View source for Hessian (algebraic curve)

''of an [[Algebraic curve|algebraic curve]] of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047150/h0471501.png" />''

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047150/h0471502.png" /> is a curve of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047150/h0471503.png" /> and class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047150/h0471504.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047150/h0471505.png" /> is the equation of a curve of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047150/h0471506.png" /> in homogeneous coordinates and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047150/h0471507.png" />, then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047150/h0471508.png" /></td> </tr></table>

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====
<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>