Strophoid

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A strophoid, also known as a logocyclic curve or a foliate, is a cubic curve generated by increasing or diminishing the radius vector of a variable point P on a straight line by the distance PA of the point from the foot of the perpendicular drawn from the origin to the fixed line.

The polar equation is

.

The Cartesian equation is

y² = x²(a − x) / (a + x),

where a is the distance of the line from the origin. The curve resembles the Folium of Descartes, and has a node between x = 0, x = a, and two branches asymptotic to the line x = −a. The curve has two more asymptotes, in the plane with complex coordinates, given by

.

A curve:

This article incorporates text from the Encyclopædia Britannica Eleventh Edition, a publication now in the public domain.