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Figure 1. The cochleoid (pdf)

A plane transcendental curve whose equation in polar coordinates is \begin{equation} \rho = a\frac{\sin\varphi}{\varphi}. \end{equation}

The cochleoid has infinitely many spirals, passing through its pole and touching the polar axis (see Fig.). The pole is a singular point of infinite multiplicity. Any straight line through the pole $O$ intersects the cochleoid; the tangents to the cochleoid at these intersection points pass through the same point.

The inverse of the cochleoid with respect to the origin is the quadratrix of Hippias.


[1] A.A. Savelov, "Planar curves" , Moscow (1960) (In Russian)
[a1] J.D. Lawrence, "A catalog of special plane curves" , Dover (1972) ISBN 0-486-60288-5 Zbl 0257.50002
How to Cite This Entry:
Cochleoid. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article