Hessian pair

In mathematics, a Hessian pair or Hessian duad, named for Otto Hesse, is a pair of points of the projective line canonically associated with a set of 3 points of the projective line. More generally, one can define the Hessian pair of any triple of elements from a set that can be identified with a projective line, such as a rational curve, a pencil of divisors, a pencil of lines, and so on.

Definition

If {A, B, C} is a set of 3 distinct points of the projective line, then the Hessian pair is a set {P,Q} of two points that can be defined by any of the following properties:

Examples

Hesse points can be used to solve cubic equations as follows. If A, B, C are three roots of a cubic, then the Hesse points can be found a roots of a quadratic equation. If the Hesse points are then transformed to 0 and ∞ by a fractional linear transformation, the cubic equation is transformed to one of the form x3 = D.

See also

References

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