In mathematics, a Hessian pair or Hessian duad, named for
Otto Hesse
Ludwig Otto Hesse (22 April 1811 – 4 August 1874) was a German mathematician. Hesse was born in Königsberg, Kingdom of Prussia, Prussia, and died in Munich, Kingdom of Bavaria, Bavaria. He worked mainly on algebraic invariants, and geome ...
, is a pair of points of the
projective line
In mathematics, a projective line is, roughly speaking, the extension of a usual line by a point called a ''point at infinity''. The statement and the proof of many theorems of geometry are simplified by the resultant elimination of special cases; ...
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 is a set of 3 distinct points of the projective line, then the Hessian pair is a set 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
In geometry, the cross-ratio, also called the double ratio and anharmonic ratio, is a number associated with a list of four collinear points, particularly points on a projective line. Given four points ''A'', ''B'', ''C'' and ''D'' on a line, the ...
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 ''x''
3 = ''D''.
See also
*
Glossary of classical algebraic geometry
The terminology of algebraic geometry changed drastically during the twentieth century, with the introduction of the general methods, initiated by David Hilbert and the Italian school of algebraic geometry in the beginning of the century, and later ...
References
*
*{{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
Projective geometry