Difference between revisions of "Hessian (algebraic curve)"
From Encyclopedia of Mathematics
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| − | ''of an [[Algebraic curve|algebraic curve]] of degree | + | {{TEX|done}} |
| + | ''of an [[Algebraic curve|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 | + | 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 |
| − | + | $$\begin{vmatrix}f_{11}&f_{12}&f_{13}\\f_{21}&f_{22}&f_{23}\\f_{31}&f_{32}&f_{33}\end{vmatrix}=0$$ | |
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). | 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). | ||
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====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J.L. Coolidge, "A treatise on algebraic plane curves" , Dover, reprint (1959)</TD></TR></table> | + | <table> |
| + | <TR><TD valign="top">[a1]</TD> <TD valign="top"> J.L. Coolidge, "A treatise on algebraic plane curves" , Dover, reprint (1959)</TD></TR> | ||
| + | </table> | ||
Latest revision as of 05:42, 9 April 2023
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
$$\begin{vmatrix}f_{11}&f_{12}&f_{13}\\f_{21}&f_{22}&f_{23}\\f_{31}&f_{32}&f_{33}\end{vmatrix}=0$$
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).
References
| [a1] | J.L. Coolidge, "A treatise on algebraic plane curves" , Dover, reprint (1959) |
How to Cite This Entry:
Hessian (algebraic curve). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hessian_(algebraic_curve)&oldid=18385
Hessian (algebraic curve). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hessian_(algebraic_curve)&oldid=18385
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article