Polar hypersurface

In algebraic geometry, given a projective algebraic hypersurface $C$ described by the homogeneous equation

$f(x_{0},x_{1},x_{2},\dots )=0$

and a point

$a=(a_{0}:a_{1}:a_{2}:\cdots )$

its polar hypersurface $P_{a}(C)$ is the hypersurface

$a_{0}f_{0}+a_{1}f_{1}+a_{2}f_{2}+\cdots =0,\,$

where $f_{i}$ are the partial derivatives of $f$.

The intersection of $C$ and $P_{a}(C)$ is the set of points $p$ such that the tangent at $p$ to $C$ meets $a$.

References

• Dolgachev, Igor V. (2012-08-16). Classical Algebraic Geometry: A Modern View (PDF) (1 ed.). Cambridge University Press. doi:10.1017/cbo9781139084437. ISBN 978-1-107-01765-8.