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Cochleoid


Cochleoid

The cochleoid, whose name means "snail-form" in Latin, was first considered by John Perks as referenced in Wallis et al. (1699). The cochleoid has also been called the oui-ja board curve (Beyer 1987, p. 215). The points of contact of parallel tangents to the cochleoid lie on a strophoid.

Smith (1958, p. 327) gives historical references for the cochleoid, but corrections to the name and date mentioned as "discussed by J. Perk Phil. Trans. 1700" (actually John Perks, as mentioned in Wallis et al. 1699 and Pedersen 1963), the separateness of papers by Falkenburg (1844) and Benthem (1844), and the spelling of the latter's name should all be noted.

In polar coordinates, the curve is given by

 r=(asintheta)/theta.
(1)

For the parametric form

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

the curvature is

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

See also

Cochleoid Inverse Curve, Conchoid, Quadratrix of Hippias

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References

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.

Cite this as:

Weisstein, Eric W. "Cochleoid." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Cochleoid.html

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