DIAFORIKH GEWMETRIA

EPIFANEIES, FULLO 2

'Askhsh 1. 'Estw kamplh c : (u), u I, pou u fusik parmetroc thc c kaiupojtoume ti h c den qei shmea kampc.An S : r(u, v) = ((u) + v(u), (u, v) I (0, +), me qrsh tou tridrou kai twnexissewn Frenet thc c, kai profanc thc kampulthtac kai thc stryhc thc c,

(i) Dexte ti h r(u, v) enai kanonik (r1(u, v) r2(u, v) 6= 0 gia kje (u, v))kai anaptxte ta diansmata r11(u, v), r12(u, v), r22(u, v) sto tredro Frenet thc c stoshmeo (u).

(ii) Dexte ti to monadiao kjeto dinusma N(u, v) thc epifneiac S (efaptmenhepifneia thc c) enai suggramik me to detero monadiao kjeto dinusma b(u) thckampplhc kai prosdiorste th kampulthta Gauss thc S.

(iii) Prosdiorste to dinusma gewdaisiakc kampulthtac kai to dinusma k-jethc kampulthtac thc kamplhc u = v thc S.

'Askhsh 2. Prosdiorste th parametrik exswsh thc epifneiac S (spera, troc)pou prorqetai ap th peristrof thc perifreiac tou kklou tou epipdou xoz mekntro to shmeo (1, 0, 0) kai aktna < 1 per ton xona twn z.

Parastste grafik thn epifneia kai prosdiorste to edoc twn shmewn P (1 +, 0, 0), Q(1 , 0, 0) thc S.'Askhsh 3. 'Estw kamplh c : (u), u I, pou u fusik parmetroc thc c kaiupojtoume ti h c den qei shmea kampc. 'Estw dinusma d tou R3 ste (u)d = 0gia kje u. Dexte ti (u) d = 0 gia kje u. Jewrome th kulindrik epifneiaS : r(u, v) = (u) + vd, (u, v) I R, me odhg thn c kai genteira parllhlh stod.

(i) Dexte ti kje shmeo thc S enai parabolik,(ii) Dexte ti h epitqunsh thc kamplhc : w(t) = r(u(t), v(t)) thc S dnetai

ap to tpo

w(t) = k(u(t))(u(t))2n(u(t)) + u

(t)t(u(t)) + v

(t)d,

pou t,n to monadiao efaptmeno kai to prto monadia kjeto dinusma thc c kai kh kampulthta thc c sta antstoiqa shmea.

(iii) Prosdiorste tic u(t), v(t) ste se kje shmeo thc , to monadiao kjetodinusma thc na enai kjeto sthn epifneia S (tte h kamplh ja enai gewdaisiakthc S).

'Askhsh 4. 'Estw h epifneia S : r (u, v) = (u, v, uv2) , u, v > 0, kai stw h kamplhC : v = u + 1, thc epifneiac. Prosdiorste thn parametrik exswsh thc kamplhckai sth sunqeia prosdiorste to dinusma kampulthtac thc kamplhc sto shmeo(1, 2, 2) kai analste to dinusma aut sto ssthma {r1, r2,N} kai prosdiorste todinusma kampulthtac kai to dinusma kjethc kampulthtac (dste touc ants-toiqouc orismoc). Prosdiorste th diejunsh thc kamplhc sto tuqao shmeo thc.

'Askhsh 5. Exetste to edoc twn shmewn thc epifneiac ek peristrofc r (x, j) =(xcosj, xsinj, f (x)), x (1, 2), j R. Poic sunjkec prpei na ikanopoie h sunrthshf ste kje shmeo thc epifneiac na enai elleiptik?

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'Askhsh 6. Prosdiorste thn kjeth kampulthta tou meshmbrino j = p4tou elleiy-

oeidoc ek peristrofc r (j,f) = (acosfcosj, acosfsinj, bsinf) sto shmeo f= p6.

Epshc prosdiorste thn kjeth kampulthta sto dio shmeo jewrntac san exswshtou elleiyoeidoc thn r (f, j) = (acosfcosj, acosfsinj, bsinf) kai exhgste giat hkampulthta allzei prshmo.

'Askhsh 7. Prosdiorste ta edh twn shmewn thc epifneiac tou knou r (u, j) =(ucosj, usinj, au) , u > 0, j R kai exetste an qei omfalik shmea. Prosdiorstethn kjeth kampulthta thc epifneiac sto shmeo

(1, p

4

)wc proc thn diejunsh (1, 1)

tic asumptwtikc dieujnseic kai tic kriec kampulthtec thc epifneiac sto shmeoaut. Na dsete touc antstoiqouc orismoc

'Askhsh 8. Prosdiorste tic asumptwtikc dieujnseic, th msh kampulthta kai thkampulthta Gauss sto tuqao shmeo thc epifneiac S : r (u, v) = (u, v, uv) , u, v R.'Askhsh 9. Prosdiorste thn exswsh thc kjethc tomc thc epifneiac r (r, t) =(r, rt2, t), r>0, t R sto shmeo r (1, 1) me diejunsh (1, 2). Poia enai h kjethkampulthta thc kjethc tomc?

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