Хиперболична спирала

Хиперболичната спирала е равнинна трансцендентна крива, известна още като реципрочна спирала. Нейното уравнение в полярни координати е $r={\frac {a}{\theta }}$ , като приликата му с уравнението на хиперболата в декартови координати обуславя избора на имената на кривата. Хиперболичната спирала е инверсна (обратна) на архимедовата спирала.

С други думи хиперболичната спирала се дефинира като геометричното място на точка, движеща се по равномерно въртящ се около полюса лъч, така че полярният ѝ радиус да е обратно пропорционален на полярния ъгъл.

Уравнения

Представянето на спиралата с параметрични уравнения е особено елегантно:

в полярни координати: $x=r\cos \theta ,y=r\sin \theta$ ,
и в декартови: $x=a{\frac {\cos t}{t}},y=a{\frac {\sin t}{t}}$ .

Тъй като

\lim _{t\to 0}x=a\lim _{t\to 0}{\cos t \over t}=\infty ,

\lim _{t\to 0}y=a\lim _{t\to 0}{\sin t \over t}=a\cdot 1=a.

хиперболичната спирала има асимптота в y = a.

Принципно има два клона, които съответстват на положителните и отрицателните стойности на $\theta$ , но поради спецификата на графиката ѝ обикновено се изобразява само единият клон на спиралата. Тя започва от безкрайността и с нарастване на аргумента се приближава, извършвайки въртеливо движение, все по-стръмно към полюса, който представлява и асимптотична точка.

Дължината на дъга между две точки от хиперболичната спирала $M_{1}(r_{1},\theta _{1}),M_{2}(r_{2},\theta _{2})$ се намира по формулата:

L=a\left[-{\frac {\sqrt {1+\theta ^{2}}}{\theta }}+\ln {(1+{\sqrt {1+\theta ^{2}}})}\right]_{\theta _{1}}^{\theta _{2}}

а лицето на повърхнината на сектора, съответстващ на дъгата $M_{1}M_{2}$ е:

S={\frac {a^{2}(r_{1}-r_{2})}{2}}

Радиусът на кривината на спиралата е равен на: $R=a/\theta$ .

Исторически факти

Хиперболичната спирала е открита през 1704 г. от Пиер де Вариньон, но независимо от него — и от Йохан Бернули.

Източници

„Лексикон Математика“, Георги Симитчиев, Георги Чобанов, Иван Чобанов, ИК Абагар, София, 1995, ISBN 954-584-146-Х
„Математический энциклопедический словарь“, Ю. В. Прохоров, „Советская энциклопедия“, Москва, 1988
„Математически енциклопедичен речник“, В. Гелерт, Х. Кестнер, З. Нойбер, ДИ Наука и изкуство, София, 1983
„Математически термини“, Н.В. Александрова, ДИ Наука и изкуство, София, 1984

Външни препратки

Read other articles:

Booboo Stewart

Booboo StewartStewart pada Oktober 2017LahirNils Allen Stewart Jr.21 Januari 1994 (umur 30)Beverly Hills, California, Amerika SerikatNama lain Booboo Stewart Booboo Torrali PekerjaanAktorTahun aktif2004–sekarang Nils Allen Booboo Stewart Jr.[1] (lahir 21 Januari 1994) adalah aktor asal Amerika Serikat. Dia dikenal karena bermain sebagai Seth Clearwater, manusia serigala di film The Twilight Saga: Eclipse, Warpath di X-Men: Days of Future Past dan Jay di franchise fil...

Caggiano

Artikel ini perlu diwikifikasi agar memenuhi standar kualitas Wikipedia. Anda dapat memberikan bantuan berupa penambahan pranala dalam, atau dengan merapikan tata letak dari artikel ini. Untuk keterangan lebih lanjut, klik [tampil] di bagian kanan. Mengganti markah HTML dengan markah wiki bila dimungkinkan. Tambahkan pranala wiki. Bila dirasa perlu, buatlah pautan ke artikel wiki lainnya dengan cara menambahkan [[ dan ]] pada kata yang bersangkutan (lihat WP:LINK untuk keterangan lebih lanjut...

كرة اليد في الألعاب الأولمبية الصيفية 2016

كرة اليد في الألعاب الأولمبية الصيفية 2016تفاصيل المسابقةالبلد المضيف البرازيلالتواريخ6–21 أغسطس 2016الفرق24الأماكن1 (في مدينة مضيفة واحدة)المراكز النهائيةالبطل الدنمارك (رجال) روسيا (سيدات)الوصيف فرنسا (رجال) فرنسا (سيدات)المركز الثالث ألمانيا (رجال) الن...

Texas Declaration of Independence

1836 proclamation of Texan independence from Mexico This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.Find sources: Texas Declaration of Independence – news · newspapers · books · scholar · JSTOR (September 2020) (Learn how and when to remove this template message) Texas Declaration of Independence1836 facsimile of the Texa...

Mikoyan MiG-29

Mikoyan MiG-29MiG-29 milik Angkatan Udara RusiaTipePesawat tempur superioritas, pesawat tempur multiperanPerancangMikoyanTerbang perdana6 Oktober 1977DiperkenalkanAgustus 1983StatusBeroperasiPengguna utamaAngkatan Dirgantara RusiaAngkatan Udara IndiaAngkatan Udara UzbekistanAngkatan Udara IranPengguna lainLihat penggunaTahun produksi1981–sekarangJumlah produksi1.600+VarianMikoyan MiG-29MMikoyan MiG-29KMikoyan MiG-35 Mikoyan MiG-29 (Rusia: Микоян МиГ-29; kode NATO: Fulcrum) adalah p...

Первородный грех

Об экономическом термине см. Первородный грех (экономика). ХристианствоБиблия Ветхий Завет Новый Завет Евангелие Десять заповедей Нагорная проповедь Апокрифы Бог, Троица Бог Отец Иисус Христос Святой Дух История христианства Апостолы Хронология христианства Ран�...

Bodhisena

Bagian dari seri tentangBuddhisme SejarahPenyebaran Sejarah Garis waktu Sidang Buddhis Jalur Sutra Benua Asia Tenggara Asia Timur Asia Tengah Timur Tengah Dunia Barat Australia Oseania Amerika Eropa Afrika Populasi signifikan Tiongkok Thailand Jepang Myanmar Sri Lanka Vietnam Kamboja Korea Taiwan India Malaysia Laos Indonesia Amerika Serikat Singapura AliranTradisi Buddhisme prasektarian Aliran Buddhis awal Mahāsāṃghika Sthaviravāda Aliran kontemporer Theravāda Mahāyāna Vajrayāna Kon...

Khaleegy (dance)

This article includes a list of general references, but it lacks sufficient corresponding inline citations. Please help to improve this article by introducing more precise citations. (July 2013) (Learn how and when to remove this template message) Arab girls dancing Khaleegy Khaleegy' or Khaliji (from Arabic خليج) is a mixture of modern style and traditional folkloric dance from the Persian Gulf countries of Iraq, Saudi Arabia, Kuwait, Oman, Bahrain, Qatar and the United Arab Emirates. Th...

Fouad Chehab

Third President of Lebanon (1902–1973) This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.Find sources: Fouad Chehab – news · newspapers · books · scholar · JSTOR (November 2020) (Learn how and when to remove this template message) Fouad Chehabفؤاد شهابChehab in 19613rd President of LebanonIn office23 September 195...

A13 road (England)

Major road in England For other uses, see A13 roads. A13Route informationLength41.7 mi[1] (67.1 km)HistoryRoute pre-dates inception of 1922 Roads list; first all-new bypass section opened 1924; last all-new section opened 1999; latest grade-separation completed 2013Major junctionsWest end A11 in Whitechapel, LondonMajor intersections A12 / A102 in Poplar, London A406 / A1020 near Barking M25 / A282 near Aveley A1089 near Grays A130 near South Benfle...

2019 FIFA Club World Cup squads

Each team in the 2019 FIFA Club World Cup had to name a 23-man squad (three of whom must be goalkeepers). FIFA announced the squads on 5 December 2019.[1][2] Al-Hilal On 11 December 2019, Nawaf Al-Abed replaced the injured Salman Al-Faraj.[3] Manager: Răzvan Lucescu Note: Flags indicate national team as defined under FIFA eligibility rules. Players may hold more than one non-FIFA nationality. No. Pos. Nation Player 1 GK KSA Abdullah Al-Mayouf 2 DF KSA Moha...

Antonio Gandini

Antonio Gandini (Modena, 20 agosto 1786 – Formigine, 10 settembre 1842) è stato un compositore italiano. Indice 1 Biografia 2 Note 3 Bibliografia 4 Voci correlate 5 Altri progetti 6 Collegamenti esterni Biografia Libretto dell'opera Ruggiero (1820) Nato a Modena nel 1786, dopo aver studiato musica sin dai suoi primi anni, non ancora ventenne, entrò a far parte dell'Accademia dei Filarmonici della città natale.[1] Evitato il servizio militare per le sue condizioni di salute,[1...

الأخلاقوري

تحتاج هذه المقالة إلى الاستشهاد بمصادر إضافية لتحسين وثوقيتها. فضلاً ساهم في تطوير هذه المقالة بإضافة استشهادات من مصادر موثوق بها. من الممكن التشكيك بالمعلومات غير المنسوبة إلى مصدر وإزالتها. (ديسمبر 2018) الأخلاقوري هو الشخص الذي يحاول الجمع بين أخلاقيات استهلاك الطعام وا...

Філософія

Частина серії проФілософіяLeft to right: Plato, Kant, Nietzsche, Buddha, Confucius, AverroesПлатонКантНіцшеБуддаКонфуційАверроес Філософи Епістемологи Естетики Етики Логіки Метафізики Соціально-політичні філософи Традиції Аналітична Арістотелівська Африканська Близькосхідна іранська Буддій�...

Ester 6

Ester 6Lima Gulungan (Megilloth) lengkap pada Kodeks Leningrad (1008 Masehi), dengan urutan: Rut, Kidung Agung, Pengkhotbah, Ratapan dan Ester.KitabKitab EsterKategoriKetuvimBagian Alkitab KristenPerjanjian LamaUrutan dalamKitab Kristen17← pasal 5 pasal 7 → Ester 6 (disingkat Est 6) adalah pasal keenam Kitab Ester dalam Alkitab Ibrani dan Perjanjian Lama di Alkitab Kristen.[1][2] Berisi riwayat Ahasyweros, raja Kekaisaran Persia, Mordekhai, Ester, dan Haman.[3&...

總加速師

此條目介紹的是對習近平的負面稱呼。关于資本主義之社會理論，请见「加速主義」。中共中央總書記习近平總加速師是對中共中央總書記、國家主席、中央軍委主席習近平的一個稱謂，暗喻習近平的執政失誤將會「加速」中國共產黨的倒台和中華人民共和國社会运作的崩潰。[1] 2020年始，社交網站Twitter開始出現「#總加速師」的主題標籤，迅速成為网络热词...

Wendell R. Anderson

American hockey player and politician (1933–2016) For the Wisconsin politician, see Wendell Abraham Anderson. Wendell R. AndersonUnited States Senatorfrom MinnesotaIn officeDecember 30, 1976 – December 29, 1978Appointed byRudy PerpichPreceded byWalter MondaleSucceeded byRudy Boschwitz33rd Governor of MinnesotaIn officeJanuary 4, 1971 – December 29, 1976LieutenantRudy PerpichPreceded byHarold LeVanderSucceeded byRudy PerpichMember of the Minnesota SenateIn officeJanuary...

Perang Tiongkok-India

Perang Tiongkok-IndiaPeta sengeketa perbatasan antara Tiongkok dan India.Tanggal20 Oktober[1] – 21 November 1962(1 bulan dan 1 hari)LokasiAksai Chin, Agensi Perbatasan Timur Laut dan AssamHasil Kemenangan Tiongkok[2] Tiongkok menguasai wilayah Aksai Chin[3]Pihak terlibat Tiongkok IndiaTokoh dan pemimpin Mao Zedong(Ketua Partai Komunis Tiongkok)[4]Liu Shaoqi(Presiden Tiongkok)[5]Zhou Enlai(Perdana Menteri Tiongkok)Lin Biao...

2013 Philadelphia Phillies season

Major League Baseball team season 2013 Philadelphia PhilliesLeagueNational LeagueDivisionEastBallparkCitizens Bank ParkCityPhiladelphiaRecord73–89 (.451)Divisional place4thOwnersBill GilesDavid MontgomeryGeneral managersRubén Amaro, Jr.ManagersCharlie Manuel (through August 16) Ryne Sandberg (interim: August 16 – September 29)TelevisionComcast SportsNet PhiladelphiaComcast Network Philadelphia PHL17 (MyNetworkTV)(Tom McCarthy, Chris Wheeler, Gary Matthews)RadioPhillies Radio Networ...

Apostolic Prefecture of Battambang

Catholic missionary jurisdiction in Cambodia Apostolic Prefecture of BattambangPraefectura Apostolica Battambangensisភូមិភាគបាត់ដំបងLocationCountry CambodiaStatisticsArea81,456 km2 (31,450 sq mi)Population- Total- Catholics(as of 2010)4,335,0004,125 (0.1%)InformationDenominationRoman CatholicSui iuris churchLatin ChurchRiteLatin RiteEstablished26 September 1968Current leadershipApostolic PrefectEnrique Figaredo Alvargonzalez, S....

/profillengkap.com/index.php/vi/Cục Kinh tế, Quân đội nhân dân Việt Nam

/profillengkap.com/index.php/article/Arnold Palmer Cup

1954 Rose Bowl

Horizon Cup

Kaliyoso, Dungaliyo, Gorontalo

Garrison, Maryland

Desakota

Kontroversi buku pelajaran sejarah Jepang

YuYu Hakusho

Marcel Schreter

Thermogravimetric analysis

Bernard Haitink discography