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 $$a$$.