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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 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