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centers of the Apollonius circles

K604 is an example of cubic having a pencil of circular polar conics as in Table 47. Compare with K393.

The circular line passes through X(351), X(523) : any point on this line has a circular polar conic :

– that of X(351) – the center of the Parry circle and also the tripolar centroid of X(6) – degenerates into the line at infinity and the radical axis of the pencil of circles passing through X(187), X(524), X(2482),

– that of X(523) is the Parry circle.

The orthic line is the tangent at X(110) to the circumcircle : any point on this line has a polar conic which is a rectangular hyperbola. These rectangular hyperbolas form a pencil having the same infinite points – that of the rectangular circum-hyperbola through X(110) – and two other common finite points lying on the radical axis above.

K604 is a nK0+ with three concurring asymptotes at X(351) but only one is real. See K150 for other cubics nK0+.