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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.

cubic

type

centers on the cubic

X

L

remarks

K392

pK

X(99), X(115)

X(2)

X(2)-X(1637)

see CL009

K393

nK0

--

X(1637)

orthic axis

see CL044

K601

pK

X(476), X(3258)

X(2)

X(2)-?

see CL009

K602

nK0

X(99), X(620)

P602

X(351)-X(2799)

see CL049

K603

nK0

X(100), X(3035)

P603

X(2804)-P603

see CL049

K604

nK0

--

X(351)

X(351)-X(523)

see CL044

K628

nK

X(1), X(1784), X(2588), X(2589)

P628

X(523)-P628

 

nK0(X468, X468)

nK0

--

X(2)

X(2)-X(6)

see CL026

nK0(X523, X523)

nK0

--

X(2)

X(2)-X(525)

see CL026

nK0(X1990, X1990)

nK0

--

X(2)

Euler line

see CL026

nK0(X57 x X527, X57)

nK0

X(1), X(1323)

X513 x X527

X(514)-?

see CL044

psK(cX113, X3260, X74)

psK

X(74), X(146), X(265)

?

?

see CL063

psK(X620, X523, X99)

psK

X(99), X(148), X(620)

?

X(2799)-?

see CL063

psK(X3035, X693, X100)

psK

X(100), X(149), X(3035)

?

X(2804)-?

see CL063

 

 

 

 

 

 

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).