查看“︁双扭线”︁的源代码

[[Image:Lemniscate of Bernoulli.svg|thumb|400px|right|[[伯努利双扭线]]，以及其二個焦點]]

'''双扭线'''（lemniscate）是[[代數幾何]]中的名詞，是指8字型或是{{math|[[∞]]}}型的[[曲線]]<ref name="lemniscatomy"/><ref name="erickson"/>，lemniscate源自拉丁文"lēmniscātus"，意思是「用緞帶裝飾」<!--from the Greek λημνίσκος meaning ribbons,--><ref name="erickson">{{citation|title=Beautiful Mathematics|series=MAA Spectrum|publisher=Mathematical Association of America|first=Martin J.|last=Erickson|year=2011|isbn=9780883855768|contribution=1.1 Lemniscate|pages=1–3|url=https://books.google.com/books?id=LgeP62-ZxikC&pg=PA1|accessdate=2017-10-28|archive-date=2016-12-04|archive-url=https://web.archive.org/web/20161204162641/https://books.google.com/books?id=LgeP62-ZxikC&pg=PA1|dead-url=no}}.</ref>，或是指[[羊毛]]（緞帶的原料）<ref name="lemniscatomy"/>。
==歷史和例子==
===Booth双扭线===
[[File:Lemniscate of Booth.png|thumb|Booth双扭线]]
Booth双扭线的研究可以追溯到西元五世紀的希臘[[新柏拉图主义]]哲學家及數學家[[普罗克洛]]，他考慮[[环面]]和一個和環面軸心平行的平面相交產生的圖形，他所觀測到的，大部份這類的截面會包括一個或是兩個[[卵形]]，不過若平面恰好和環面的內表面[[相切]]，其圖形會是一個8字型的圖案，普罗克洛稱為[[腳銬]]或是{{le|Hippopede|Hippopede}}。lemniscate這個名字最早是在十七世紀出現，19世紀的數學家{{le|James Booth|James Booth (mathematician)}}也曾研究此一曲線<ref name="lemniscatomy">{{citation
 | last = Schappacher | first = Norbert
 | contribution = Some milestones of lemniscatomy
 | location = New York
 | mr = 1483331
 | pages = 257–290
 | publisher = Dekker
 | series = Lecture Notes in Pure and Applied Mathematics
 | title = Algebraic Geometry (Ankara, 1995)
 | volume = 193
 | year = 1997}}</ref>。

Booth双扭线可以定義為[[四次函數|四次多項式]]<math>(x^2 + y^2)^2 - cx^2 - dy^2</math>的零集，其中參數''d''為負值。若參數''d''為正值，會得到{{le|Booth卵形|oval of Booth}}。

===伯努利双扭线===
[[File:Lemniskate bernoulli2.svg|thumb|伯努利双扭线]]
[[乔瓦尼·多梅尼科·卡西尼]]在1680年研究一系列的曲線，現今稱為[[卡西尼卵形线]]，是所有到兩個定點（圖形的[[焦點]]）距離乘積為常數的點形成的[[軌跡]]。在非常特殊的條件下（兩點距離的一半等於上述常數的平方根），所得的就是双扭线。

[[約翰·白努利]]在1694年研究卡西尼卵形线中的双扭线（現今稱為[[伯努利双扭线]]，如上圖），他找到這曲線和[[戈特弗里德·莱布尼茨]]稍早提出的[[等時降線]]的關係。此曲線是多項式<math>(x^2 + y^2)^2 - 2a^2 (x^2 - y^2)</math>的零集。約翰·伯努利的哥哥[[雅各布·伯努利]]也在同一年研究該曲線，並且給了lemniscate的名稱<ref name="bos">{{citation
 | last = Bos
 | first = H. J. M.
 | contribution = The lemniscate of Bernoulli
 | location = Dordrecht
 | mr = 774250
 | pages = 3–14
 | publisher = Reidel
 | series = Boston Stud. Philos. Sci., XV
 | title = For Dirk Struik
 | url = https://books.google.com/books?id=OvK9orJNezwC&pg=PA3
 | year = 1974
 | isbn = 9789027703934
 | accessdate = 2017-10-28
 | archive-date = 2016-12-04
 | archive-url = https://web.archive.org/web/20161204072203/https://books.google.com/books?id=OvK9orJNezwC&pg=PA3
 | dead-url = no
 }}.</ref>，伯努利双扭线也可以定義為所有由到兩定點距離之乘積為定值（兩定點之間距離平方的四分之一）的點的軌跡<ref>{{citation
 | last1 = Langer | first1 = Joel C.
 | last2 = Singer | first2 = David A.
 | doi = 10.1007/s00032-010-0124-5
 | issue = 2
 | journal = Milan Journal of Mathematics
 | mr = 2781856
 | pages = 643–682
 | title = Reflections on the lemniscate of Bernoulli: the forty-eight faces of a mathematical gem
 | volume = 78
 | year = 2010}}</ref>。伯努利双扭线是特殊的Booth雙扭線，滿足<math>d=-c</math>的條件，且其對應的環面內圓和環面截面的圓大小相同<ref name="lemniscatomy"/>。{{le|雙扭線橢圓函數|Lemniscatic elliptic function}}是針對伯努利双扭线，類似[[橢圓函數]]的函數，而[[高斯常數]]可用來定義雙扭線常數（Lemniscate常數），在計算伯努利双扭线[[弧長]]時會用到。

===赫羅諾雙紐線===
[[File:Lemniscate-of-Gerono2.svg|thumb|280px|赫羅諾雙紐線：x<sup>4</sup>&minus;x<sup>2</sup>+y<sup>2</sup>=0的解集合<ref>{{Cite web |url=http://www.mathematische-basteleien.de/acht.htm |title=Achtkurve. |accessdate=2017-10-28 |archive-date=2017-10-29 |archive-url=https://web.archive.org/web/20171029065139/http://www.mathematische-basteleien.de/acht.htm |dead-url=no }}</ref>]]
赫羅諾雙紐線（lemniscate of Gerono）也稱為[[8字型線]]或惠更斯雙紐線（lemniscate of Huygens），是四次方程式<math>y^2-x^2(a^2-x^2)=0</math>的解集合ref>{{citation|title=An elementary treatise on cubic and quartic curves|first=Alfred Barnard|last=Basset|publisher=Deighton, Bell|year=1901|pages=171–172|url=https://books.google.com/books?id=T40LAAAAYAAJ&pg=PA171|contribution=The Lemniscate of Gerono|accessdate=2017-10-28|archive-date=2016-12-04|archive-url=https://web.archive.org/web/20161204074607/https://books.google.com/books?id=T40LAAAAYAAJ&pg=PA171|dead-url=no}}.</ref><ref>{{citation|title=Newton's Principia for the common reader|first=S|last=Chandrasekhar|publisher=Oxford University Press|year=2003|isbn=9780198526759|page=133|url=https://books.google.com/books?id=qomP58txKQwC&pg=PA133|accessdate=2017-10-28|archive-date=2016-12-04|archive-url=https://web.archive.org/web/20161204072134/https://books.google.com/books?id=qomP58txKQwC&pg=PA133|dead-url=no}}.</ref>。{{le|Viviani曲線|Viviani's curve}}是由圓柱和圓相交所形成的三維曲線，也是8字型的曲線，赫羅諾雙紐線是Viviani曲線在特定平面上的投影<ref>{{citation|first1=Luisa Rossi|last1=Costa|first2=Elena|last2=Marchetti|contribution=Mathematical and Historical Investigation on Domes and Vaults|pages=73–80|title=Aesthetics and architectural composition : proceedings of the Dresden International Symposium of Architecture 2004|year=2005|location=Mammendorf|publisher=Pro Literatur|editor1-last=Weber|editor1-first=Ralf|editor2-last=Amann|editor2-first=Matthias Albrecht}}.</ref>。

===其他===
其他有8字型的代數曲線有
*[[魔鬼曲線]]，是由四次方程<math>y^2 (y^2 - a^2) = x^2 (x^2 - b^2)</math>所定義的曲線，曲線中有一部份為8字型<ref>{{citation|title=The Universal Book of Mathematics: From Abracadabra to Zeno's Paradoxes|first=David|last=Darling|publisher=John Wiley & Sons|year=2004|isbn=9780471667001|contribution=devil's curve|pages=91–92|url=https://books.google.com/books?id=HrOxRdtYYaMC&pg=PA91|accessdate=2017-10-28|archive-date=2016-12-04|archive-url=https://web.archive.org/web/20161204143654/https://books.google.com/books?id=HrOxRdtYYaMC&pg=PA91|dead-url=no}}</ref>。
*[[瓦特曲線]]是由機械[[連桿]]形成的8字型曲線，瓦特曲線是六次方程<math>(x^2+y^2)(x^2+y^2-d^2)^2+4a^2y^2(x^2+y^2-b^2)=0</math>的解集合，伯努利双扭线是瓦特曲線中的一個特例。

==相關條目==
* [[日行跡]]，一年之中太陽在中午位置所形成8字形的軌跡
* [[洛伦茨吸引子]],三維動態系統，其外形類似双扭线
* {{le|多項式雙紐線|Polynomial lemniscate}}，複數多項式絕對值的水平集
* {{le|廣義圓錐曲線|Generalized conic}}
* [[双叶线]]

==參考資料==
{{Reflist}}

==外部連結==
{{commonscat|Lemniscate}}
* {{springer|title=Lemniscates|id=p/l058130}}
[[Category:数学术语]]
[[Category:曲線]]