View source for Cochleoid

{{short description|1=Spiral curve of the form r = a*sin(θ)/θ}}
[[File:Cochleoid with a=1.svg|thumb|upright=1.25|<math>r=\frac{\sin \theta}{\theta}, -20<\theta<20</math>]]
[[File:Kochleoide als inverse zur quadratrix.svg|thumb|upright=1.25|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>r=\frac{a \sin \theta}{\theta},</math>
the [[Cartesian equation]]
:<math>(x^2+y^2)\arctan\frac{y}{x}=ay,</math>
or the [[parametric equation]]s
:<math>x=\frac{a\sin t\cos t}{t}, \quad y=\frac{a\sin^2 t}{t}.</math>

The cochleoid is the [[inverse curve]] of [[Quadratrix of Hippias|Hippias' quadratrix]].<ref>Heinrich Wieleitner: ''Spezielle Ebene Kurven''. Göschen, Leipzig, 1908, pp. [https://archive.org/details/spezielleebenek01wielgoog/page/n277 256-259] (German)</ref>

==Notes==
<references/>

==References==
* {{cite book | author=J. Dennis Lawrence | title=A catalog of special plane curves | publisher=Dover Publications | year=1972 | isbn=0-486-60288-5 | page=[https://archive.org/details/catalogofspecial00lawr/page/192 192] | url-access=registration | url=https://archive.org/details/catalogofspecial00lawr/page/192 }}
*[https://www.encyclopediaofmath.org/index.php/Cochleoid ''Cochleoid''] in the [[Encyclopedia of Mathematics]]
*Liliana Luca, Iulian Popescu: [http://www.utgjiu.ro/rev_mec/mecanica/pdf/2011-01/3_luca.pdf ''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.&nbsp;222–227 ([https://www.jstor.org/stable/2299243 JSTOR])
* [[Howard Eves]]: ''A Graphometer''. The Mathematics Teacher, Vol. 41, No. 7 (November 1948), pp.&nbsp;311–313 ([https://www.jstor.org/stable/27953353 JSTOR])

== External links ==
{{commons category}}
*[http://www.2dcurves.com/spiral/spiralc.html cochleoid] at 2dcurves.com
* {{MathWorld|title=Cochleoid|urlname=Cochleoid}}

[[Category:Plane curves]]


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