Cochleoid

$r={\frac {\sin \theta }{\theta }},-20<\theta <20$

cochleoid (solid) and its polar inverse (dashed)

A flexible pole is fixed upright at one end and bent over to always form a circular arc. The other end then traces out a Cochleoid.

In geometry, a cochleoid is a snail-shaped curve similar to a strophoid which can be represented by the polar equation

$r={\frac {a\sin \theta }{\theta }},$

the Cartesian equation

$(x^{2}+y^{2})\arctan {\frac {y}{x}}=ay,$

or the parametric equations

$x={\frac {a\sin t\cos t}{t}},\quad y={\frac {a\sin ^{2}t}{t}}.$

The cochleoid is the inverse curve of Hippias' quadratrix.

References

• J. Dennis Lawrence (1972). A catalog of special plane curves. Dover Publications. p. 192. ISBN 0-486-60288-5.
• Cochleoid in the Encyclopedia of Mathematics
• Liliana Luca, Iulian Popescu: A Special Spiral: The Cochleoid. Fiabilitate si Durabilitate - Fiability & Durability no 1(7)/ 2011, Editura "Academica Brâncuşi", Târgu Jiu, ISSN 1844-640X
• Roscoe Woods: The Cochlioid. The American Mathematical Monthly, Vol. 31, No. 5 (May, 1924), pp. 222–227 (JSTOR)
• Howard Eves: A Graphometer. The Mathematics Teacher, Vol. 41, No. 7 (November 1948), pp. 311–313 (JSTOR)

External links

• cochleoid at 2dcurves.com
• Weisstein, Eric W. "Cochleoid". MathWorld.