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BIFOLIUM
Curve studied by de Longchamps in 1886 and Brocard in
1887.
Other names: double folium, bifoliate curve. 
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