Cochleoid

Short description: Spiral curve of the form r = a*sin(θ)/θ

[math]\displaystyle{ r=\frac{\sin \theta}{\theta}, -20\lt \theta\lt 20 }[/math]

cochleoid (solid) and its polar inverse (dashed)

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

[math]\displaystyle{ r=\frac{a \sin \theta}{\theta}, }[/math]

the Cartesian equation

[math]\displaystyle{ (x^2+y^2)\arctan\frac{y}{x}=ay, }[/math]

or the parametric equations

[math]\displaystyle{ x=\frac{a\sin t\cos t}{t}, \quad y=\frac{a\sin^2 t}{t}. }[/math]

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. https://archive.org/details/catalogofspecial00lawr/page/192. 
• 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". http://mathworld.wolfram.com/Cochleoid.html.