bicircular quartic

The bicircular quartic is a bicircular algebraic curve that is a quartic.
The curve is the cyclic of a conic.

When the conic has the Cartesian equation x² / l + y² / m = 1, the fixed point is (n, o), and the power of inversion is p, then l, m, n, o and p can be written as function of a, b, c, d and e ¹⁾.

p=0
When the power of inversion (p) is equal zero, the curve is called rational: a bicircular rational quartic.
This curve is the pedal of the circle with respect to one of its points.

We know the following species:

• a=b, d=0 or l=m, o=0: limacon of Pascal
  • with c²=2a³/27, e=-a²/12 or with l=n², b=0: cardioid
• b=1, c=d=e=0 or n=o=0: hippopede

p<>0
For p <> 0, the the following curves can be distinguished:

• a=b, d=0 or l=m, o=0: Cartesian oval
  Now the curve is the cyclic of a circle
• a=-b, c=d=0 or l+m+n²+o²-p=0: Cassinian oval
• a<>b, d=0 or l<>m, o=0: plane spiric curve

The bicircular quartic can also be written in a tripolar equation:

f r₁ + g r₂ + h r₃ = 0.

notes

1) As follows:
a = - 2 (2l+n²+o²-p)
b = - 2 (2m+n²+o²-p)
c = 4ln
d = 4mo
e = (n²+o²-p)² - 4 (ln²+mo²)