˆklash tou R 2 wc proc thn ε θ. ShmeÐwseic H anˆklash wc proc ton ˆxona Ox orðzetai apì thn sqèsh ( ) ( ) ( ) x 1 0 x Re 0 (x, y) = = y 0 1 y 2. H anˆklash wc proc mia eujeða ε θ pou dièrqetai apì thn arq twn axìnwn kai sqhmatðzei gwnða θ ( π, π ] me ton jetikì hmiˆxona Ox orðzetai apì thn sqèsh 2 2 ( ) ( ) cos 2θ sin 2θ x Re θ (x, y) =. sin 2θ cos 2θ y Prˆgmati, Re θ = Ro θ Re 0 Ro θ. Epomènwc Re θ (x, y) = = ( ) ( ) ( ) ( ) cos θ sin θ 1 0 cos( θ) sin( θ) x = sin θ cos θ 0 1 sin( θ) cos( θ) y ( ) ( ) cos 2θ sin 2θ x sin 2θ cos 2θ y 3. Upˆrqei mia 1-1 antistoiqða metaxô twn anaklˆsewn Re θ, θ ( π, π ], wc proc tic 2 2 ( ) a b eujeðec pou dièrqontai apì thn arq (0, 0) kai twn pinˆkwn thc morf c me b a a 2 + b 2 = 1 (apì to Je rhma 1.3.5). 4. To sônolo twn peristrof n tou R 2 gôrw apì to (0, 0) katˆ gwnða θ ( π, π] kai twn anaklˆsewn wc proc eujeðec pou dièrqontai apì to (0, 0) me prˆxh sônjeshc apeikonðsewn ( ) a b eðnai omˆda isìmorfik me thn omˆda O(2, R) ìlwn twn pinˆkwn twn morfwn ( ) b a a b, ìpou a b a 2 + b 2 = 1.
30 28 Prìtash Kˆje anˆklash tou R 2 eðnai isometrða. Apìdeixh. 'Estw Re 0 (x, y) : R 2 R 2 anˆklash wc proc ton ˆxona Ox. Gia shmeða ū = (x u, y u ) kai v = (x v, y v ) tou R 2 èqoume Re 0 (ū) = ( x u, y u ) Re 0 ( v) = ( x v, y v ) Epomènwc Re 0 (ū) Re 0 ( v) = ( x u + x v ) 2 + (y u y v ) 2 = (x u x v ) 2 + (y u y v ) 2 = ū v. 'Ara, Re 0 eðnai isometrða. 'Estw ìti ε : y = ax + b eðnai mia eujeða tou R 2 diaforetik apì ton ˆxona Ox kai θ ( π, π ) me tan θ = a. 2 2 Tìte h anˆklash a ε wc proc thn ε paristˆnetai wc sônjesh isometri n wc ex c: a ε = T b Ro θ Re 0 Ro θ T b,, ìpou b = (0, b). 'Estw ìti ε : x = b. Tìte h anˆklash a ε wc proc thn ε paristˆnetai wc sônjesh isometri n wc ex c: a ε = T b Ro π Re 0 Ro 2 π T 2 b, ìpou b = (b, 0). Sunep c, kˆje anˆklash wc proc eujeða eðnai isometrða wc sônjesh isometri n. 4.2 IsometrÐec tou R 2. Je rhma Kˆje isometrða T : R 2 R 2 pou af nei stajer thn arq twn axìnwn (T ( 0) = 0) èqei tic akìloujec idiìthtec: 1. T (ā) = ā, gia kˆje ā R T (ā) T ( b) = ā b, gia kˆje ā, b R 2 (T af nei analloðwto to eswterikì ginìmeno). 3. T eðnai grammik apeikìnish. Apìdeixh. 1. Gia 0 = (0, 0) paðrnoume T (ā) = T (ā) 0 = T (ā) T ( 0) = ā 0 = ā. 2. 'Estw a, b R 2. Tìte ā b 2 = ā 2 + b 2 2ā b T (ā) T ( b) 2 = T (ā) 2 + T ( b) 2 2T (ā) T ( b) Epeid T eðnai isometrða, T (ā) T ( b) = ā b. Apì thn idiìthta 1, pou apodeðxame T (ā) = ā kai T ( b) = b. 'Ara, T (ā) T ( b) = ā b.
31 29 3. 'Estw a, b R 2 kai λ R. Tìte apì tic idiìthtec 1 kai 2 pou apodeðxame prokôptei ìti: T (λā) λt (ā) 2 = T (λā) 2 +λ 2 T (ā) 2 2λT (λā) T (ā) = λā 2 +λ 2 ā 2 2λλā ā = 0. 'Ara, T (λā) = λt (ā). EpÐshc T (ā+ b) (T (ā)+t ( b)) 2 = T (ā+ b) 2 2T (ā+ b) T (ā) 2T (ā+ b) T ( b)+ T (ā)+t ( b) 2 Apì tic idiìthtec 1 kai 2 pou apodeðxame prokôptei ìti: T (ā + b) 2 = ā + b 2 = ā 2 + 2ā b + b 2, T (ā + b) T (ā) = (ā + b) ā = ā 2 + ā b, T (ā + b) T ( b) = (ā + b) b = b 2 + ā b, T (ā) + T ( b) 2 = T (ā) 2 + T ( b) 2 + 2T (ā) T ( b) = ā 2 + b 2 + 2ā b. Apì ta parapˆnw T (ā + b) (T (ā) + T ( b)) 2 = 0. 'Ara, T (ā + b) = (T (ā) + T ( b)). Sunep c T eðnai grammik apeikìnish. Je rhma ( ) Gia kˆje grammik isometrða T : R 2 R 2 upˆrqei orjog nioc pðnakac c11 c C = 12 gia ton opoðon T ( v) = C v gia kˆje v R c 21 c Apìdeixh. Jètoume ē 1 = (1, 0) kai ē 2 = (0, 1). Epeid {ē 1, ē 2 } eðnai bˆsh tou R 2, èqoume T (ē 1 ) = c 11 ē 1 + c 21 ē 2 (4.1) T (ē 2 ) = c 12 ē 1 + c 22 ē 2. 'Estw v = (x, y) R 2 kai T ( v) = x ē 1 + y ē 2. Epeid T eðnai grammik, T ( v) = T (xē 1 + yē 2 ) = xt (ē 1 ) + yt (ē 2 ) = (4.2) = x(c 11 ē 1 + c 21 ē 2 ) + y(c 12 ē 1 + c 22 ē 2 ) = = (xc 11 + yc 12 )ē 1 + (xc 21 + yc 22 )ē 2 Epomènwc x = xc 11 + yc 12 (4.3) y = xc 21 + yc 22 ( ) ( ) ( ) c11 c Gia C = 12 x x oi (4.3) grafontai c 21 c 22 y = C, isodônama T ( v) = C v. y Epomènwc T ( 0) = C 0 = 0 gia thn isometrða T. Apì to Je rhma èpetai ìti T (ē 1 ) = ē 1, T (ē 2 ) = ē 2 kai T (ē 1 ) T (ē 2 ) = ē 1 ē 2. 'Ara, c c 2 21 = 1, c c 2 22 = 1 kai c 11 c 12 + c 21 c 22 = 0. Dhlad o C eðnai orjog nioc.
32 30 Je rhma Gia kˆje isometrða T : R 2 R 2 upˆrqei orjog nioc 2 2 pðnakac C kai ā R 2, tètoia ste T ( v) = C v + ā gia kˆje v R 2. Apìdeixh. 'Estw T : R 2 R 2 mia isometrða. An T ( 0) = 0, tìte T eðnai grammik isometrða. Epomènwc upˆrqei orjog nioc pðnakac C gia ton opoðon T ( v) = C v gia kˆje v R 2. An T ( 0) = ā 0, tìte h apeikìnish S( v) = T ( v) ā gia kˆje v R 2 eðnai isometrða gia thn opoða S( 0) = T ( 0) ā = ā ā = 0. 'Ara, S eðnai grammik isometrða. Epomènwc S( v) = C v, ìpou C orjog nioc pðnakac. 'Ara, T ( v) = S( v) + ā = C v + ā gia kˆje v R 2. Pìrisma Kˆje isometrða eðnai afinikìc metasqhmatismìc. Pìrisma Kˆje isometrða orðzetai monos manta apì tic eikìnec A, B, C tri n mh suneujeiak n shmeðwn A, B, C. Pìrisma To sônolo ìlwn twn isometri n tou R 2 pou af noun stajer thn arq (0, 0) me thn prˆxh sônjeshc apeikonðsewn eðnai omˆda isomorfik me thn pollaplasiastik omˆda twn orjog niwn pinˆkwn O(2, R). Je rhma Kˆje isometrða enìc epipèdou mporei na parastajeð wc sônjesh kˆpoiwn apì tic parakˆtw isometrðec: metaforˆ, anˆklash wc proc ton Ox kai peristrof gôrw apì thn arq. Apìdeixh. 'Estw T : R 2 R 2 mia isometrða. Tìte upˆrqei orjog nioc 2 2 pðnakac C kai ā R 2, tètoia ste T ( v) = C v + ā gia kˆje v R 2. Epomènwc T ( v) = S( v) + ā, ìpou ( S( v) = C v. ) ( ) cos θ sin θ cos θ sin θ O C èqei mða apì tic morfèc, ìpou θ ( π, π]. sin θ cos θ sin θ cos θ ( ) cos θ sin θ C = antistoiqeð sthn peristrof gôrw apì thn arq katˆ gwnða θ. sin θ cos θ ( ) ( ) ( ) cos θ sin θ cos θ sin θ 1 0 C = = antistoiqeð sthn anˆklash wc proc sin θ cos θ sin θ cos θ 0 1 ton Ox pou akoloujeðtai apì thn peristrof kata th gwnða θ gôrw apì thn arq. 'Ara, S antistoiqeð sthn peristrof sth sônjesh anˆklashc kai peristrof c. Epomènwc T = S + ā eðnai sônjesh kˆpoiwn apì tic parakˆtw isometrðec: metaforˆ, anˆklash wc proc ton Ox kai peristrof gôrw arq. ShmeÐwsh To eswterikì ginìmeno twn ā = (a 1,..., a n ), b = (b 1,..., b n ) R n eðnao o arijmìc ā b = a 1 b a n b n. 'Opwc kai sthn perðptwsh tou R 2, apodeiknôetai ìti ta Jewr mata 4.2.1, kai isqôoun kai sto R n.
33 AnalloÐwtec eukleðdeiwn metasqhmatism n. Epeid kˆue eukleðdeioc metasqhmatismìc eðnai isometrða, apì ton orismì thc isometrðac prokôptei ìti Kˆje eukleðdeioc metasqhmatismìc: diathreð ta m kh twn eujôgrammwn tmhmˆtwn. diathreð tic apostˆseic. Epeid kˆje eukleðdeioc metasqhmatismìc eðnai sônjesh peristrof c, anˆklashc kai metaforac, kai epeid oi peristrofèc, oi anaklˆseic kai oi metaforèc diathroôn tic gwnðec, prokôptei ìti Kˆje eukleðdeioc metasqhmatismìc diathreð tic gwnðec metaxô twn eujei n. Epeid kˆje eukleðdeioc metasqhmatismìc eðnai afinikìc, oi analloðwtec twn afinik n metasqhmatism n eðnai analloðwtec twn eukleðdeiwn metasqhmatism n. Epomènwc Kˆje eukleðdeioc metasqhmatismìc apeikonðzei eujeðec se eujeðec, apeikonðzei parˆllhlec eujeðec se parˆllhlec eujeðec, diathreð ton aplì lìgo twn tri n suneujeiak n shmeðwn. ApodeiknÔetai epðshc ìti Kˆje eukleðdeioc metasqhmatismìc diathreð ton bajmì tou polliwnômou. apeikonðzei mia èlleiyh se èlleiyh me thn Ðdia ekkentrìthta. apeikonðzei mia uperbol se uperbol me thn Ðdia ekkentrìthta. apeikonðzei mia parabol se parabol me thn Ðdia estiak parˆmetro.
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