Conical surface

An elliptic cone, a special case of a conical surface

In geometry, a conical surface is a three-dimensional surface formed from the union of lines that pass through a fixed point and a space curve.

Definitions

A (general) conical surface is the unbounded surface formed by the union of all the straight lines that pass through a fixed point — the apex or vertex — and any point of some fixed space curve — the directrix — that does not contain the apex. Each of those lines is called a generatrix of the surface. The directrix is often taken as a plane curve, in a plane not containing the apex, but this is not a requirement.[1]

In general, a conical surface consists of two congruent unbounded halves joined by the apex. Each half is called a nappe, and is the union of all the rays that start at the apex and pass through a point of some fixed space curve.[2] Sometimes the term "conical surface" is used to mean just one nappe.[3]

Special cases

If the directrix is a circle , and the apex is located on the circle's axis (the line that contains the center of and is perpendicular to its plane), one obtains the right circular conical surface or double cone.[2] More generally, when the directrix is an ellipse, or any conic section, and the apex is an arbitrary point not on the plane of , one obtains an elliptic cone[4] (also called a conical quadric or quadratic cone),[5] which is a special case of a quadric surface.[4][5]

Equations

A conical surface can be described parametrically as

,

where is the apex and is the directrix.[6]

Conical surfaces are ruled surfaces, surfaces that have a straight line through each of their points.[7] Patches of conical surfaces that avoid the apex are special cases of developable surfaces, surfaces that can be unfolded to a flat plane without stretching. When the directrix has the property that the angle it subtends from the apex is exactly , then each nappe of the conical surface, including the apex, is a developable surface.[8]

A cylindrical surface can be viewed as a limiting case of a conical surface whose apex is moved off to infinity in a particular direction. Indeed, in projective geometry a cylindrical surface is just a special case of a conical surface.[9]

See also

References

  1. ^ Adler, Alphonse A. (1912), "1003. Conical surface", The Theory of Engineering Drawing, D. Van Nostrand, p. 166
  2. ^ a b Wells, Webster; Hart, Walter Wilson (1927), Modern Solid Geometry, Graded Course, Books 6-9, D. C. Heath, pp. 400–401
  3. ^ Shutts, George C. (1913), "640. Conical surface", Solid Geometry, Atkinson, Mentzer, p. 410
  4. ^ a b Young, J. R. (1838), Analytical Geometry, J. Souter, p. 227
  5. ^ a b Odehnal, Boris; Stachel, Hellmuth; Glaeser, Georg (2020), "Linear algebraic approach to quadrics", The Universe of Quadrics, Springer, pp. 91–118, doi:10.1007/978-3-662-61053-4_3, ISBN 9783662610534
  6. ^ Gray, Alfred (1997), "19.2 Flat ruled surfaces", Modern Differential Geometry of Curves and Surfaces with Mathematica (2nd ed.), CRC Press, pp. 439–441, ISBN 9780849371646
  7. ^ Mathematical Society of Japan (1993), Ito, Kiyosi (ed.), Encyclopedic Dictionary of Mathematics, Vol. I: A–N (2nd ed.), MIT Press, p. 419
  8. ^ Audoly, Basile; Pomeau, Yves (2010), Elasticity and Geometry: From Hair Curls to the Non-linear Response of Shells, Oxford University Press, pp. 326–327, ISBN 9780198506256
  9. ^ Giesecke, F. E.; Mitchell, A. (1916), Descriptive Geometry, Von Boeckmann–Jones Company, p. 66

Read other articles:

Kapal induk kelas Nimitz

Kapal induk kelas Nimitz USS Nimitz (CVN 68), kapal utama dari kelas kapal induk ini, sedang transit ke San Diego sebelum berlabuh di Naval Air Station North Island. Tentang kelas Nama:Kapal induk kelas NimitzPembangun:Newport News ShipbuildingOperator: Angkatan Laut Amerika SerikatDidahului oleh:Kelas Kitty Hawk dan kelas EnterpriseDigantikan oleh:Kelas Gerald R. FordBiaya:US$10,1 Miliar (Rp156,62 Triliun)Beroperasi:3 Mei 1975 Tentang kelas Selesai:10Aktif:10 Ciri-ciri umum Jenis Kapal indu...

 

Deportasi anak-anak Białystok

Anak-anak BiałystokLukisan anak-anak Białystok karya Otto UngarAuschwitzBiałystokTheresienstadtDeportasi anak-anak Białystok (Polandia)Tanggal21 Agustus–7 Oktober 1943PelakuHeinrich HimmlerAdolf EichmannOrganisasiBadan Keamanan Utama ReichCampAuschwitz II-BirkenauGhettoBiałystokTheresienstadtKorban±1.200[a] anak-anakPenyintasTidak ada Pada tanggal 21 Agustus 1943, ketika Ghetto Białystok pelan-pelan dikosongkan, sekitar 1.200[a] anak-anak Yahudi dinaikkan ke kereta ap...

 

Pembelajaran berbasis masalah

Kelompok yang terlibat dalam pembelajaran berbasis masalah di Rumah Sakit Gigi Sydney Pembelajaran berbasis masalah (Inggris: problem-based learningcode: en is deprecated atau disingkat PBL) adalah suatu pembelajaran yang menyuguhkan berbagai situasi bermasalah yang autentik dan bermakna kepada siswa yang berfungsi sebagai landasan bagi investigasi dan penyelidikan siswa.PBL membantu siswa untuk mengembangkan keterampilan untuk belajar secara mandiri, keterampilan penyelidikan dan keterampila...

2020 Iowa House of Representatives election

Not to be confused with 2020 United States House of Representatives elections in Iowa. 2020 Iowa House of Representatives election ← 2018 November 3, 2020 2022 → All 100 seats in the Iowa House of Representatives51 seats needed for a majority   Majority party Minority party   Leader Linda Upmeyer(retired) Todd Prichard Party Republican Democratic Leader since January 14, 2016 January 14, 2019 Leader's seat 54th district 52nd district Last electi...

 

Patrick Moore

Patrick MooreLahirPatrick Alfred Caldwell-Moore(1923-03-04)4 Maret 1923[1]Pinner, Middlesex, InggrisMeninggal9 Desember 2012(2012-12-09) (umur 89)Selsey, West Sussex, InggrisDikenal atasThe Sky at NightPenghargaanOrdo Imperium BritaniaKarier ilmiahInstitusiRoyal Air ForceBBCMenginspirasiBrian MayChris Lintott Situs websirpatrickmoore.combanguniverse.com/sirpatrickmoore Sir Patrick Alfred Caldwell-Moore CBE, FRS, FRAS (4 Maret 1923 – 9 Desember 2012)[1] ad...

 

Hormuz Island

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. (February 2022) (Learn how and when to remove this template message) Island in Hormozgān, IranHormuz IslandIslandSatellite photo of Hormuz IslandHormuz IslandCoordinates: 27°04′03″N 56°27′36″E / 27.06750°N 56.46000°E / 27.06750; 56.46000Country IranProvinceHormozgānArea&...

QWERTY

Untuk lagu, lihat QWERTY (lagu). Tata letak QWERTY Keyboard komputer laptop menggunakan tata letak QWERTY QWERTY adalah salah satu jenis tata letak tombol-tombol pada papan tombol. Tata letak QWERTY ini pertama kali digunakan pada sebuah mesin ketik buatan E. Remington pada tahun 1874. Dinamakan demikian karena tombol-tombol huruf Q, W, E, R, T, dan Y berada secara berurutan seperti terlihat dalam baris paling atas dari papan ketik ini (yaitu yang dipakai pada kebanyakan papan tombol komputer...

 

Llantrisant Guildhall

Municipal Building in Llantrisant, Wales Llantrisant GuildhallNative name Neuadd y Dref LlantrisantLlantrisant GuildhallLocationCastle Green, LlantrisantCoordinates51°32′31″N 3°22′30″W / 51.5419°N 3.3751°W / 51.5419; -3.3751Built1773Architectural style(s)Georgian style Listed Building – Grade IIOfficial nameThe GuildhallDesignated18 August 2000Reference no.23943 Shown in Rhondda Cynon Taf Llantrisant Guildhall (Welsh: Neuadd y Dref Llantrisant) is a ...

 

Kevin Love

Kevin Love Love con la maglia dei Cleveland Cavaliers Nazionalità  Stati Uniti Altezza 203 cm Peso 114 kg Pallacanestro Ruolo Ala grande / centro Squadra  Miami Heat CarrieraGiovanili Lake Oswego High School2007-2008 UCLA Bruins39 (681)Squadre di club 2008-2014 Minnesota T'wolves364 (6.989)2014-2023 Cleveland Cavaliers489 (7.663)2023- Miami Heat76 (647)Nazionale 2010-2012 Stati Uniti22 (163)Palmarès  Olimpiadi Oro Londra 2012  Mondiali Oro Turchia 20...

Marvel vs. Capcom 3: Fate of Two Worlds

Marvel vs. Capcom 3: Fate of Two WorldsvideogiocoCapitan America contro RyuPiattaformaXbox 360, PlayStation 3 Data di pubblicazione 18 febbraio 2011 15 febbraio 2011 18 febbraio 2011 GenerePicchiaduro a incontri OrigineGiappone SviluppoCapcom, Eighting PubblicazioneCapcom Modalità di giocoGiocatore singolo, multigiocatore SupportoDVD-ROM, Blu-ray Disc Fascia di etàPEGI: 12 SerieMarvel vs. Capcom Preceduto daMarvel vs. Capcom 2: New Age of Heroes Seguito daUltimate Marvel vs. Ca...

 

新罕布什尔州

坐标：43°11′38″N 71°34′21″W / 43.1938516°N 71.5723953°W / 43.1938516; -71.5723953 此條目需要补充更多来源。 (2017年5月21日)请协助補充多方面可靠来源以改善这篇条目，无法查证的内容可能會因為异议提出而被移除。致使用者：请搜索一下条目的标题（来源搜索：新罕布什尔州 — 网页、新闻、书籍、学术、图像），以检查网络上是否存在该主题的更多可靠来源...

 

世界政區索引

此條目可能包含不适用或被曲解的引用资料，部分内容的准确性无法被证實。 (2023年1月5日)请协助校核其中的错误以改善这篇条目。详情请参见条目的讨论页。 各国相关 主題列表 索引 国内生产总值 石油储量 国防预算 武装部队（军事） 官方语言 人口統計 人口密度 生育率 出生率 死亡率 自杀率 谋杀率 失业率 储蓄率 识字率 出口额 进口额 煤产量 发电量 监禁率 死刑 国债 ...

A California Christmas

2020 film A California ChristmasFilm posterDirected byShaun PiccininoWritten byLauren SwickardProduced by Daniel Aspromonte Ali Afshar Lauren Swickard Starring Lauren Swickard Josh Swickard CinematographyBrad RushingEdited byBrett HedlundMusic byJamie ChristophersonProductioncompanyESX EntertainmentRelease date December 14, 2020 (2020-12-14) Running time106 minutesCountryUnited StatesLanguageEnglish A California Christmas is a 2020 Christmas film directed by Shaun Paul Piccinin...

 

Boris Donskoy

Boris Donskoy(1915) Boris Mikhailovich Donskoy (Russian: Борис Михайлович Донской; 1894 – August 10, 1918) was a Russian revolutionary. He was a member of the Left Socialist-Revolutionary party as SR-maximalist. Donskoy became widely known for his assassination of German Field Marshal and military governor of Ukraine Hermann von Eichhorn in the summer of 1918. He was convicted of murder by a German military court and executed by hanging. Biography Early life Boris was ...

 

Hiba Abu Nada

Hiba Abu NadaPotret Hiba Abu NadaNama asalهبة أبو ندىLahirHiba Kamal Saleh Abu Nada(1991-06-24)24 Juni 1991Mekkah, Arab SaudiMeninggal20 Oktober 2023(2023-10-20) (umur 32)Khan Yunis, Jalur GazaPekerjaanPenyair, novelis, and ahli Gizi Hiba Kamal Abu Nada (24 Juni 1991 – 20 Oktober 2023) adalah seorang penyair, novelis, ahli gizi, dan Wikimedian dari Palestina.[1][2] Novelnya Oxygen is not for the dead memenangkan tempat kedua dalam Penghargaan Sharjah untu...

مجموعات دافعي الضرائب

يفتقر محتوى هذه المقالة إلى الاستشهاد بمصادر. فضلاً، ساهم في تطوير هذه المقالة من خلال إضافة مصادر موثوق بها. أي معلومات غير موثقة يمكن التشكيك بها وإزالتها. (مارس 2016) هذه المقالة يتيمة إذ تصل إليها مقالات أخرى قليلة جدًا. فضلًا، ساعد بإضافة وصلة إليها في مقالات متعلقة بها. (...

 

Prince Ferdinando, Duke of Genoa (1884–1963)

For his grandfather, see Prince Ferdinando, Duke of Genoa (1822–1855). Duke of Genoa Prince Ferdinando of SavoyDuke of GenoaPrince of UdineBorn(1884-04-21)21 April 1884TurinDied24 June 1963(1963-06-24) (aged 79)BordigheraSpouseMaria Luisa Alliaga GandolfiNamesFerdinando Umberto Filippo Adalberto di SavoiaHouseHouse of Savoy (Genoa branch)FatherThomas, 2nd Duke of GenoaMotherPrincess Isabella of Bavaria Prince Ferdinando in 1917 with the mayor of New York, John Purroy Mitchel. Prince Fe...

 

後小松天皇

後小松天皇 後小松天皇像（雲龍院蔵）第100代天皇北朝第6代天皇 在位期間1382年5月24日 - 1412年10月5日永徳2年4月11日 - 応永19年8月29日即位礼 1383年1月31日（永徳2年12月28日）大嘗祭 1383年12月10日（永徳3年11月16日）元号 永徳至徳嘉慶康応明徳応永時代 室町時代・南北朝時代征夷大将軍 足利義満→義持先代 北朝：後円融天皇南朝：後亀山天皇次代 称光天皇誕生 1377年8月1日（...

UFOキャッチャー

NEWUFOキャッチャー UFOキャッチャー（正式名：UFO CATCHER、読み：ユーフォーキャッチャー）は、株式会社セガ フェイブ（2015年3月までと2020年4月から2024年3月まではセガ、2015年4月から2020年3月まではセガ・インタラクティブ）が製造・販売する、プライズゲームに分類されるアーケードゲーム機およびシリーズ名。本項では、セガ フェイブが運営しているオンラインクレ...

 

Alemania

República Federal de AlemaniaBundesrepublik Deutschland  (alemán) Estado miembro de la Unión EuropeaBandera Escudo Lema: Einigkeit und Recht und Freiheit[1]​(en alemán: «Unidad y Justicia y Libertad») Himno: Das Lied der Deutschen(en alemán: «La canción de los alemanes») ¿Problemas al reproducir este archivo?      Alemania     Resto de la Unión Europea     Resto de EuropaCapital(y ciudad más poblada) ...