Reiss relation explained
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
where the sum is over the points of intersection of
C and
L, and
fx,
fxy and so on stand for partial derivatives of
f .This can also be written as
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
- Akivis, M. A.; Goldberg, V. V.: Projective differential geometry of submanifolds. North-Holland Mathematical Library, 49. North-Holland Publishing Co., Amsterdam, 1993 (chapter 8).
