Reiss Relation

In algebraic geometry, the Reiss relation, introduced by , is a condition on the second-order elements of the points of a plane algebraic curve meeting a given line.

Statement

If ''C'' is a complex plane curve given by the zeros of a polynomial ''f''(''x'',''y'') of two variables, and ''L'' is a line meeting ''C'' transversely and not meeting ''C'' at infinity, then
:$\sum\frac=0$
where the sum is over the points of intersection of ''C'' and ''L'', and ''f''_''x'', ''f''_''xy'' and so on stand for partial derivatives of ''f'' .
This can also be written as
:$\sum\frac=0$
where κ is the curvature of the curve ''C'' and θ is the angle its tangent line makes with ''L'', and the sum is again over the points of intersection of ''C'' and ''L'' .

References

*
*{{Citation , last1=Segre , first1=Beniamino , title=Some properties of differentiable varieties and transformations: with special reference to the analytic and algebraic cases , publisher= Springer-Verlag , location=Berlin, New York , series=Ergebnisse der Mathematik und ihrer Grenzgebiete , isbn=978-3-540-05085-8 , mr=0278222 , year=1971 , volume=13
*Akivis, M. A.; Goldberg, V. V.: Projective differential geometry of submanifolds. North-Holland Mathematical Library, 49. North-Holland Publishing Co., Amsterdam, 1993 (chapter 8).

Theorems in algebraic geometry
Algebraic curves