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equation of the hessian


infinite points of the Kjp cubic

Any cubic of the form u x^3 + v y^3 + w z^3 + k x y z = 0 has its inflexions on the sidelines of ABC since its hessian is a cubic of the same form namely u x^3 + v y^3 + w z^3 + l x y z = 0. When k = 0, the hessian degenerates into the sidelines of ABC.

K598 is the only cubic of this form that is a stelloid i.e. a cubic with three concurring asymptotes (here at the Lemoine point K) making 60° angles with one another. These asymptotes are parallel to those of K024 therefore parallel to the sidelines of the Morley triangle.

The hessian of K598 is a focal cubic with focus K and it is the only proper focal cubic of this same form. Its axis is the orthic axis (L) and its real asymptote (A) is the homothetic of (L) under h(K, 2).

The satellite (S) of the line at infinity in K598 is the homothetic of (L) under h(K, 2/3).

These two cubics meet at their nine common inflexions which lie on the sidelines of ABC. Those on BC are Fa, F'a, F"a with coordinates of the form :

0 : (b^2/SB)^(1/3) : – (c^2/SC)^(1/3)
0 : j (b^2/SB)^(1/3) : – j^2 (c^2/SC)^(1/3)
0 : j^2 (b^2/SB)^(1/3) : – j (c^2/SC)^(1/3)

where j is one of the complex cube root of the unity. Naturally, one only of these three points is real and is denoted Fa.

The polar conics of any point on the orthic axis (L) in both cubics are rectangular hyperbolas that meet on the quartic Q083 which is also a stelloid.

The poles of the line at infinity in K598 are the foci of the inconic with center K, perspector H sometimes called ellipse-K when ABC is acutangle.