矢 (幾何)

圓弧的矢（sagitta，有時縮寫成sag^[1]）或弓形高^[2]是指該圓弧對應的弦之中點到弧之中點的距離^[3]。在建築學中，矢廣泛用於計算跨越一定高度和距離所需的弧度，並且在光學中用於評估球面鏡或透鏡的厚度。矢的英文sagitta來自拉丁文sagitta意思是「箭」。

三角函數的正矢函數正是得名於矢^[4]^[5]，在割圓八線中，矢對應到正矢。

矢與弓形高是相似的概念，差別僅在矢專指一個弧中點到弧兩端連線之中點的那條線，而弓形高是弓形的高。矢與弧相關，而弓形高與弓形相關。

公式

在下列等式中， $s$ 代表矢（弓形高）， $r$ 為圓的半徑， $l$ 為圓弧兩端點的距離，也就是弦長。其中半弦 ${\tfrac {1}{2}}l$ 和弓心距 $r-s$ 正好是直角三角形的兩條邊，半徑 $r$ 剛好是其斜邊，根據勾股定理則有：

r^{2}=\left({\tfrac {1}{2}}l\right)^{2}+\left(r-s\right)^{2}.

由此可以推導出矢 $s$ 、弦 $l$ 和半徑 $r$ 的關係式：

{\begin{aligned}s&=r-{\sqrt {r^{2}-{\tfrac {1}{4}}l^{2}}},\\[10mu]l&=2{\sqrt {2rs-s^{2}}},\\[5px]r&={\frac {s^{2}+{\tfrac {1}{4}}l^{2}}{2s}}={\frac {s}{2}}+{\frac {l^{2}}{8s}}.\end{aligned}}

矢也可以透過正矢函數來計算出來。若圓弧對應的圓心角為 $Δ$ ，令 $Δ = 2 θ$ ，則矢為：

s=r\operatorname {versin} \theta =r\left(1-\cos \theta \right)=2r\sin ^{2}{\frac {\theta }{2}}.

近似值

當矢相對於半徑很小時，可以使用以下公式來近似計算^[3]：

s\approx {\frac {l^{2}}{8r}}.

或者，如果矢長（弓形高）很小，且已知矢長、半徑和弦長，則可以透過以下公式來估計計弧長：

a\approx l+{\frac {2s^{2}}{r}}\approx l+{\frac {8s^{2}}{3l}},

其中， $a$ 是弧長。這個公式為中國數學家沈括所知，兩個世紀後，郭守敬提出了一個更準確的公式。^[6]

參見

參考文獻

^ Shaneyfelt, Ted V. 德博士的 Notes About Circles, ज्य, & कोज्य: What in the world is a hacovercosine?. Hilo, Hawaii: University of Hawaii. [2015-11-08]. （原始内容存档于2015-09-19）.
^ 弓形高. Yahoo奇摩字典. [2023-12-28]. （原始内容存档于2023-12-28）.
^ ^3.0 ^3.1 Woodward, Ernest. Geometry - Plane, Solid & Analytic Problem Solver. Problem Solvers Solution Guides. Research & Education Association (REA). December 1978: 359. ISBN 978-0-87891-510-1.
^ van Brummelen, Glen Robert. Heavenly Mathematics: The Forgotten Art of Spherical Trigonometry. Princeton University Press. 2013 [2015-11-10]. ISBN 9780691148922. 0691148929.
^ sagitta. 牛津英語詞典 (第三版). 牛津大學出版社. 2005-09 （英语）.
^ Needham, Noel Joseph Terence Montgomery. Science and Civilisation in China: Mathematics and the Sciences of the Heavens and the Earth 3. Cambridge University Press. 1959: 39. ISBN 9780521058018.

Read other articles:

The Emperor's Tomb

For the video game, see Indiana Jones and the Emperor's Tomb. The Emperor's Tomb AuthorJoseph RothOriginal titleDie KapuzinergruftTranslatorJohn HoareCountryNetherlandsLanguageGermanPublisherDe GemeenschapPublication date1938Published in English1984Pages231 The Emperor's Tomb (German: Die Kapuzinergruft) is a 1938 novel by the Austrian writer Joseph Roth. The Overlook Press published an English translation by John Hoare in 1984.[1] The novel was adapted into the 1971 film Tr...

Komořany

Cadastral subdivision of Prague 12 in Czech RepublicKomořanyCadastral subdivision of Prague 12Courtyard of Komořany ChateauLocation of Komořany in PragueCoordinates: 49°59′25″N 14°24′38″E / 49.9902°N 14.4106°E / 49.9902; 14.4106CountryCzech RepublicRegionPragueArea • Total2.08 km2 (0.80 sq mi)Population (31. 12. 2015[1]) • Total1,941 • Density930/km2 (2,400/sq mi)Time zoneUTC+1 (CET)...

Синелобый амазон

Синелобый амазон Научная классификация Домен:ЭукариотыЦарство:ЖивотныеПодцарство:ЭуметазоиБез ранга:Двусторонне-симметричныеБез ранга:ВторичноротыеТип:ХордовыеПодтип:ПозвоночныеИнфратип:ЧелюстноротыеНадкласс:ЧетвероногиеКлада:АмниотыКлада:ЗавропсидыКласс:Пт�...

Reciprocating engine

Engine utilising one or more reciprocating pistons See also: Pneumatic motor and Hydraulic motor Internal combustion piston engine Components of a typical, four-stroke cycle, internal combustion, gasoline piston engine. C. CrankshaftE. Exhaust camshaftI. Intake camshaftP. PistonR. Connecting rodS. Spark plugW. Water jacket for coolant flowV. Valves A reciprocating engine, also often known as a piston engine, is typically a heat engine that uses one or more reciprocating pistons to convert hig...

Kouklen

Cet article est une ébauche concernant une localité bulgare. Vous pouvez partager vos connaissances en l’améliorant (comment ?) selon les recommandations des projets correspondants. KuklenVue de Kuklen.GéographiePays BulgarieOblast PlovdivMunicipalité Kouklen (chef-lieu)Superficie 31 524 km2Altitude 393 mCoordonnées 42° 02′ 03″ N, 24° 47′ 16″ EDémographiePopulation 6 089 hab. (2024)Densité 0,2 hab./km2 (2024)Fonct...

The Foreman Went to France

1942 British filmThe Foreman Went to FranceOriginal UK quad format posterDirected byCharles FrendScreenplay byLeslie ArlissJohn DightonAngus MacPhailStory byJ. B. PriestleyProduced byMichael BalconStarringClifford EvansTommy TrinderConstance CummingsGordon JacksonCinematographyWilkie CooperEdited byRobert HamerMusic byWilliam WaltonProductioncompanyEaling StudiosDistributed byUnited Artists Ltd (UK)[1]Release date 22 June 1942 (1942-06-22) (UK) Running time87 minut...

Северный морской котик

Северный морской котик Самец Научная классификация Домен:ЭукариотыЦарство:ЖивотныеПодцарство:ЭуметазоиБез ранга:Двусторонне-симметричныеБез ранга:ВторичноротыеТип:ХордовыеПодтип:ПозвоночныеИнфратип:ЧелюстноротыеНадкласс:ЧетвероногиеКлада:АмниотыКлада:Синапси...

Interleague play

Regular-season baseball games between the American and National Leagues This article is about Major League Baseball interleague play. For Nippon Professional Baseball interleague play, see Interleague play (NPB). Interleague play logo Interleague play in Major League Baseball refers to regular-season baseball games played between an American League (AL) team and a National League (NL) team. Interleague play was first introduced during the 1997 Major League Baseball season. Prior to that, matc...

Pembebasan Petrus

Antonio de Bellis, Pembebasan Santo Petrus. Pembebasan Santo Petrus adalah sebuah peristiwa yang disebutkan dalam Kisah Para Rasul, pasal 12 dimana Santo Petrus diselamatkan dari penjara oleh seorang malaikat. Meskipun disebutkan dalam sebuah cuplikan tekstual pendek, kisah tersebut memberikan kebangkitan diskusi teologi dan telah menjadi subyek karya seni. Penjelasan Alkitab Bartolomé Esteban Murillo, Pembebasan St. Petrus Bagian dari serial tentangSanto Petrus Dalam Perjanjian Baru Berjala...

Super Mario's Wacky Worlds

Cancelled 1990 video game Video gameSuper Mario's Wacky WorldsThe first stage, Greek 1Developer(s)NovaLogic[1]Publisher(s)Philips Interactive Media[2]Designer(s)Marty FoulgerProgrammer(s)John BrooksSilas WarnerArtist(s)Nina StanleySeriesMarioPlatform(s)Philips CD-iReleaseCancelledGenre(s)Adventure, platformMode(s)Single-player Super Mario's Wacky Worlds is a canceled Mario platform video game developed by NovaLogic for the Philips CD-i system. The game was conceived as a sequ...

Murmansk (2015 icebreaker)

Russian icebreaker Murmansk under construction in Helsinki, Finland, in June 2015 History Russia NameMurmansk (Мyрманск) NamesakeMurmansk OwnerRosmorport[1] Port of registrySaint Petersburg[2] Ordered 23 February 2012 (Vyborg Shipyard)[5] 18 December 2012 (Arctech Helsinki Shipyard)[6] BuilderArctech Helsinki Shipyard (Helsinki, Finland)[9] Cost About RUB 4 billion[8] About 100 million euro[9] Yard number509[2] Laid down26...

List of federal judges appointed by Warren G. Harding

List of judges President Warren G. Harding. This article is part of a series aboutWarren G. Harding Early life Harding Home Legacy Memorials Electoral history Harding Railroad Car Death Presidential Center Political rise Lieutenant Governorship 1910 race for governor 1914 Senate race 29th President of the United States Presidency timeline Transition Inauguration Cabinet Judiciary Supreme Court candidates Executive actions Foreign Washington Naval Conference Depression (1920–1921) Railroad S...

Rodale, Inc.

Defunct American corporation Rodale, Inc.StatusDefunct, 2017Founded1930FounderJ. I. RodaleSuccessorHearst (magazines)Crown Publishing Group (books)Country of originUnited StatesHeadquarters locationEmmaus, Pennsylvania, U.S.Publication typesMagazines, booksOfficial websiterodale.com Rodale, Inc. (/ˈroʊdeɪl/), was an American publisher of health and wellness magazines, books, and digital properties headquartered in Emmaus, Pennsylvania, with a satellite office in New York City.[1] T...

9th Hussar Regiment (France)

This article includes a list of references, related reading, or external links, but its sources remain unclear because it lacks inline citations. Please help improve this article by introducing more precise citations. (February 2013) (Learn how and when to remove this message) 9th Hussar Regiment 9e régiment de hussardsFrench 9th hussarsActive1792–1979Country FranceBranchArmyTypeCavalryRoleHussarsAnniversariesSaint George, Saint-GeorgesEngagementsFrench Revolutionary Wars Napoleonic ...

Polonius

Character in Hamlet Fictional character PoloniusHamlet characterPolonius Behind the Curtain by Jehan Georges Vibert, 1868Created byWilliam ShakespeareIn-universe informationAffiliationKing ClaudiusChildrenOphelia (daughter) Laertes (son) Polonius is a character in William Shakespeare's play Hamlet. He is the chief counsellor of the play's ultimate villain, Claudius, and the father of Laertes and Ophelia. Generally regarded as wrong in every judgment he makes over the course of the play,[1...

2016–17 Little Rock Trojans women's basketball team

Intercollegiate basketball season 2016–17 Little Rock Trojans women's basketballSun Belt regular season championsWNIT, Second RoundConferenceSun Belt ConferenceRecord25–9 (17–1 Sun Belt)Head coachJoe Foley (14th season)Assistant coaches Robert Dallimore Alicia Cash Steve Wiedower Home arenaJack Stephens CenterSeasons← 2015–162017–18 → 2016–17 Sun Belt Conference women's basketball standings vte Conf Overall Team W L PCT W L ...

Crocifisso degli Esuli

Crocifisso degli EsuliIl Crocifisso degli Esuli con il baldacchino da esposizioneAutoreScuola senese DataMetà del XV secolo MaterialeLegno Dimensioni205×100 cm UbicazioneCollegiata di Santa Maria Assunta, San Ginesio Il Crocifisso degli Esuli, conosciuto anche come Crocifisso Ligneo o Crocifisso di San Ginesio, è una piccola scultura lignea raffigurante la crocifissione di Gesù, databile alla metà del XV secolo, tra il 1450 e il 1458 circa, opera di alcuni ignoti artisti della scuol...

Kingdom of Iraq

Independent Iraqi monarchy (1932–1958) Hashemite Kingdom of Iraqالمملكة العراقية الهاشمية (Arabic)al-Mamlakah al-ʿIrāqiyyah ʾal-Hāshimyyah1932–1958 Flag Coat of arms Anthem: السلام الملكيAs-Salam al-MalakiThe Royal SaluteCapitaland largest cityBaghdadOfficial languagesArabicCommon languagesArabicAssyrianKurdishPersianEnglishReligion Islam (80%)Christianity (15%)Judaism (2%)Yazidism (2%)Mandaeism (1%)Demonym(s)IraqiGovernmentUnitar...

コンピュータ

この記事は検証可能な参考文献や出典が全く示されていないか、不十分です。出典を追加して記事の信頼性向上にご協力ください。（このテンプレートの使い方）出典検索?: コンピュータ – ニュース · 書籍 · スカラー · CiNii · J-STAGE · NDL · dlib.jp · ジャパンサーチ · TWL（2014年4月）コンピュータさまざまなコンピュータ。(1)20...

نادي كلية جامعة دبلن

نادي كلية جامعة دبلن تأسس عام 1895 البلد جمهورية أيرلندا الدوري الدوري الأيرلندي الدرجة الأولى [لغات أخرى]‏ المدرب مارتن راسيل (1 يناير 2009–23 فبراير 2014) الموقع الرسمي الموقع الرسمي تعديل مصدري - تعديل نادي كلية جامعة دبلن لكرة القدم (بالإنجليزية: Un...