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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.

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*.

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*.

- Edge, W. L. (1978), "Bring's curve",
*Journal of the London Mathematical Society*,**18**(3): 539–545, doi:10.1112/jlms/s2-18.3.539, ISSN 0024-6107, MR 0518240 - Inoue, Naoki; Kato, Fumiharu (2005), "On the geometry of Wiman's sextic",
*Journal of Mathematics of Kyoto University*,**45**(4): 743–757, doi:10.1215/kjm/1250281655, ISSN 0023-608X, MR 2226628