Polar hypersurface

In algebraic geometry, given a projective algebraic hypersurface [math]\displaystyle{ C }[/math] described by the homogeneous equation

[math]\displaystyle{ f(x_0,x_1,x_2,\dots) = 0 }[/math]

and a point

[math]\displaystyle{ a = (a_0:a_1:a_2: \cdots) }[/math]

its polar hypersurface [math]\displaystyle{ P_a(C) }[/math] is the hypersurface

[math]\displaystyle{ a_0 f_0 + a_1 f_1 + a_2 f_2+\cdots = 0, \, }[/math]

where [math]\displaystyle{ f_i }[/math] are the partial derivatives of [math]\displaystyle{ f }[/math].

The intersection of [math]\displaystyle{ C }[/math] and [math]\displaystyle{ P_a(C) }[/math] is the set of points [math]\displaystyle{ p }[/math] such that the tangent at [math]\displaystyle{ p }[/math] to [math]\displaystyle{ C }[/math] meets [math]\displaystyle{ a }[/math].

References

• Dolgachev, Igor V. (2012-08-16). Classical Algebraic Geometry: A Modern View (1 ed.). Cambridge University Press. doi:10.1017/cbo9781139084437. ISBN 978-1-107-01765-8. https://dept.math.lsa.umich.edu/~idolga/CAG.21.pdf.