Polar hypersurface

In algebraic geometry, given a projective algebraic hypersurface ${\displaystyle C}$ described by the homogeneous equation

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

and a point

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

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

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

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

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