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Reference : Stuyvaert, Point remarquable dans le plan d'une cubique, Nouvelles Annales de Mathématiques, Série. 3, 18 (1899), pp. 275-285.

In general, there is one and only one point S whose polar conic in a cubic is a (possibly degenerate) circle. In such case, we call S the Stuyvaert point of the cubic.

When the cubic is circular, it is the singular focus. When the cubic is a K+, it is the common point of the asymptotes and the circle splits into the line at infinity and another line.

On the other hand, a Stuyvaert cubic SK is a cubic having a pencil of circular polar conics.

In other words, there is a line L (we call the circular line of the cubic) such that the polar conic of each of its points is a circle. These circles are obviously in a same pencil. SK must be a K+ with asymptotes (one only is real) concurring at X and then the polar conic of X must split into the line at infinity and the radical axis of the circles.

The poloconic of the line at infinity splits into two perpendicular lines secant at X which are the bisectors of the circular line L and the orthic line. Recall that this orthic line is the locus of points whose polar conic is a rectangular hyperbola (it would be undefined if the cubic be a K60+ which is not the case here).

The following table gathers together several examples of these cubics.

Notes :

P602 =(b-c) (b+c) (2 a^4-2 a^2 b^2+b^4-2 a^2 c^2+c^4) : : , SEARCH=8.660881599853399

P603 = a (a-b-c) (b-c) (2 a^3-2 a^2 b-a b^2+b^3-2 a^2 c+4 a b c-b^2 c-a c^2-b c^2+c^3) : : , SEARCH=19.08170495120384

P628 = (b-c) (-2 a^4+a^2 b^2+b^4+a^2 c^2-2 b^2 c^2+c^4) : : , SEARCH=-2.260132844466736

These points are now X(11123), X(11124), X(11125) in ETC (2016-12-07).