Cochleoid

Spiral curve of the form r = a*sin(θ)/θ
r = sin ⁡ θ θ , − 20 < θ < 20 {\displaystyle r={\frac {\sin \theta }{\theta }},-20<\theta <20}
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

r = a sin ⁡ θ θ , {\displaystyle r={\frac {a\sin \theta }{\theta }},}

the Cartesian equation

( x 2 + y 2 ) arctan ⁡ y x = a y , {\displaystyle (x^{2}+y^{2})\arctan {\frac {y}{x}}=ay,}

or the parametric equations

x = a sin ⁡ t cos ⁡ t t , y = a sin 2 ⁡ t t . {\displaystyle 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.[1]

Notes

  1. ^ Heinrich Wieleitner: Spezielle Ebene Kurven. Göschen, Leipzig, 1908, pp. 256-259 (German)

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)
Wikimedia Commons has media related to Cochleoid.
  • cochleoid at 2dcurves.com
  • Weisstein, Eric W. "Cochleoid". MathWorld.


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