EQUILATERAL TREFOIL

 Curve studied by G. de Longchamps in 1884. Other name: Longchamps trisectrix.

 The asymptotes form an equilateral triangle. Polar equation: . Cartesian equation:  or . (nota: (0,0) is an isolated point of the algebraic curve). Cartesian parametrization:  (). Rational cubic with an isolated point (O).

The equilateral trefoil is:

- an epispiral with 3 branches
 - the inverse of the regular trifolium with respect to its centre - therefore, it is also the reciprocal polar of the negative pedal of the trifolium, namely, the deltoid - the curve obtained as the locus of the intersection points between two tangents at P and Q to a circle with centre O, the angle being equal to twice the angle . Explanation: the line (PQ) envelopes the deltoid, polar of the trefoil.
- the cissoid with respect to O of the hyperbola with equation:  and the line (to be checked!!!!!) (see cissoid of Zahradnik).

 - the planar section of a sinusoidal cone

As indicated by its second name, it is a trisectrix.

 This curve is quite close to the curve with polar equation (inverse of the torpedo), with Cartesian equation: or also , the asymptotes of which this time form an isosceles right triangle. Ditto with the curve with polar equation , Cartesian equation or also .

The equilateral trefoil and the Humbert cubic (similar shape, but the asymptotes of which intersect) are the only cubics with a rotation symmetry of order 3 (see Goursat curve).