Hessian (algebraic curve)

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of an algebraic curve of degree $n$

The set of points whose conic polars can be split into two straight lines, as well as the set of double points of the first polars. The Hessian of a non-singular curve of degree $n$ is a curve of degree $3(n-2)$ and class $3(n-2)(3n-7)$. If $f=0$ is the equation of a curve of degree $n$ in homogeneous coordinates and if $f_{ik}=\partial^2f/\partial x_i\partial x_k$, then


is the equation of the Hessian. The Hessian of a non-singular curve of degree 3 in characteristic not equal to three intersects the curve at nine ordinary points of inflection. Named after O. Hesse (1844).



[a1] J.L. Coolidge, "A treatise on algebraic plane curves" , Dover, reprint (1959)
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Hessian (algebraic curve). Encyclopedia of Mathematics. URL:
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article