Polar hypersurface
In algebraic geometry, given a projective algebraic hypersurface C {\displaystyle C} described by the homogeneous equation f ( x 0 , x 1 , x 2 , … ) = 0 {\displaystyle f(x_{0},x_{1},x_{2},\dots )=0} and a point a = ( a 0 : a 1 : a 2 : ⋯ ) {\displaystyle a=(a_{0}:a_{1}:a_{2}:\cdots )} its polar hypersurface P a ( C ) {\displaystyle P_{a}(C)} is the hypersurface a 0 f 0 + a 1 f 1 + a 2 f 2 + ⋯ = 0 , {\displaystyle a_{0}f_{0}+a_{1}f_{1}+a_{2}f_{2}+\cdots =0,\,} where f i {\displaystyle f_{i}} are the partial derivatives of f {\displaystyle f} . The intersection of C {\displaystyle C} and P a ( C ) {\displaystyle P_{a}(C)} is the set of points p {\displaystyle p} such that the tangent at p {\displaystyle p} to C {\displaystyle C} meets a {\displaystyle a} .