BICIRCULAR QUARTIC

 (Reduced) Cartesian equation: .

The bicircular quartics are the cyclic curves the deferent (or, to use the vocabulary of this last link, the initial curve) of which is a centred conic, in other words, they are the envelopes of the circles the centres of which describe a centred conic and such that a fixed point has a constant power with respect to these circles.PB: C or D = 0

If the deferent is  (so, is centred on O), the fixed point is (a, b), the power is p and , then the equation of the bicircular quartic is the equation above, with

In some cases, the power p obtained is complex (for example if C = D = 0 and E < 0).

Examples:
(Remember that the reference circle, or circle of inversion, is the circle with centre W and radius ).

 Type Condition related to A, B, C, D, E providing the generality of the example. Condition related to L, M, N, a, b providing the generality of the example NSC related to the reference circle and the deferent providing the generality of the example. rational bicircular quartic the reference circle is reduced to a point or tangent to the deferent Cartesian curve A = B, D = 0 L = M, b = 0, The deferent is a circle complete Cartesian oval L = M = R2, b = 0 the deferent is a circle and p < 0, or p ³ 0 and the deferent and reference circles do not intersect. limaçon of Pascal L = M, b = 0, p = 0 the deferent is a circle and the reference circle is reduced to a point or is tangent to the deferent circle cardioid , b = 0 the deferent is a circle and the reference circle reduces to a point on this circle plane spiric A  B and D = 0 L ¹ M and b = 0 the deferent is not a circle and the reference circle is centred on an axis of the deferent spiric of Perseus A  B and C = D = 0 L ¹ M and a = b = 0 the deferent is not a circle and the reference circle is centred on the centre of the deferent Cassinian oval B = –A and C =D =0 L +M +N =0 and a = b = 0 the reference circle is the Monge circle of the deferent. Booth curve C = D = E = 0 a = b = N = 0 The reference circle is reduced to the centre of the deferent lemniscate of Bernoulli B = –A and C =D = E = 0 M = – L and a = b = N = 0 The deferent is a rectangular hyperbola and the reference circle is reduced to the centre of it

The bicircular quartics are the inverses of the Cartesian ovals; more precisely, if we take , then the quartic above is the image of the Cartesian oval:  by an inversion of pole , transforming  into  and  into .
Therefore, Cartesian ovals are special cases of bicircular quartics.

The curve is not empty iff the three sums  are not simultaneously positive nor negative; this condition is equivalent to saying that among the three coefficients a’, b’, g’, the couple of them with largest absolute value have opposite signs (not strict inequalities).