Bifoliate Curve

The bifoliate curve is the quartic curve given by the Cartesian equation

(1)

and the polar equation

(2)

for .

It has a cusp at the origin .

The area of the bifoliate is given by

			(3)
			(4)
			(5)
			(6)

(OEIS A093954).

For , the two loops have parametric equations

			(7)
			(8)

where traces the upper loop and traces the lower loop. The total perimeter is

			(9)
			(10)

(OEIS A118289), where and .

A closed form in Lauricella functions is

(11)

where , , , , and , ..., are the roots of

(12)

The value uses the Euler integral representation with principal branches along . The substitution gives this representation.

Taking as the parameter, the bifoliate has curvature and tangential angle given by

			(13)
			(14)