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蜗线


Cochleoid

蜗线,其名称在拉丁语中意为“蜗牛形状”,最早由约翰·珀克斯在沃利斯等人(1699)的参考文献中提出。蜗线也被称为占卜板曲线(Beyer 1987,第 215 页)。与蜗线平行切线的接触点位于箕舌线上。

史密斯(1958,第 327 页)给出了蜗线的历史参考,但应注意对名称和日期的更正,即“J. Perk Phil. Trans. 1700”讨论过(实际上是约翰·珀克斯,如沃利斯等人 1699 年和佩德森 1963 年所述),Falkenburg (1844) 和 Benthem (1844) 的论文的独立性,以及后者的名字的拼写。

极坐标中,该曲线由下式给出

 r=(asintheta)/theta.
(1)

对于参数形式

x=(asintcost)/t
(2)
y=(asin^2t)/t,
(3)

曲率

 kappa(t)=(2sqrt(2)t^3[2t-sin(2t)])/([1+2t^2-cos(2t)-2tsin(2t)]^(3/2)).
(4)

另请参阅

蜗线反曲线蚌线希皮亚斯割圆曲线

使用 Wolfram|Alpha 探索

参考文献

Benthem, A. "De Slakkenlijn of Cochleoïde." Nieuw Arch. Wisk. 10, 76-80, 1884.Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 215, 1987.Falkenburg, C. "Die Cochleoïde." Archiv der Math. u. Physik 70, 259-267, 1884.Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 192 and 196, 1972.Luca, L.; Ghimisi, S.; and Popescu, I. "Studies Regarding the Movement on the Cochleoid." Advanced Materials Res. 463-464, 147-150, 2012.MacTutor History of Mathematics Archive. "Cochleoid." http://www-groups.dcs.st-and.ac.uk/~history/Curves/Cochleoid.html.Pedersen, O. "Master John Perks and his Mechanical Curves." Centaurus 8, 1-18, 1963.Smith, D. E. History of Mathematics, Vol. 2: Special Topics of Elementary Mathematics. New York: Dover, p. 327, 1958.Wallis, D.; Gregory, D.; and Caswell, J. "A Letter of Dr. Wallis to Dr. Sloan, concerning the Quadrature of the Parts of the Lunula of Hippocrates Chius, performed by Mr. John Perks; with the further Improvements of the same, by Dr. David Gregory, and Mr. John Caswell." Philos. Trans. 21, 411-418, 1699.

请这样引用

Weisstein, Eric W. “蜗线。” 来自 MathWorld—— Wolfram Web 资源。 https://mathworld.net.cn/Cochleoid.html

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