
deerghavrithaakrithiyilulla geodesics

thrikonaakrithiyilulla shringalakalude pariharavumaayi jiodesiyumaayi bandhappett oru deerghavrithathile jiyodesiksinekkurichu patanam uyarnnuvannu. bhoomiyude aakruthi alppam paranna golamaaya "oolbettu ellipsoid" kondu nannayi kanakkaakkappedunnu. oru samathala uparithalathile nerrekhaykku samaanamaayi valanja uparithalathile rund poyintukalkkidayilulla ettavum cheria paathayaan geodesic. athinaal oru deerghavrithaakrithiyilulla trikona shringalayude pariharam golaakrithiyilulla trikonamithiyile oru koottam vyaayaamangalaanu.
bhoomiye oru golamaayi kanakkaakkukayaanenkil, geodesics valia vrithangalaanu (avayellaam adanjirikkunnu), prasnangal golaakrithiyilulla trikonamithiyilekku kurayunnu. ennirunnaalum, bhoomiyude bhramanathinte phalam alpam oolbettu deerghavrithavumaayi saamyamullathaanennu kaanichu eee saahacharyathil, bhoomadhyarekhayum meridian maathramaan lalithamaaya adanja geodesics. koodaathe, madhyarekhayile rund poyintukalkkidayilulla ettavum cheria paatha madhyarekhayiloode kadannupokanamennilla. avasaanamaayi, deerghavritham oru trioxial deerghavrithamaayi (moonnu vyathyastha semi aaxukalulla) maarunnathin kooduthal aswasthamaakkukayaanengil, moonnu geodesics maathrame adaykkukayulloo.
viplavathinte deerghavrithathekkurichu geodesics.
geodesics nirvachikkaan niravadhi vazhikalundu. oru uparithalathile rund poyintukalkkidayilulla ettavum cheria paathayaan lalithamaaya nirvachanam. ennirunnaalum, poojyamaaya geodesic vakrata-i ulla paathakalayi avaye nirvachikkunnath palappozhum kooduthal upayogapradamaanu.iii.valanja uparithalathile nerrekhakalude analog. deerghavrithaakrithiyilulla uparithalathiludaneelam sanjarikkunna geodesics ulkkollunna eee nirvachanam mattu roottukal kooduthal nerittullathum parasparam murichukadakkunnatho veendum kandethunnathumaya pathakal ulkkollunnathumaayathil avaa arambha poyintilekku madangaan thudangunnu. oru geodesicsinte cheru bhagangal ippozhum avayude anthima bindukkalkkidayilulla ettavum cheria paathayaan, ennaal geodesics aagolathalathil ettavum kuranjathaayirikkanamennil (i.iii. saadhyamaaya ellah vazhikalilum ettavum cheruth). aagolathalathil ettavum hrasvamaaya oro pathayum oru geodesic aanu, pakshe tirichalla.
pathinettaam noottaandinte avasaanathode, viplavathinte deerghavritham (spheroid enna padavum upayogikkunnu) bhoomiyude roopathinu nannayi amgeekarikkappetta oru ekadesha kanakkaayirunnu. thrikonaakrithiyilulla shringalakalude krameekaranatthil ellah alavukalum oru reference deerghavrithamaayi kuraykkukayum golaakrithiyilulla trikonamithiyile oru vyaayaamamaayi thathfalamaayundaakunna dvimaana prashnam pariharikkukayum cheiyunnu.
vividha geodesic prasnangal rund tharangalil onnaayi kuraykkaan kazhiyum. rund pooyintukal pariganikkuka akshaamshathilum rekhaamshathilum akshaamshathilum rekhaamshathilum (chithram kaanuka. 1). asimuthukalum rund anthima bindukkalumulla neelamullathaanu bandhippikkunna geodesic (muthal). saadhaaranayaayi pariganikkunna rund geodesic prasnangal ivayaanu
athippazhathil ninnu kaanaan kazhiyum. 1, eee prashnangalkk oru koonn nalkiyirikkunna trikonathe, nerittulla prashnathinum vipareeta prashnathinum, athinte rund vashangalkkum, pariharikkuka ennathaanu.
oru golathe sambandhichidatholam eee prasnangalkkulla pariharangal golaakrithiyilulla trikonamithiyile lalithamaaya vyaayaamangalaanu, avayude pariharam oru golaakrithiyilulla trikonathe pariharikkunnathinulla soothravaakyangal nalkunnu. (grettu sarkkil naavigeshanekkurichulla lekhanam kaanuka.)
oru deerghavrithaakrithiyilulla viplavathinu, jiodesikkine nirvachikkunna swabhaavasavisheshatha sthiraamkam kandethi. geodesixinte paathakalkk oru chittayaya pariharam nalki (1808lum 1810lum thudarnnulla prabandhangal).
nerittulla prashnathinulla poornna pariharam (computational tableukalum thayyarakkiya udaaharanavum upayogichu poortthiyaakkuka) nalkunnu.
pathinettaam noottaandil geodesics saadhaaranayaayi "ettavum cheria varikal" ennu vilikkapettu.
geodesic line" (yadhaarthathil, oru vakram) enna padam srishtichatha
nose designarons settu ligney leeii nome dii "ligney geodesic" [eee variye nammal "geodesic line" ennu vilikkum].
eee padavali engleeshil "geodesic line" allengil "geodetic line" aayi avatharippichu, udaaharanathinu,
nammal ippol vivarikkunna reethiyilo trikonamithikalil ninnu anumaanikkunna reethiyilo kandethiyittulla oru variye nammal soochippicha maarggangaliloode "geodetic" allengil "geodesic line" ennu vilikkunnu bhoomiyude uparithalathil athinte rund attangalkkidayil varaykkaan kazhiyunna ettavum cheruthaanu ithinu; athinaal aaa rund pooyintukal thammilulla dooratthinte sariyaaya yatra alavukol ithaanu.
mattu mekhalakal ithu sweekarikkunnathil "geodesic line", palappozhum "geodesic" ennu churukkipparayunnu, ithinu munganana nalki.
eee vibhaagam revolution deerghavrithaakrithiyilulla (oolbettu, prolate) prashnathe kaikaaryam cheiyunnu. oru trioxial ellipsoidile prashnam adutha vibhaagathil ulppeduthiyittundu.
oru geodesic samavaakyangal.
ivide oru jiodesikkinulla samavaakyangal vikasippicheduthittundu; derivation athine aduthu pinthudarunnu. ,,,,,, koodaathe eee samavaakyangalude vyulppannam nalkukayum cheiyunnu.
bhoomadhyarekhayude vyaasavum dhruva ardha akshavumulla oru deerghavrithaakrithiyilulla bhramanapatham pariganikkuka. parannathum, vikendrathayum, randaamathe vikendrathayum nirvachikkuka
deerghavrithaakrithiyilulla oru paathayude oru praadhamika bhaagathinu neelam undakatte. athippazhangalil ninnu. 2 umm 3 umm, athinte asimat aanenkil, athu athumaayi bandhappettirikkunnave nammal kaanunnu.
vakrathayude meridian aaram evideyaanu, akshamsha vrithathinte aaram, vakrathayude saadhaarana aaram.
athinaal praadhamika vibhaagam nalkunnathu
allengil
evide, lagranchian function throo aandu ennivaye aasrayichirikkunnu. evaykkidayilulla oru ekapaksheeyamaaya paathayude neelam
evideyaanu samthrupthiyude pravartthanam. ettavum cheria paatha allengil geodesic ennathu minimise cheiyunna function kandethuka ennathaanu. ithu vyathiyaanangalude calculusile oru vyaayaamamaanu, beltramy identityaanu minimising avastha nalkunnathu,
iii. kyu. kalkk pakaramaayi upayogikkukayum cheyyuka. nalkunnu
 oru jyaamitheeya nirmmaanam upayogichu eee bandham kandethi; samanamaaya oru vyulppannam avatharippikkunnath. eee bandham verthirikkunnathu nalkunnu
ithu, eqsumaayi chernnu. , oru geodesicinu saadhaarana deferential samavaakyangalude oru samvidaanathilekku nayikkunnu
paaraameettar akshaamshatthinte adisthaanathil namukk prakadippikkaan kazhiyum,
appol clairottinte bandham maarunnu
trikonatthinte rund vashangalumayi bandhapetta golaakrithiyilulla trikonamithiyude saine niyamamaanith (chithram kaanuka. 4),,, avayude vipareeta konukalum.
moonnaam vashathe, "golaakrithiyilulla kamaanatthinte neelam", "golaakrithiyilulla rekhamsham" enniva ulppedunna konumaayi bandhapetta bandham kandethunnathinu, madhyarekhayil ninnu aarambikkunna oru jiodesikkine prathinidheekarikkunna trikonathe pariganikkunnath upayogapradamaanu; amshakam kaanuka. 5. eee chithratthil, ooksilari golathe paraamarsikkunna variableukal paraantheesukalil kaanichirikkunna deerghavrithathinte anubandha alavukalode kaanikkunnu.
subscriptukalillaath alavukal ekapaksheeyamaaya binduvine soochippikkunnu; geodesic vadakkott bhoomadhyarekha kadakkunna pooyint, athinte uthbhavamaayi upayogikkunnu,
ananthamaayi neengikkond vasham neettiyittundengil (amjooram kaanuka. 6), namukk labikkum
eqsukal samyojippikkuka. koodaathe deferential samavaakyangal nalkunnu 
athu thammilulla bandham
nalkunna
angane jiyodesikkinte deferential samavaakyangal maarunnu
eee rund deferential samavaakyangalilum swatantra paaraameteraayi upayogikkukayum athuvazhi integralukalaayi prakadippikkukayum cheyyuka ennathaanu avasana ghattam. saine niyamam konukalilum aakruthiyile golaakrithiyilulla trikonathilum prayogikkuka. 5 sammanangal
asimat evideyaanu?
ithu samavaakyathilekku maattisthaapikkunnathum phalam samyojippikkunnathum nalkunnu
evide
integralinte paridhikal thiranjedukkunnathu angane aanu. ardha akshangalulla deerghavrithathile kamaanatthinte samavaakyathin thulyamaanu eee samavaakyam ennu choondikkaanichu. eee samavaakyam prakadippikkunnathin, nammal ezhuthunnu
clairottinte bandhatthilninnu athu pinthudarunnu.
eee vilavu
bhoomadhyarekha kadakkumbol, integralukalude paridhikal thiranjedukkunnu.
ithu sahaayagolam upayogichu oru jiyodesikkinte paathayude pariharam poortthiyaakkunnu. eee upakaranathiloode oru valia vritham oru deerghavrithaakrithiyilulla oru jiodesikkilekku krithyamaayi mapp cheyyaan kazhiyum.
oru bhouma deerghavrithathil (cheria parannukidakkunna) geodesics ekadeshamaayi kanakkakkaan niravadhi vazhikalundu; ivayil chilath bhoomishaasthraparamaaya doorathekkurichulla lekhanathil vivarichittundu. ennirunnaalum, iva saadhaaranayaayi sankeernathayil krithyamaaya parihaarathinulla reethiyumaayi thaarathamyappeduthaavunnatha.
geodesicsinte swabhaavam.
figue. meridian (pacha), bhoomadhyarekha (chuvapp) enniva ulkkollunna lalithamaaya adanja geodesics 7 kaanikkunnu. (ivide "lalitham" enna yogyatha ardhamaakkunnathu geodesic oru edapedal svayam-intersection illaathe svayam adaykkunnu ennaanu.) muun vibhaagathil nalkiyirikkunna geodesixinte samavakyangalil ninnaan ithu pinthudarunnathu.
matellaa geodesics athippazhangalaal tharamthirickunnu. bhoomadhyarekhayil aarambikkunna oru geodesic kaanikkunna 8 umm 9 umm. bhoomadhyarekhaye chutti geodesic aandolanangal nadathunnu. bhoomadhyarekha crossingukale "nodukal" ennum paramavadhi allengil ettavum kuranja akshamsha pooyintukale "vertisukal" ennum vilikkunnu; vertisukalude parametric akshamshangal nalkunnathu rekhamsham vardhikkunnathinu munbu geodesic akshaamshathil oru poornna aandolanam poortthiyaakkunnu. athinaal, bhoomadhyarekhayude oro thudarchayaaya vadakkottulla crossingle (amjooram kaanuka. 8), bhoomadhyarekhayude poornna sarkyoottinekkaal ekadesham kuravaanu (oru prolate deerghavrithathinu, eee alavu negatteevu aanu, koodaathe oru poornna sarkyoottinekkaal kooduthal poortthiyaakkunnu; ampathy kaanuka. 10). eathaandu ellah moolyangalkkum, geodesic rund sheershaka akshaamshangalkkidayillu deerghavrithathinte aaa bhagam niraykkum (chithram kaanuka. 9).
deerghavritham mathiyaya alavil oblettu aanenkil, i.iii.,, lalithamaaya clossed geodesicsinte mattoru klaas saadhyamaan. atharam rund geodesics athippazhangalil chithreekarichirikkunnu. 11 umm 12 umm. ivideyum bhoomadhyarekhayile asimuth, pachaykku (marupadi. neela) geodesic (rep) aayi thiranjedukkunnu. ), angane geodesic 2 poortthiyaakkunnu (respa. 3) deerghavrithathinte oru sarcutil bhoomadhyarekhayekkurichulla poornnamaaya aandolanangal.
figue. 13 geodesics (neelayil) avaa ettavum cheria paathakalayi avasaanikkunna pooyint vare onniladhikam aayi purathuvarunnu. (deerghavrithaakrithiyilulla phalangal vardhippikkunnathinaayi parannathaakkunnath vardhippichu.) (pachayil) sthirathayude valavukalayi kaanikkunnu, avaa geodesic vrithangale kendreekarichirikkunnu. eathu uparithalathilum geodesics geodesic vrithavum samakonukalil murichukadakkunnuvennu kaanichu. chuvanna rekha katt locus aanu, onniladhikam (eee saahacharyathil rund) hrasvamaaya geodesics ulla pooyintukalude locus. oru golatthil, katt locus oru poyintaanu. oru ooblettu deerghavrithathil (ivide kaanichirikkunnu), ithu akshamsha vrithathinte oru bhagamaanu, ithu antipodal pooyintil kendreekarichirikkunnu. katt lokkasinte rekhamsha vyaapthi ekadesham aanu. undengil
 bhoomadhyarekhayilaanu sthithicheyyunnathu, eee bandham krithyamaanu, athinte anantharafalamaayi bhoomadhyarekha ettavum cheria geodesic maathramaan. oru prolate deerghavrithaakrithiyilulla sthalathinu, katt locus aanti-meridiyante oru bhagamaanu, ithu antipodal pooyintil athunnathinumumbu meridian geodesics ettavum cheria paathakalayi maarunnu ennaanu ithinartham.
geodesicsinte deferential propertykal.
geodesics ulppedunna vividha prashnangalkk aswastharaakumbol avarude perumaattam ariyendathundu. trikonamitha krameekaranangalkkum geodesics pinthudarunna signalukalude bhauthika gunangal nirnnayikkunnathinum ithu upayogapradamaanu. oru reference geodesic, paaraameterais cheythathum athil ninnu kurachu akaleyulla randaamathe geodesic pariganikkuka. gaus-jacoby samavaakyam anusarikkunnuvennu kaanichu
gausiyan vakrata evideyaanu? randaamathe kramam, rekheeyavum ekathaanavumaaya deferential samavaakyam enna nilayil, athinte pariharam rund swatantra parihaarangalude aakethukayaayi prakadippikkaam.
evide
alavu "kuranja neelam" ennum "geodesic skeyil" ennum vilikkappedunnu.
avarude adisthaana nirvachanangal amshakalikayil chithreekarichirikkunnu. 14.
viplavathinte deerghavrithaakrithiyilulla oru gossian vakrata
 eee kesil gass-jacoby samavaakyam pariharichu, ithu integralukalaayi pravarthippikkukayum prakadippikkukayum cheiyunnu.
athippazhathil ninnu nammal kaanunnath pole. 14 (mukalile upa-chithram), asimuthukal thammil vyathyaasappettirikkunna ore pooyintil aarambikkunna rund geodesixinte verthiriv. deerghavrithaakrithiyilullathu oru adanja uparithalathil, eathaandu poojyathil aandolanamcheyyunnu. poojyamaayitheerunna pooyint arambha pooyintilekkulla pooyint konjugettu aanu. neelathinum, neelathinum edayilulla oru geodesicinu, ettavum cheria paathayaakunnathin, athu jacoby vyavasthaye thripthippeduthanam, athil pooyint konjugettu illa. eee vyavastha thripthikaramallenkil, hrasvamaaya oru "aduthulla" paathayund (geodesic aayirikkanamennilla). athinaal, jacoby avastha jiyodesikkinte oru praadeshika swathaanu, ithu geodesic aagola hrasvamaaya paathayaayirikkunnathin aavashyamaaya oru vyavastha maathramaan. oru geodesic ettavum cheria paathayaayirikkunnathin aavasyamaayathum mathiyayathumaaya vyavasthakal ivayaanu
geodesicsinte cover.
oru prathyeka pooyintil ninnulla geodesics katt locasinu mukaliloode thudarukayaanenkil athippazhathil chithreekaricha oru cover roopappedunnu. 15. ivide geodesics ilam neelayil kaanichirikkunnu. (geodesics avayude aadya bhagam antipodal poyintinadutthaayi maathrame kaanikkoo, thudarnnullavaykkalla.) chila geodesic vrithangal pacha niratthil kaanichirikkunnu; iva kavaril kaspukal roopappeduthunnu. muricha locus chuvappil kaanichirikkunnu. envalappu ennathu oru geodesicil eathu poyintilaanu ennu kandethunnathiloode envalappile pooyintukal kanakkakkaam. cover uthppaadippikkunna eee nakshatram polulla roopathe austroid ennu vilikkunnu.
astroidinu purathu rund geodesics oro pooyintilum murichukadakkunnu; athinaal eee pooyintukalkkum idayil rund geodesics (deerghavrithathinte chuttalavinte ekadesham pakuthi neelamulla) undu. rund pooyintukalkkidayil oru valia vrithathil "hraswavum" "neelamullathumaaya" roottukal ulla golatthile saahacharyavumayi ithu poruthappedunnu. austroidinullil naalu geodesics oro pooyintilum murichukadakkunnu. atharam naalu geodesics amshakathil kaanichirikkunnu. 16 ivide geodesics vardhichuvarunna dairkhyathinte kramatthil ennappedunnu. (eee chithram athippazhathinte athe sthaanam upayogikkunnu. 13, athe projectionil varachirikkunnu.) hrasvamaaya rund geodesics "sthirathayulla", athaayath.iii., athinaal, cheruthaanu rund pooyintukale bandhippikkunna aduthulla pathayilla; mattu rund asthiramaanu. ettavum cheria variyil (aadyathethu) maathrame ulloo. chithratthil pacha niratthil kaanichirikkunna kavarinodu ellah geodesics sparsikkunnu.
geodesic vrithangalude (baahya) parinaamamaan austroid. athupole, geodesic vrithangal aastroyidinte involt aanu.
oru geodesic polygoninte vistheernnam.
geodesic pooligoon ennathu geodesics vashangalulla oru polygonanu. ithu oru "golaakrithiyilulla bahubhujathin" samaanamaan, athinte vashangal valia vrithangalaanu. oru geodesic segmentinum madhyarekhaykkum edayilulla vistheernnam aadyam kanakkukoottunnathiloode atharam oru bahubhujathinte vistheernnam kandethaam, i.iii., amjeerathile chathushkonatthinte vistheernnam. 1. eee pradesham arinjukazhinjaal, oru bahubhujathinte vistheernnam bahubhujathinte ellah arikukalil ninnumulla sambhavanakal samgrahichukondu kanakkakkaam.
ivide eee pradeshathinaayulla oru padaprayogam thaazhepparayunna reethiyil vikasippicheduthittundu. deerghavrithaakrithiyilulla eathenkilum adanja pradeshatthinte vistheernnam
uparithala vistheernnathinte oru gadakavum gausiyan vakrathayum evideyaanu. ippol gass-boonnet sidhaantham oru geodesic pooligoon avasthakalil prayogikkunnu.
evide
ithu geodesic adhikavum sheershakathile baahya konum aanu. samavaakyathe gunanam cheyyumbol, rachayithaavinte aaram evideyaanu, ithu samavaakyathil ninnu kuraykkumbol labikkum
oru deerghavrithathinte moolyam pakaram vachirikkunna sthalathu.
eee soothravaakyam chathushkonatthil prayogikkukayum, integral ovar cheiyunnathu nalkukayum cheiyunnu.
integral geodesic line mukalilaayirikkumbol (athinaal athu parokshamaayi oru functionaanu). integral cheruthum saadhuthayullathumaaya oru sreniyaayi prakadippikkaam.
oru geodesic polygoninte vistheernnam athinte arikukalkk mukaliloode samgrahichaanu nalkunnathu. eee phalam paligonil oru dhruvam ulppedunnilla ennu vyavastha cheiyunnu; angane cheythaal, thukayilekku cherkkanam. arikukal avayude konukal kondu vyakthamaakkiyittundel, geodesic adhikathinu soukaryapradamaaya oru padaprayogam
nerittullathum vipareethavumaaya prashnangalkk pariharam.
geodesic prasnangal pariharikkunnathiloode geodesic sahaaya golatthilekku mapp cheyyukayum grettu-sarkkil naavigeshanil anubandha prashnam pariharikkukayum cheiyunnu.
praadhamika" golaakrithiyilulla trikonathe amshakathil pariharikkumbol. 5, kwaadrantal trikonangalkkaayulla naappiyarinte niyamangal upayogikkaam,
jiyodesikkinte mappingil doorathinum rekhaamshathinum vendiyulla integralukal vilayiruthunnathu ulppedunnu. koodaathe iva paaraamettarine aasrayichirikkunnu.
nerittulla prashnam kaikaaryam cheiyunnathu neraayathaan, kaaranam nalkiyirikkunna alavukalil ninnu nerittu nirnnayikkaan kazhiyum; oru sample kanakkukoottalinayi, kaanuka.
vipareeta prashnatthinte kaaryatthil, nalkiyirikkunnu; ithu anjjaathamaayathinaal thulyamaaya golaakrithiyumaayi eluppathil bandhappedutthaan kazhiyilla. athinaal, prashnatthinte pariharam aavarthichu kandethanam (root finding); vishadaamshangalkku kaanuka.
geodetic aaplikkeshanukalil, cheruthaanenkil, integralukal saadhaaranayaayi oru seereesaayi vilayiruthappedunnu. ekapaksheeyamaayi, integralukal (3), (4) enniva samkhyaa quadrature vazhiyo deerghavrithaakrithiyilulla integralukalude adisthaanathil prakadippikkunnathiloode kandethaam.
 nerittullathum vipareethavumaaya prashnangalkk pariharangal nalkunnu; iva parannukidakkunna moonnaam kramatthilekku nadathunna oru shreni vipuleekaranathe adisthaanamaakkiyullathum dablyujies 84 deerghavrithathinu ekadesham krithyatha nalkunnathumaanu; ennirunnaalum vipareeta reethi mikkavarum antipodal poyintukalkku othucherunnathil paraajayappedunnu. aaram kramatthilekkulla vipuleekaranangal thudarunnu, ithu poornnamaaya eratta krithyatha krithyatha nalkunnathinum vipareeta prashnatthinte pariharam mechappeduthunnathinum paryaapthamaan, angane athu ellah saahacharyangalilum othucherunnu. ekapaksheeyamaaya parannukidakkunna deerghavrithangalil prayogikkaan kazhiyunna deerghavrithaakrithiyilulla integralukal upayogikkunnathinulla reethi vipuleekarikkunnu.
oru trioxial deerghavrithaakrithiyilulla geodesics.
oru deerghavrithaakrithiyilulla viplavathinu geodesic prashnam pariharikkunnath ganithashaasthraparamaayi neraayathaan samamiti kaaranam, geodesicsinu "chalanatthinte sthiraangam" undu, clairottinte bandham prashnam kwadrecharaayi kuraykkaan anuvadikkunnu. pathonpathaam noottaandinte thudakkathil (legender, ooriyaani, bezel muthalaayavarude srishtikalkkoppam.), viplavathinte deerghavrithathil geodesixinte gunangalekkurichu poornnamaaya dhaaranayundaayirunnu.
maruvashathu, oru triraksha deerghavrithaakrithiyilulla (moonnu asamamaya achuthandukalulla) geodesicsinu chalanatthinte vyakthamaaya sthiraankamilla, athinaal pathonpathaam noottaandinte aadya pakuthiyil pariharikkappedaatha oru velluvili niranja prashnathe prathinidheekarikkunnu. sradheyamaaya oru prabandhatthil, eee prashnam kwadrecharilekku kuraykkaan anuvadikkunna chalanatthinte oru sthiratha kandethi.
trioxial ellipsoid coordinate sisttam.
nirvachicha deerghavritham pariganikkuka
deerghavrithaakruthiyil kendreekarichullathum, saamanyatha nashtappedaathe ullathumaaya cartician kordinettukal evideyaanu.
(trioxial) deerghavrithaakrithiyilulla kordinettukal (trioxial deerghavrithaakrithiyilulla akshaamshavum trioxial deerghavrithaakrithiyilulla rekhaamshavum ulla) upayogichukondu nirvachichirikkunnathu
paridhiyil, oru oolbettu deerghavrithaakrithiyilulla parametric akshaamshamaayi maarunnu, athinaal chinnatthinte upayogam munbathe vibhagangalumaayi poruthappedunnu. ennirunnaalum, mukalil nirvachicha golaakrithiyilulla rekhaamshathil ninnu ithu "vyathyasthamaan".
sthirathayude grid lineukal (neelayil), (pachayil) enniva athippazhathil nalkiyirikkunnu. 17. iva oru oorthoogoonal coordinate sistamaanu grid lineukal samakonukalil murichukadakkunnu. deerghavrithathinte pradhana bhagangal chuvappil kaanichirikkunnu. moonnaamathe pradhana vibhaagam, varikalum allengil. eee varikal naalu pokkil bindukkalil (avayil randennam eee chithratthil drishyamaanu) kandumuttunnu, avide vakrathayude pradhana trijyakal thulyamaanu. ivideyum eee vibhaagathile mattu chithrangalilum deerghavrithaakruthiyude paaraamettarukal undu, ithu mukalilulla oru pooyintil ninnu oru orthographic projectionil kaanunnu.
deerghavrithaakrithiyilulla kordinettukalude grid lineukal moonnu bhagangalaayi vyaakhyaanikkaam.
vyathyastha vazhikal
jacobyude pariharam.
deerghavrithaakrithiyilulla kordinettukalil prakadippikkunna geodesic samavaakyangal verthirikkaavunnavayaane jacoby kaanichu. thante suhruthum ayalkkaranumaaya besselinode addeham thante kandethal vivarichathenganeyenn ivide parayaam.
 innale munpu, "moonnu asamamaya achuthandukalulla deerghavrithaakrithiyilulla" geodesic lainukalude prashnam njaan naalayi kurachu. avaa lokathile ettavum lalithamaaya soothravaakyangalaanu, abelian integralukal, 2 akshangal thulyamaayi sajjeekarichaal avaa ariyappedunna deerghavrithaakrithiyilulla integralukalaayi maarunnu.<bii. aarr.>
conigsburg, 28 desember. '38.
jacoby nalkia pariharam ithaanu
jacoby soochippikkunnathu pole "konile oru function konile oru functionu thulyamaanu. eee rund functionukalum abelian integralukal maathramaan..."rund sthiraangangal laayaniyil prathyakshappedunnu. integralukalude thaazhnna paridhikal jiyodesikkinte prarambha pooyintaayi kanakkaakkukayum geodesixinte disha nirnnayikkukayum cheythaal saadhaaranayaayi poojyamaanu. ennirunnaalum, pokkil pooyintil ninnu aarambikkunna geodesicsil, pokkil pooyintil namukk dishayundu, athu nirnnayikkunnu.
sthiraamkam ingane prakadippikkaam
geodesic sthirathayude varikal upayogichu undaakkunna koonn evideyaanu? paridhiyil, ithu parichithamaaya clairt bandhatthilekku kurayunnu. jacobyude phalatthinte oru vyulppannam nalkunnathu; pothuvaaya chathuraakrithiyilulla prathalangalkkayi addeham kandethiya pariharam nalkunnu.
trioxial geodesicsinte sarway.
oru trioxial ellipsoidil, moonnu lalithamaaya clossed geodesics maathrameyulloo, elypsoidinte moonnu pradhana vibhagangal, aandu.
mattu geodesics sarway cheiyunnathinu, madhya pradhana vibhaagathe lamba konukalil murichukadakkunna geodesics pariganikkunnath soukaryapradamaanu. atharam geodesics athippazhangalil kaanichirikkunnu. 18-22, avaa athe deerghavrithaakrithiyilulla paaraamettarukalum athippazhathinte athe kazhcha dishayum upayogikkunnu. 17. koodaathe, eee moonnu pradhana deerghavrithangalum oro roopathilum chuvappil kaanichirikkunnu.
arambha pooyint, pinne, geodesic deerghavrithathe "sarkkampolar" ardhathil valayam cheiyunnu. bhoomadhyarekhayude vadakkum thekkum geodesic aandolanangal nadathunnu; oro aandolanathilum ithu deerghavrithathinu chuttumulla oru poornna sarkyoottinekkaal alpam kuravu poortthiyaakkunnu, athinte falamaayi, saadhaarana saahacharyathil, rund akshaamsharekhakalaal chuttapetta pradesham geodesic niraykkunnu. rund udaaharanagal athippazhangalil nalkiyittundu. 18 umm 19 umm. chithram 18 praayogikamaayi viplavathinte oru oolbettu deerghavrithathinte athe perumaattam kaanikkunnu (kaaranam); athippazhavumaayi thaarathamyappeduthuka. 9.
ennirunnaalum, arambha pooyint uyarnna akshaamshathilaanenkil (figure. 18) ithinte falamaayundaakunna vaikalyangal vyakthamaan. oru sarkkampolar jiodesikkilekkulla ellah sparsanangalum deerghavrithathe murichukadakkunna konfokkal single-sheett hyperboloidine sparsikkunnu.
arambha pooyint, pinne, geodesic deerghavrithathe "transpolar" ardhathil valayam cheiyunnu. geodesic deerghavrithathinu kizhakkum padinjaarum aandolanangal nadathunnu; oro aandolanathilum ithu deerghavrithathinu chuttumulla oru poornna sarkyoottinekkaal alpam kooduthal poortthiyaakkunnu. saadhaarana saahacharyathil, ithu rund rekhaamsharekhakalum athirthi pradeshavum geodesic niraykkunnathinu kaaranamaakunnu.
ellah meridians geodesics aanenkil; athinte prabhavam atharam geodesics kizhakkum padinjaarum aandolanathinu kaaranamaakunnu.
rund udaaharanagal athippazhangalil nalkiyittundu. 20 umm 21 umm. dhruvathinaduthulla jiyodesikkinte sangocha paridhi aprathyakshamaakunnu; eee saahacharyathil, deerghavritham oru prolate deerghavrithaakrithiyilullathum athippazhavumaayi maarunnu. 20 ennam athippazhathinu samanamaayirikkum. 10 (athinte vashath karakkunnu). oru transpolar jiodesikkilekkulla ellah sparsanangalum deerghavrithathe murichukadakkunna konfokkal double sheett hyperboloidine sparsikkunnu.
arambha pooyint, (oru pokkil pooyint), (geodesic deerghavrithathe lamba konukalil vidunnu) aanenkil, geodesic ethir pokkil pooyintine aavarthichu murichukadakkukayum athinte prarambha poyintilekku madangukayum cheiyunnu. ennirunnaalum, oro sarkyoottilum athu murichukadakkunna koonn aakruthiyil kaanichirikkunnathupole deerghavrithathil geodesic lakshanangalillaathavidham sthithicheyyunnathinodu kooduthal adukkunnu. 22. orotta geodesic deerghavrithathile oru pradesham niraykkunnilla. pokkil geodesixilekkulla ellah sparsanangalum pokkil pooyintukalil deerghavrithathe murichukadakkunna konfokkal hyperbola sparsikkunnu.
pokkil geodesic niravadhi rasakaramaaya gunangal aasvadikkunnu.
oru jiyodesikkinte arambha pooyint oru pokkil pooyintallengil, athinte cover rund kaspukalum mattu rund kaspukalum ulla oru austroid aanu. kaspukalkkidayilulla variyude bhagamaanu katt locus.
aaplikkeshanukal.
nerittullathum vipareethavumaaya geodesic prasnangal orikkal cheytha jiodesiyil kendra punk vahikkunnilla. golaakrithiyilulla trikonamithiyile dvimaana prashnamaayi geodetic nettvarkkukalude krameekaranam pariharikkunathinupakaram, eee prasnangal ippol trimana reethikaliloodeyaanu pariharikkappedunnathu.
ennirunnaalum, bhouma geodesics ippozhum niravadhi mekhalakalil oru pradhana punk vahikkunnu
kuranja pravartthana thathwam anusarichu, bhauthikashaasthrathile pala prashnangalum geodesicsinu samanamaaya oru veriyashanal prashnamaayi roopappeduthaam. vaasthavathil, geodesic prashnam uparithalathil neengaan nirbandhithamaaya oru kanikayude chalanathinu thulyamaanu, pakshe allaathapaksham oru sakthikkum vidheyamalla.
ekkaaranathaal, revolution deerghavrithaakrithiyilulla deerghavrithaakrithiyilulla deerghavrithaakrithiyilulla deerghavrithaakrithiyilulla deerghavrithaakrithiyilulla deerghavrithaakrithiyilulla deerghavrithaakrithiyilulla deerghavrithaakrithiyilulla deerghavrithaakrithiyilulla deerghavrithaakrithiyilulla deerghavrithaakrithiyilulla deerghavrithaakrithiyilulla deerghavrithaakrithiyilulla deerghavrithaakrithiyilulla deerghavrithaakrithiyilulla deerghavrithaakrithiyilulla deerghavrithaakrithiyilulla deerghavrithaakrithiyilulla deerghavrithaakrithiyilulla deerghavrithaakrithiyilulla deerghavrithaakrithiyilulla deerghavrithaakrithiyilulla deerghavrithaakrithiyilulla deerghavrithaakrithiyilulla deerghavrithaakrithiyilulla deerghavrithaakrithiyilulla deerghavrithaakrithiyilulla deerghavrithaakrithiyilulla deerghavrithaakrithiyilulla deerghavrithaakrithiyilulla deerghavrithaakrithiyilulla deerghavrithaakrithiyilulla deerghavrithaakrithiyilulla deerghavrithaakrithiyilulla deerghavrithaakrithiyilulla deerghavrithaakrithiyilulla deerghavrithaakrithiyilulla deerghavrithaakrithiyilulla deerghavrithikal, deerghakaalamaayulla deerghakaalamaayulla udaaharanangalil iva ulppedunnu