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:
- P and Q are the roots of the Hessian of the binary cubic form with roots A, B, C.
- P and Q are the two points fixed by the unique projective transformation taking A to B, B to C, and C to A.
- P and Q are the two points that when added to A, B, C form an equianharmonic set (a set of 4 points with cross-ratio a cube root of 1).
- P and Q are the images of 0 and ∞ under the projective transformation taking the three cube roots of 1 to A, B, C.
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 as 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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- {{Citation | last1=Inoue | first1=Naoki | last2=Kato | first2=Fumiharu | title=On the geometry of Wiman's sextic | url=http://projecteuclid.org/euclid.kjm/1250281655 |mr=2226628 | year=2005 | journal=Journal of Mathematics of Kyoto University | issn=0023-608X | volume=45 | issue=4 | pages=743–757| doi=10.1215/kjm/1250281655 | doi-access=free }}