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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).
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© Robert FERRÉOL  2017