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Cochleoid

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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.

Notes

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

References

External links


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