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BIFOLIUM

Curve studied by de Longchamps in 1886 and Brocard in 1887.
Other name: double folium.

 
Polar equation: 
.
Cartesian equation:  .
Rational Cartesian parametrization: .
Rational circular quartic.

Let (C) be the circle passing through O, A(a,0) and B(0, b), and a variable line passing through O and intersecting (C) in P, whose projection on Ox is H. The associated bifolium is the locus of the projection M of H on the line (OP).
 
 

Bifoliums are the pedal curves of deltoids with respect to one of their points (here O); see the links between the bifolium and the deltoid at folium.

When a = 0, i.e. when the pedal is taken with respect to a cusp of the deltoid, we get the regular bifolium.
When b = 0, i.e. when the pedal is taken with respect to a vertex of the deltoid, we get the simple folium.
 
 
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© Robert FERRÉOL  2017