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} : \dots)

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

a_{0} f_{0} + a_{1} f_{1} + a_{2} f_{2} + \dots = 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 $a$ .

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