The sphere of radius a has the geographical form. r (,)=(acoscos,acossin,asin) T =(p(u)cos v, p(u)sin v,q(u) ) T.

Transcription

1 Che 5. Dieeil Geome o Sces 5. Sce i meic om I 3D sce c be eeseed b. Elici om z =. Imlici om z = 3. Veco om = o moe geel =z deedig o wo mees. Emle. he shee o dis hs he geoghicl om =coscoscossisi Emle. he clide bil o he ce = b i he -le hs he om = b - < < Emle 3. Sce o eolio b oig ce = q b bo he z-is cos = R z = si si cos =cos si q q b Secicl cses e: os wih = R + cos si A coe wih = m - ge ecos o he sce e d. Hece i oml is gie b ssmig o-sigl oi. I he sce is gie imlicil z = he

2 5. Meic oeies Disce o he sce is mesed b A d d d dz d d z z B A A d d d d ds whee G F F E A A B i sdd oio his is he s dmel om o he sce: Gd Fd E ds he i ge ˆ log he ce = is / ˆ B A he legh o he segme o he ce om = o = is d B d s / I wo ces i i i iesec gle o he sce he / / ˆ ˆ cos ds ds B B B A A da

3 A eleme e da will be gie b da [ ] [ d ] d d We he EG F deb B hs A B R R d o Regio R o he sce. 5.3 Ces Reclled h o geel sce ce sˆ d sˆ s knˆ whee Nˆ is he icil oml o he ce o o cose wih he oml o he sce. I lies o he sce he so h Dieeiig gi Noice h he sce oml is eedicl o d S ˆ kn D 3

4 he igh-hd side is he secod dmel om o he sce L M D is sdd oio. M N he oml ce o he ce i he sce is deied o be S D B Noe h ˆ ˆ N cos i.e. i is he comoe o i he diecio o. he ohe comoe o i he ge le is kow s he geodesic ce g becse o ohogoli i hs he mgide g si A sce ce o which g = ee oi is clled geodesic s sigh s ossible o he sce. Coside ow d s cio o he diecio d d d Le he L E M N F G LMN EFG As chges diecio i he sce will chiee mimm d miimm le less L: M: N = E: F: G i h cse is ideede o sch locios e clled mbilic ois d Seig he icil diecios d he coesodig icil d ces k e goeed b FN GM EN GL EM FL d EG F k LG NE FM k LN M 4

5 he solios sis EF G LG FM NE k k EG F LN M de kk EG F de D B K k k is clled he Gssi Ce. H k k is he me Ce Gemi Ce. I he wo diecios coesod o d d d d d d d d d d he E F G d d d d d d d d heeoe he wo icil diecios e ohogol. Some geomeicl meig o he ces e he ollowig. A sce is clled miiml i H = eewhee. A miiml sce wih bod l hs he smlles sce e mog ll sces wih bod l. I he icil diecios e ke s he meic ces he F M d L N k k E G ce i ohe diecio is he gie b LN E G K k k EG EG EG k cos k si whee is he gle bewee d which is kow s he Ele s oml. 3 I K> oi P o he sce he P is elliic oi. As k k he he sme sig so ll he sce is bedig he sme w i ll diecios. 5

6 4 I K< oi P he i is hebolic oi. ge diecios P c bed w o owds he ge le. 5 I K= he eihe Oe icil ce ol is zeo. he oi P is bolic oi d oe o he icil diecio is sigh e P. b Boh icil ce e zeo. he oi is secil e o mbilic oi d is l. Noe: Isoled l ois c eis o sces which is om l. E.g. he z 3 3 P moke sddle sce 6 Coside oi P o he sce z = B chge o coodies choose he oigi P d es log he icil diecios P lso z-is i he diecio o he sce oml P he he sce hs eqio z = locl o wih = - is he ge le d z 3 3 O # "! 6

7 Now kig A P = d E F G he icil diecios e ohogol. Also Hece z L M N 3 3 L M N O 3 3 k k O hese coics jsi he emiolog sed. he simles sce o which ll hee cses % o Gssi ce K occ is he os. $ 5.4 Secil cses 5.4. Deeloble sces Coside sce i R 3 which is cosced b moig sigh lie his so clled led sce hs he om 7

8 whee is he osiio eco o oi o gie lie is he diecio o he moig lie geeos o i sigh lies e sed o joi wo gie ces he Emles e clides d coes. Now we look o codiios so h led sce c be olled io l le wiho disoio i.e. disces e eseed. I led sce is deeloble he ll he geeos eell lie o le heeoe he e eihe llel o iesec oe ohe. Now he iesecio o wo geeos d is goeed b 8

9 I cse ll he geeos e llel he boe codiio is lso sisied. heeoe i is he codiio o led sce becomes deeloble. As heeoe N M D B de LN M K o deeloble sces. de de B I cse he led sce is goeed b wo ces he codiio becomes his c is sed i he ge le mehod o geeig deeloble sce ssig hogh wo ces. 9

10 5.4. Eeloe o sce ces Regdig s mil o ces deedig o mee. hee m eis ce e which is geil o ee ce he meic le. Sch ce i i eiss is clled he eeloe o he mil o ces. I ems o he oigil mees i imlies is llel o d he sce oml is o deied hese ois: I cse o he deeloble sce he geeos will he eeloe i he e o llel. A eeloe sisies so log s he geeos e o llel hece he locio o commo ge occs d he eqio o he eeloe o he geeos is e