邊 (幾何)

图为三角形的三边AB、BC和CA,位于三角形的每两个顶点之间
图为一个正方形,它有4条边

图为一个立方体,它是多面体的一种,每个边都与多面体中的两个面相接

图为一个超立方体,它是四維凸正多胞体的一种,每个边都与多面体中的三个面相接

幾何學中,是指幾何形狀中連接頂點的幾何結構。在一般常見的幾何圖形多邊形、多面體和多胞體中,邊是連接兩個頂點的線段[1],而邊長指這線段的長度。而在一些較複雜的空間中的幾何結構中,邊有可能連接2個以上的頂點,例如複數空間中的複多胞形[2]。在多邊形中,邊是位於多邊形邊界上的線段,又可以稱為邊緣[3]。而在多面體或更高維度的多胞形中,邊是面相交的線段[4]。而穿過幾何結構內部的線段不能稱為邊,其稱為對角線

種類

邊依照所屬的幾何結構會有不同的特性。

角的邊

一個角,其中藍色和紅色線段為角的邊,在圖中的角中,藍色的邊稱為始邊,紅色的邊稱為終邊。

角是由两条有公共端点的射线组成的几何对象。这两条射线叫做角的[5]。在有向角中,角的兩條邊皆有不同的稱呼。通常稱有向角起始的邊為始邊、另一條邊則稱為終邊,而始邊終邊相同的角稱為同界角[6]

多邊形的邊

多邊形中,邊是位於多邊形邊界上的線段,又可以稱為邊緣[3]。一般情況下,多邊形的邊數會與頂點數相等。在一些特殊的多邊形中,特定的編會被依照其特性命名,例如在梯形中,一組平形的邊通常稱為底邊[7],求面積時三角形的與高垂直的邊也稱為底邊,其餘兩邊則稱側邊。[8]

多面體的邊

立方體中的一條邊。

在多面體中,邊是多面體中兩個面互相相交的線段,通常稱為,表示物體兩面相接的部分[9]。而其所對應的二面角,即物體邊緣的接角又稱為稜角稜子[10][11]。邊通常不包括其角本身,而稜則會包括其接角,然而這些詞彙在英語中皆稱為Edge,而稜圖(Edge figure)探討的則為稜角的特性,而非只探討邊本身。[12]。一般情況下,多面體的邊數可透過歐拉特徵數計算得出。任何凸多面體表面的歐拉特徵皆符合下列等式:

其中V是頂點數、E是邊數、F是面數。這個等式稱為歐拉恆等式[13],由此可知,邊的數量恆比頂點和面的數量的總和小2。例如,立方體有8個頂點和6個面,因此根據歐拉恆等式可以得到有立方體有12條邊。

多胞形的邊

多胞形是指多邊形、多面體、多胞體等幾何結構再任意維度的類比,因此多邊形也是一種多胞形。在多邊形中,兩條邊會交會在一個點上,更精確地說在維度為d維的d維凸多胞形中,會有至少d條邊交會在1個頂點上,例如前述的多邊形是一種二維多胞形,因此每個頂點至少都是2條邊的交會點[14],這個現象稱為巴林斯基定理英语Balinski's theorem,類似地,在多面體中,每條邊都至少是2個二維面的交線[15],而在四維或更高維多胞體中會有三個或更多個二維面在每個邊上相交。

複空間多胞形的邊

一種由6個三元邊組成的複多邊形,其塗上藍色與紅色的部分為這個複多邊形的邊,其在施萊夫利符號中可以用3{4}2表示。

在實數空間中,邊可以視為一種在實數線上的封閉圖形,其可以由兩個端點來定義。類似地,複空間的邊可以也可以視為在以構成的「線」中的點集,其可以視為位於阿干特图英语Argand diagram(x,y)=x+iy中的點集。而複空間的邊可以視為連接位於同一個阿干特平面上多個頂點的多邊形[16],這個多邊形其不存在邊,而是這個邊連結了這些頂點。這種結構稱為複空間線段。與實空間線段不同,由於複數不存在自然序,因此不能定義內部,換句話說即無法定義複空間的邊上的點。[2]

這種由3個或三個以上的頂點組成,且並未定義哪幾個頂點要兩兩相連,只定義了一個表示需要相連之頂點的集合所組成的邊,在圖論中有對應的概念,為超邊

由n個頂點組成的邊稱為n元邊或n元稜。

三元邊

三元邊又稱三元稜是一種位於複數空間中的邊,其可以視為實數空間中的線段在複數空間的類比。這種結構無法存於實空間,在實空間中,三元棱對應的幾何結構為三角形。這種幾何結構在施萊夫利符號中可以用3{}來表示。[16]

這種特殊的邊出現於莫比烏斯-坎特八邊形中。[2]

圖論中的邊

在圖論中,邊是連接兩個圖節點的抽象數學物件,而非如同多邊形一般擁有具體的線段也不存在邊長。然而,任何多面體都可以透過其骨架或邊的骨架找到一個對應的邊與頂點的圖英语n-skeleton,在該圖中的頂點可以對應到多面體的幾何頂點,該圖中的邊也可以對應到多面體的幾何邊。[18]反過來說,三維多面體的骨架圖可以透過斯坦尼茨定理英语Steinitz's theorem表達成3頂點連通的平面圖。[19]

其他用法

在高維凸多胞形理論中,維度為d的d維凸多胞形中,其(d-1)維的元素稱為維面、(d-2)維的元素稱為維脊或維邊或維稜、(d-3)維的元素稱為維峰。 因此,多邊形的邊同時也是其維面、三維凸多面體的邊同時也是其維脊、四維凸多胞體的邊同時也是其維峰。[20]

參見

參考文獻

  1. ^ Ziegler, Günter M., Lectures on Polytopes, Graduate Texts in Mathematics 152, Springer, Definition 2.1, p. 51, 1995 [2019-09-14], (原始内容存档于2019-06-12) .
  2. ^ 2.0 2.1 2.2 Shephard, G.C.; Regular complex polytopes, Proc. London math. Soc. Series 3, Vol 2, (1952), pp 82–97.
  3. ^ 3.0 3.1 Weisstein, Eric W. (编). "Polygon Edge". at MathWorld--A Wolfram Web Resource. Wolfram Research, Inc. [2019-09-14] (英语). 
  4. ^ Weisstein, Eric W. (编). "Polytope Edge". at MathWorld--A Wolfram Web Resource. Wolfram Research, Inc. [2019-09-14] (英语). 
  5. ^ Sidorov, L. A., Angle, Hazewinkel, Michiel (编), 数学百科全书, Springer, 2001, ISBN 978-1-55608-010-4 
  6. ^ 四‧ 有向角, 三角函數的定義. math.fcu.edu. [2019-09-15]. (原始内容存档于2014-01-04). 
  7. ^ 《中學數學實用辭典》P.210 ISBN 957-603-093-5 九章出版
  8. ^ 《圖解數學辭典》天下遠見出版 P.37 三角形 ISBN 986-417-614-5
  9. ^ 【稜】. 教育部重編國語辭典修訂本. [2019-09-15]. 
  10. ^ 【稜角】. 教育部重編國語辭典修訂本. [2019-09-15]. 
  11. ^ 【稜子】. 教育部重編國語辭典修訂本. [2019-09-15]. 
  12. ^ Klitzing, Richard. Klitzing:Vertex figures,etc.. bendwavy.org. [2019-09-15]. (原始内容存档于2011-08-08). 
  13. ^ Richeson, David S.; Euler's Gem: The Polyhedron Formula and the Birth of Topology. Princeton University Press 2008.
  14. ^ Balinski, M. L., On the graph structure of convex polyhedra in n-space, Pacific Journal of Mathematics, 1961, 11 (2): 431–434 [2019-09-14], MR 0126765, doi:10.2140/pjm.1961.11.431, (原始内容存档于2019-05-11) .
  15. ^ Wenninger, Magnus J., Polyhedron Models, Cambridge University Press: 1, 1974 [2019-09-14], ISBN 9780521098595, (原始内容存档于2015-03-21) .
  16. ^ 16.0 16.1 Complex Regular Polytopes,[17] 11.1 Regular complex polygons p.103
  17. ^ Coxeter, H.S.M., Regular Complex Polytopes, Cambridge University Press, 1991, ISBN 0-521-39490-2 
  18. ^ Senechal, Marjorie, Shaping Space: Exploring Polyhedra in Nature, Art, and the Geometrical Imagination, Springer: 81, 2013 [2019-09-14], ISBN 9780387927145, (原始内容存档于2014-01-07) .
  19. ^ Pisanski, Tomaž; Randić, Milan, Bridges between geometry and graph theory, Gorini, Catherine A. (编), Geometry at work, MAA Notes 53, Washington, DC: Math. Assoc. America: 174–194, 2000, MR 1782654 . See in particular Theorem 3, p. 176页面存档备份,存于互联网档案馆).
  20. ^ Seidel, Raimund, Constructing higher-dimensional convex hulls at logarithmic cost per face, Proceedings of the Eighteenth Annual ACM Symposium on Theory of Computing (STOC '86): 404–413, 1986, doi:10.1145/12130.12172 .

外部連結

Read other articles:

Lokasi Distrik Kazuno di Prefektur Akita. Lokasi munisipalitas yang ada di Distrik Kazuno, Prefektur Akita1. – Kosakawarna hijau - cakupan wilayah distrik saat iniwarna kuning - bekas wilayah distrik pada awal zaman Meiji Distrik Kazuno (鹿角郡code: ja is deprecated , Kazuno-gun) adalah sebuah distrik yang terletak di Prefektur Akita, Jepang. Per 1 Oktober 2020, distrik ini memiliki estimasi jumlah penduduk sebesar 4.780 jiwa dan kepadatan penduduk sebesar 23,70 orang per km². Distrik i...

 

 

Jinshi Hanzi tradisional: 進士 Hanzi sederhana: 进士 Alih aksara Mandarin - Hanyu Pinyin: jìnshì - Wade-Giles: chin⁴-shih⁴ Jinshi (Hanzi: 進士; Pinyin: jìnshì) adalah tingkat tertinggi dan terakhir dalam ujian kenegaraan pada masa Kekaisaran Tiongkok.[1][2] Ujian ini biasanya diadakan di ibu kota, di dalam istana, sehinga disebut juga Ujian Ibu kota. Mereka yang berhasil lulus ujian tingkat akhir ini dinamakan jinshi, terkadang diterjemahkan menjadi Sarj...

 

 

Public university in Charlottesville, Virginia, US UVa redirects here. For other uses, see Uva. University of VirginiaTypePublic research universityEstablishedJanuary 25, 1819; 205 years ago (January 25, 1819)[1]FounderThomas JeffersonAccreditationSACSAcademic affiliationsAAUORAUSCHEVURASea-grantSpace-grantEndowment$13.6 billion (2022)[2]Budget$1.91 billion (2020)[3][a]PresidentJames E. RyanProvostIan BaucomAcademic staff3,265 (Fall 2019)[...

This article does not cite any sources. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.Find sources: Eska TV – news · newspapers · books · scholar · JSTOR (February 2024) (Learn how and when to remove this template message) Television channel Eska TVCountryPolandOwnershipOwnerZPR Media Group (formerly), Telewizja Polsat (Present)Sister channelsEska TV ExtraEska Rock TVHistoryLa...

 

 

Commune in Hauts-de-France, FranceSeclinCommuneSeclin Town Hall Coat of armsLocation of Seclin SeclinShow map of FranceSeclinShow map of Hauts-de-FranceCoordinates: 50°32′56″N 3°01′49″E / 50.5489°N 3.0303°E / 50.5489; 3.0303CountryFranceRegionHauts-de-FranceDepartmentNordArrondissementLilleCantonFaches-ThumesnilIntercommunalityMétropole Européenne de LilleGovernment • Mayor (2020–2026) François-Xavier Cadart[1]Area117.42 k...

 

 

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: Inter Nashville FC – news · newspapers · books · scholar · JSTOR (April 2018) (Learn how and when to remove this message) This article needs to be updated. Please help update this article to reflect recent events or newly available information. (May 2022) Socce...

Shoe with a tall, thin heel For other uses, see Stiletto (disambiguation). 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: Stiletto heel – news · newspapers · books · scholar · JSTOR (April 2019) (Learn how and when to remove this message) Stiletto heels can be found on almost every type of shoes, such as th...

 

 

Disambiguazione – Indiana Jones and the Temple of Doom rimanda qui. Se stai cercando il videogioco tratto dal film, vedi Indiana Jones and the Temple of Doom (videogioco). Indiana Jones e il tempio maledettoShorty (Ke Huy Quan), Willie (Kate Capshaw) e Indiana Jones (Harrison Ford) in una scena del filmTitolo originaleIndiana Jones and the Temple of Doom Lingua originaleinglese Paese di produzioneStati Uniti d'America Anno1984 Durata119 min Rapporto2,35:1 Genereavventura, azion...

 

 

ХристианствоБиблия Ветхий Завет Новый Завет Евангелие Десять заповедей Нагорная проповедь Апокрифы Бог, Троица Бог Отец Иисус Христос Святой Дух История христианства Апостолы Хронология христианства Раннее христианство Гностическое христианство Вселенские соборы Н...

Stasiun Abeyamakōen安部山公園駅Stasiun Abeyamakōen pada Agustus 2007Lokasi4 Yugawa-shinmachi, Kokuraminami, Kitakyushu, Fukuoka(北九州市小倉南区湯川新町4丁目)JepangKoordinat33°50′39″N 130°54′15″E / 33.84417°N 130.90417°E / 33.84417; 130.90417Koordinat: 33°50′39″N 130°54′15″E / 33.84417°N 130.90417°E / 33.84417; 130.90417Operator JR KyushuJalur■ Jalur Utama NippōSejarahDibuka1987PenumpangFY20...

 

 

Scottish novelist, surgeon, critic and playwright, 1721–1771 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: Tobias Smollett – news · newspapers · books · scholar · JSTOR (October 2017) (Learn how and when to remove this message) Tobias SmollettBornTobias George Smollett(1721-03-19)19 March 1721Dalquhurn (...

 

 

United States historic placeCrow Creek SiteU.S. National Register of Historic PlacesU.S. National Historic Landmark Aerial view of the siteShow map of South DakotaShow map of the United StatesNearest cityChamberlain, South DakotaCoordinates43°58′48″N 99°19′54″W / 43.98000°N 99.33167°W / 43.98000; -99.33167Builtcirca 1100 ADNRHP reference No.66000710Significant datesAdded to NRHPOctober 15, 1966[1]Designated NHLJuly 19, 1964[2]...

Baseball player Zack CoxCox with the Arkansas RazorbacksThird basemanBorn: (1989-05-09) May 9, 1989 (age 34)Campbellsville, KentuckyBats: LeftThrows: Right Zackary Kendrick Cox (born May 9, 1989) is an American retired professional baseball third baseman. Amateur career Cox was born in Campbellsville, Kentucky. He attended Pleasure Ridge Park High School in the Pleasure Ridge Park neighborhood of Louisville, and played for the school's baseball team. Cox was selected in the 20th round o...

 

 

World Weightlifting Championship Main article: 2021 World Weightlifting Championships 2021 World Weightlifting ChampionshipsMenWomen55 kg45 kg61 kg49 kg67 kg55 kg73 kg59 kg81 kg64 kg89 kg71 kg96 kg76 kg102 kg81 kg109 kg87 kg+109 kg+87 kgvte The men's 96 kilograms competition at the 2021 World Weightlifting Championships was held on 13 and 14 December 2021.[1] Schedule Date Time Event 13 December 2021 21:30 Group D 14 December 2021 08:30 Group C 13:00 Group B 16:00 Group A Medalists Ev...

 

 

This article needs to be updated. Please help update this article to reflect recent events or newly available information. (April 2019) The Ecuadorian Constitution requires that all children attend school until they achieve a “basic level of education,” which is estimated at nine school years. The Human Rights Measurement Initiative (HRMI)[1] finds that Ecuador is fulfilling only 83.4% of what it should be fulfilling for the right to education based on the country's level of inco...

Angelique KerberAngelique Kerber nel 2021Nazionalità Germania Polonia Altezza173 cm Peso68 kg Tennis Carriera Singolare1 Vittorie/sconfitte 673 – 363 (64.96%) Titoli vinti 14 Miglior ranking 1ª (12 settembre 2016) Ranking attuale ranking Risultati nei tornei del Grande Slam  Australian Open V (2016)  Roland Garros QF (2012, 2018)  Wimbledon V (2018)  US Open V (2016) Altri tornei  Tour Finals F (2016)  Giochi olimpici (2016) Doppio1 Vittorie/sconfit...

 

 

Metro station in Chongqing, China Cuiyun翠云General informationLocationYubei District, ChongqingChinaOperated byChongqing Rail Transit Corp., LtdLine(s)     Line 3Platforms2 side platformsConstructionStructure typeElevatedOther informationStation code3/34HistoryOpened8 October 2011 (2011-10-08)Services Preceding station Chongqing Rail Transit Following station The EXPO Gardentowards Yudong Line 3 Changfulutowards Terminal 2 of Jiangbei Airport Cuiyun ...

 

 

Study of the timing of biological events Not to be confused with Phrenology, Phonology, or Phenomenology. Phenology is the study of periodic events in biological life cycles and how these are influenced by seasonal and interannual variations in climate, as well as habitat factors (such as elevation).[1] Examples include the date of emergence of leaves and flowers, the first flight of butterflies, the first appearance of migratory birds, the date of leaf colouring and fall in deciduous...

Orang Han (漢族 atau 汉族)Zhou GongGong FuziLaoziQu YuanZhuge LiangWang XizhiTang TaizongLi BaiQin ShihuangHan WudiWu ZetianYue FeiWen TianxiangQin LiangyuKoxingaSun Yat-senJumlah populasi1.310.000.000 (satu miliar tiga ratus sepuluh juta)19,73% populasi dunia(estimasi)Daerah dengan populasi signifikan Tiongkok1.317.541.842[1] Taiwan22.575.365[2] Hong Kong7.393.410[3] Singapura2.984.936[4] Makau433.641[5] Thailand9.053.240[6] Malaysia7.590.500[7] Am...

 

 

Lok Sabha constituency in West Bengal Purulia WB-35Lok Sabha constituencyInteractive Map Outlining Purulia Lok Sabha ConstituencyConstituency detailsCountryIndiaRegionEast IndiaStateWest BengalAssembly constituenciesBaghmundiManbazarBalarampurKashipurParaJoypurPuruliaEstablished1957Total electors18,23,120 (2024)[1]ReservationNoneMember of Parliament18th Lok SabhaIncumbent Jyotirmoy Singh Mahato Party Bharatiya Janata PartyElected year2024 Purulia Lok Sabha constituency is one of the p...