Hessian (algebraic curve)

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of an algebraic curve of degree

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 is a curve of degree and class . If is the equation of a curve of degree in homogeneous coordinates and if , 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)
How to Cite This Entry:
Hessian (algebraic curve). A.B. Ivanov (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098