Cochleoid

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

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

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

cochleoid (solid) and its polar inverse (dashed)

the Cartesian equation

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

or the parametric equations

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

The cochleoid is the inverse curve of Hippias' quadratrix.^[1]