Teorema egrégio

Uma consequência do **Theorema Egregium** é que a Terra não pode ser representada em um mapa sem distorção. A projeção de Mercator, mostrada aqui, preserva ângulos⁣, mas não preserva área.

O Teorema Egrégio (do latim Theorema Egregium, "teorema notável") é um resultado fundamental em geometria diferencial demonstrado por Carl Friedrich Gauss que trata da curvatura das superfícies. O teorema afirma que a curvatura gaussiana de uma superfície fica completamente determinada pela medição de ângulos, distâncias e suas proporções na própria superfície, sem qualquer referência à forma particular segundo a qual a superfície esteja situada no ambiente do espaço tridimensional euclidiano. Assim, a curvatura gaussiana é um invariante intrínseco das superfícies.

O resultado foi publicado por Carl Friedrich Gauss em 1827 juntamente com outras importantes ideias geométricas, tais como a curvatura gaussiana.^[1]^[2]

O teorema é "notável" porque a definição inicial da curvatura gaussiana faz uso direto da posição que a superfície ocupa no espaço. Deste modo, é bastante surpreendente o fato de que o resultado final não depende de sua imersão apesar de todas as deformações submetidas.

Em termos matemáticos modernos, o teorema pode ser enunciado como segue:

A curvatura gaussiana de uma superfície é invariante sob isometrias locais.

Aplicações elementares

Animação mostrando a deformação de um helicoide em um catenoide

Uma esfera de raio R tem uma curvatura gaussiana constante que é igual a 1/R². Ao mesmo tempo, a curvatura gaussiana de um plano é zero. Como um corolário do teorema egrégio, não se pode embrulhar uma esfera com um pedaço de papel sem amassá-lo. Reciprocamente, a superfície de uma esfera não pode ser desdobrada em uma superfície plana, sem distorcer as distâncias. Se alguém fosse pisar em uma casca de ovo vazia, suas bordas teriam que se quebrar ao expandir antes de poder ser achatada. Matematicamente falando, uma esfera e um plano não são isométricos, nem mesmo localmente. Este fato é de grande importância para a cartografia: ele implica que é impossível criar um mapa perfeito da Terra, mesmo que seja para um pedaço pequeno de sua superfície. Portanto, toda projeção cartográfica distorcerá necessariamente pelo menos algumas distâncias.^[a]

A catenóide e o helicóide são duas superfícies bem diferentes em sua aparência. Apesar disso, pode-se deformar continuamente cada uma delas na outra: elas são localmente isométricas. Segue do teorema notável que sob esta deformação a curvatura gaussiana em quaisquer dois pontos correspondentes do catenóide e do helicóide é sempre a mesma. Desse modo uma isometria consiste simplesmente de torcer e entortar uma superfície sem qualquer amassamento ou rasgo interno, em outras palavras, sem qualquer tensão, compressão ou cisalhamento extra.

Notas

↑ O conceito de aplicações geodésicas foi uma das motivações principais para que Gauss fizesse suas "investigações sobre as superfícies curvas".

Referências

↑ do Carmo, M. P. (2005). Geometria Diferencial das Curvas e Superfícies. Rio de Janeiro: SBM. ISBN 9788583370246
↑ Carl Friedrich Gauss, Disquisitiones generales circa superficies curvas 1827 Oct. 8 (in Latin), http://gdz.sub.uni-goettingen.de/no_cache/dms/load/img/?IDDOC=139389

Este artigo sobre matemática é um esboço. Você pode ajudar a Wikipédia expandindo-o.

Read other articles:

Boissise-la-Bertrand

Peta infrastruktur dan tata guna lahan di Komune Boissise-la-Bertrand. = Kawasan perkotaan = Lahan subur = Padang rumput = Lahan pertanaman campuran = Hutan = Vegetasi perdu = Lahan basah = Anak sungaiBoissise-la-BertrandNegaraPrancisArondisemenMelunKantonLe Mée-sur-SeineAntarkomuneCommunauté d'agglomération Melun-Val de SeinePemerintahan • Wali kota (2008-2014) Marie-Claude Pignoux • Populasi1889Kode INSEE/pos77039 /&...

Loose Change (film)

Artikel ini sebatang kara, artinya tidak ada artikel lain yang memiliki pranala balik ke halaman ini.Bantulah menambah pranala ke artikel ini dari artikel yang berhubungan atau coba peralatan pencari pranala.Tag ini diberikan pada Mei 2016. Loose ChangeSampul DVDLoose Change 9/11: An American CoupSutradaraDylan AveryProduserKorey RoweDylan AveryJason BermasMatthew BrownDitulis olehDylan AveryPenata musikDJ SkoolyPenyuntingDylan AveryDistributorMicrocinema InternationalTanggal rilis 13 A...

Daftar film Bosnia-Herzegovina

(Artikel ini adalah daftar lengkap dari film-film yang diproduksi di Bosnia dan Herzegovina. Negara tersebut mulai berdiri pada April 1992. Untuk daftar A-Z lihat Kategori:Film Bosnia dan Herzegovina) Karena Bosnia dan Herzegovina menjadi bagian dari Yugoslavia sebelum kemerdekaannya pada 1992, semua film di wilayah tersebut disebut sebagai produksi asal Yugoslavia, dan tidak dianggap sebagai produksi negara tersendiri secara terpisah 1994-1999 Judul Sutradara Pemeran Genre Catatan 1994 Bosn...

Kawasaki Ninja ZX-25R

Artikel ini sebatang kara, artinya tidak ada artikel lain yang memiliki pranala balik ke halaman ini.Bantulah menambah pranala ke artikel ini dari artikel yang berhubungan atau coba peralatan pencari pranala.Tag ini diberikan pada Januari 2023. Kawasaki Ninja ZX-25R (ZX250E/F/G)ProdusenKawasaki Motorcycle & Engine CompanyPerusahaan indukKawasaki Heavy IndustriesProduksi2020–sekarangPerakitanIndonesia: Bekasi, Jawa Barat (Kawasaki Motor Indonesia)[1]Thailand: Rayong (Kawasaki Mot...

Loriot

Artikel ini sebatang kara, artinya tidak ada artikel lain yang memiliki pranala balik ke halaman ini.Bantulah menambah pranala ke artikel ini dari artikel yang berhubungan atau coba peralatan pencari pranala.Tag ini diberikan pada Desember 2022. Vicco von BülowLoriotVicco von Bülow, 1971LahirBernhard-Viktor Christoph-Carl von Bülow(1923-11-12)12 November 1923Brandenburg an der Havel, Provinsi Brandenburg, Negara Bebas Prusia, Republik WeimarMeninggal22 Agustus 2011(2011-08-22) (umur 8...

سفارة فنزويلا في بولندا

سفارة فنزويلا في بولندا فنزويلا بولندا الإحداثيات 52°12′27″N 21°01′04″E / 52.207608333333°N 21.017875°E / 52.207608333333; 21.017875 البلد بولندا المكان وارسو الاختصاص بولندا الموقع الالكتروني الموقع الرسمي تعديل مصدري - تعديل سفارة فنزويلا في بولندا هي أرفع تمثيل دبلوماسي&...

Dorothy Lamour

American actress and singer (1914–1996) Dorothy LamourLamour in 1945BornMary Leta Dorothy Slaton(1914-12-10)December 10, 1914New Orleans, Louisiana, U.S.DiedSeptember 22, 1996(1996-09-22) (aged 81)Los Angeles, California, U.S.Resting placeForest Lawn Memorial ParkOccupationsActresssingerYears active1933–1995Known forThe Jungle PrincessRoad to SingaporeHigh, Wide and HandsomeThe HurricaneRoad to BaliDisputed PassageSpouses Herbie Kay (m. 1935;...

Aeone

Aeone Données clés Surnom Vikki, Vikki Watson Nom de naissance Aeone Victoria Watson Naissance 1959Liss, Royaume-Uni Activité principale Chanteuse Activités annexes Compositrice Genre musical Pop Instruments Piano Années actives Depuis 1985 Site officiel www.aeone.com modifier Aeone Victoria Watson, plus connue sous le mononyme d’Aeone (née en 1959 à Liss) est une chanteuse britannique. Carrière Au début de sa carrière, sous le nom de Vikki Watson, elle représente le Royaume-Uni...

Anna Chapman

Anna Beatdown AttackАнна ЧапманAnna Chapman, Juni 2010LahirAnna Vasil’yevna Kushchyenko23 Februari 1982 (umur 42)Volgograd, Uni SovietTempat tinggalMoskow, RusiaNama lainAnna Kushchenko Anya Kuschenko Anya ChapmanWarga negaraRusiaBritania (dicabut)PekerjaanPengusaha, konsultan penjualan independen, wirausahawan, dan agen Federasi RusiaDikenal atasKeterlibatan dengan Program Ilegal RusiaGugatan kejahatanKonspirasi untuk bertindak sebagai agen yang melanggar hukum dari pe...

Henry L. Benning

Confederate Army general (1814–1875) For the American sugar industry executive, see Henry Arthur Benning. Henry L. BenningPortrait of Gen. Henry Lewis Benning by Bjorn EgeliBirth nameHenry Lewis BenningNickname(s)Old RockBorn(1814-04-02)April 2, 1814Columbia County, Georgia, U.S.DiedJuly 10, 1875(1875-07-10) (aged 61)Columbus, Georgia, U.S.BuriedLinwood CemeteryColumbus, Georgia, U.S.AllegianceConfederate States of AmericaService/branchConfederate States ArmyYears of service1861�...

Deutsche Schule Istanbul

Private international high school in Istanbul, TurkeyDeutsche Schule IstanbulÖzel Alman LisesiAddressŞahkulu Bostanı Sk. 8 34420 BeyoğluIstanbulTurkeyCoordinates41°01′40″N 28°58′32″E / 41.02778°N 28.97556°E / 41.02778; 28.97556InformationSchool typePrivate international high schoolEstablished1 May 1868PrincipalGerman: Hans BrügmannTurkish: S. Didem VeyisoğluTeaching staff51 German, 36 TurkishEnrollment640Color(s) Blue-WhiteWebsiteds-istanb...

Mikhail Prokhorov

Mikhail Dmitrievich ProkhorovLahirМихаил Дмитриевич Прохоров3 Mei 1965 (umur 59)Moskwa, RSFSR, Uni SovietKebangsaanRusiaPekerjaanKetua ForbesKekayaan bersihUS$11,4 miliar (Forbes)(2021)Situs webmdprokhorov.ru Mikhail Dmitrievich Prokhorov (IPA: [mʲɪxɐˈil ˈdmʲitrʲɪjɪvʲɪtɕ ˈproxərə]) (bahasa Rusia: Михаи́л Дми́триевич Про́хоров) (lahir 3 Mei 1965), adalah seorang miliarder Rusia, dianggap sebagai salah seorang oli...

War of the Robots (adventure book)

2007 novel by Trevor Baxendale The topic of this article may not meet Wikipedia's notability guideline for books. Please help to demonstrate the notability of the topic by citing reliable secondary sources that are independent of the topic and provide significant coverage of it beyond a mere trivial mention. If notability cannot be shown, the article is likely to be merged, redirected, or deleted.Find sources: War of the Robots adventure book – news · newspapers...

Magheramason

Presbyterian Church, Magheramason Magheramason (from Irish Machaire Measáin)[1][2] is a small village and townland in County Tyrone, Northern Ireland. The village sits near the County Londonderry/County Tyrone border, 5 miles (8.0 km) from the city of Derry and 9 miles (14 km) from the town of Strabane. In the 2001 census, it had a population of 393 people. It lies within the Derry City and Strabane District Council area.[3] Religion In the 1870s, Presbyter...

Antaios (magazine)

Not to be confused with Antaeus (magazine). AntaiosEditor-in-chiefPhilipp Wolff-WindeggEditorsMircea EliadeErnst JüngerCategoriescultural magazineFrequencybi-monthlyCirculation3000 (1959), 1200 (1971)PublisherErnst Klett Verlag [de]FounderErnst Klett [de]First issue1959Final issue1971CountryWest GermanyBased inStuttgartLanguageGerman This article is part of a series onConservatism in Germany Ideologies Agrarian Christian democracy Liberal Ordo Ritter School Monarchi...

List of sports rivalries in the United Kingdom

This article uses bare URLs, which are uninformative and vulnerable to link rot. Please consider converting them to full citations to ensure the article remains verifiable and maintains a consistent citation style. Several templates and tools are available to assist in formatting, such as reFill (documentation) and Citation bot (documentation). (August 2022) (Learn how and when to remove this message) The Old Firm derby between Celtic and Rangers is widely considered the greatest in British ...

Marius Trésor

Marius Trésor Trésor bersama Prancis pada tahun 1974Informasi pribadiNama lengkap Marius Paul Trésor[1]Tanggal lahir 15 Januari 1950 (umur 74)Tempat lahir Sainte-Anne, GuadeloupeTinggi 1,82 m (5 ft 11+1⁄2 in)Posisi bermain BekKarier junior1958–1969 Juventus de Sainte-AnneKarier senior*Tahun Tim Tampil (Gol)1969–1972 Ajaccio 92 (1)1972–1980 Marseille 253 (8)1980–1984 Bordeaux 93 (3)Total 438 (12)Tim nasional1971–1983 Prancis 65 (4) * Penampilan d...

Orbital angular momentum multiplexing

Optical multiplexing technique Multiplexing Analog modulation AM FM PM QAM SM SSB Circuit mode(constant bandwidth) TDM FDM / WDM SDMA Polarization Spatial OAM Statistical multiplexing(variable bandwidth) Packet switching Dynamic TDMA FHSS DSSS OFDMA SC-FDM MC-SS Related topics Channel access methods Medium access control vte Orbital angular momentum multiplexing is a physical layer method for multiplexing signals carried on electromagnetic waves using the orbital angular momentum (OA...

Telstra

Australian telecommunications company Not to be confused with Japanese telecommunications company au (mobile phone company) or Telstar. Telstra Group LimitedTelstra Corporate Centre in MelbourneTrade nameTelstraFormerly List Australian Telecommunications Commission (1979–1989) Australian Telecommunications Corporation (1989–1991) Australian and Overseas Telecommunications Corporation (1991–1993) Telstra Corporation Limited (1993–2023) Company typePublicTraded asASX: TLSS&P/AS...

Guru's Jazzmatazz, Vol. 2: The New Reality

1995 studio album by GuruJazzmatazz, Volume II: The New RealityStudio album by GuruReleasedJuly 18, 1995 (1995-07-18)Recorded1994–1995StudioMain sessions D&D Studios (New York, NY) Firehouse Studio (New York, NY) Additional sessions Platinum Island Studios (New York, NY) Matrix Studios (London, UK) Echo Studios (Los Angeles, CA) Unique Studios (New York, NY) EMI Studios (London, UK) The Hit Factory (New York, NY) Brooklyn Sound (Los Angeles, CA) GenreJazz rapR...