Bicircular quartic

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).

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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