Reiss relation
# Reiss relation

In algebraic geometry, the Reiss relation, introduced by Reiss (1837), 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

    $\sum {\frac {f_{xx}f_{y}^{2}-2f_{xy}f_{x}f_{y}+f_{yy}f_{x}^{2}}{f_{y}^{3}}}=0$

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 (Griffiths & Harris 1994, p. 675). This can also be written as

    $\sum {\frac {\kappa }{\sin(\theta )^{3}}}=0$

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 (Griffiths & Harris 1994, p. 677).

