Reiss relation - Alchetron, The Free Social Encyclopedia

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

∑ f x x f y 2 − 2 f x y f x f y + f y y 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

∑ κ sin ⁡ ( θ ) 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).

References

Reiss relation Wikipedia

(Text) CC BY-SA