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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
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 = R^{2}, 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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© Robert FERRÉOL 2017