CU-Cu1 The CU-Hessian

Definition

The Hessian of a cubic is the locus of points P for which the P-Polar Conic CU-P-Co1 is degenerated in two lines.

It is also a cubic and it intersects the Reference Cubic CU in 9 points, which are the 9 inflection points (flexpoints CU-9P1) of the cubic.

Further description

The locus of points P for which the CU-Polar Conic (CU-P-Co1) degenerates into 2 lines (meeting at Q) is called the Hessian HE (which is also a cubic).

Also the CU-Q-Polar Conic will degenerate into 2 lines meeting at P.

P and Q are referred to as corresponding points.

The tangents to HE at P and Q share the same tangential R.

Let S be the corresponding point of R.

S is called the complementary point of P and Q.

The CU-P-Polar Line is the same as the HE-Q-tangent.

See [16b], page 24.

There is also the Hessian matrix. It is a square matrix of second-order partial derivatives of a scalar-valued function, or scalar field. It describes the local curvature of a function of many variables. The Hessian matrix was developed in the 19th century by the German mathematician Ludwig Otto Hesse (1811-1874) and later named after him. The Hessian Matrix is used to find the equation of the Hessian Curve by calculating the determinant of the Hessian Matrix.

|  d² f       d² f         d² f   |

| dx²       dx dy      dx dz |

|                                        |

CU-Hessian     =   DET  |  d² f        d² f       d² f    |   =   0

| dy dx      dy²      dy dz  |

|                                        |

|  d² f        d² f        d² f   |

| dz dx     dz dy      dz²   |

Where f represents the equation of CU in (x,y,z).

Construction

Besides calculating this curve it is also possible to construct the Hessian as follows.

1. Given Reference Cubic CU.

2. Choose 3 random points P, Q, R (not necessary on CU) such that their Polar Conics mutually intersect in 4 real points. For the construction of a P-Polar Conic, see CU-P-Co1.

3. Draw the Diagonal Triangles of the 3 QA’s formed by the mutual intersection points of the 3 Polar Conics.

4. The cubic constructed from the 3×3 vertices of the Diagonal Triangles will be the Hessian of CU.

CU-Cu1 Hessian-50-BipartiteCubic

See [66], QPG-messages #1923 and #1944 and [17b].

See for more points on the Hessian also [66], QPG-messages #2105-#2107.

Properties

• CU (Reference Cubic) and HE (Hessian) intersect in the 9 inflection points of the reference cubic. Only three of these inflection points are real collinear points, the other 6 points are always imaginary points.
• The Hessian itself is also a cubic and has therefore its own inflection points but they are the same 9 points.
• Any cubic is the Hessian of 3 other cubics named Prehessians (not necessary real). See [66] QPG#1945. See [BG, Isocubics].
• Any cubic shares its Hessian with 2 other cubics. See [66] QPG#1945. See [BG, Isocubics].
• The initial cubic, it’s 3 Prehessians as well as it’s Hessian and the 2 other cubics sharing this Hessian form the Syzygetic pencil of cubics through the 9 flexes, 3 of them being real and aligned and the 6 others imaginary. See QPG#1945. See [17b] and also [75].

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