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KIEPERT CURVE
Curve studied by W. Roberts and L. Kiepert in 1870.
Friedrich Wilhelm August Ludwig Kiepert (18461934): German mathematician. Other name of this curve: lemniscate with three poles. 
Tripolar equation:
where (ABC) is the equilateral triangle A(d),
B(dj),
C(dj²).
Polar equation: with . Cartesian equation: . Tricircular sextic of genus 1. 

The Kiepert curve is the sinusoidal spiral of order 3, it is therefore a special case of Cassinian curve; it is to the equilateral triangle what the lemniscate of Bernoulli is to the bipoint.
Its inverse with centre O is the Humbert cubic.
Compare to the regular trifolium.
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© Robert FERRÉOL 2017