Reiss relation

Template:Short description In algebraic geometry, the Reiss relation, introduced by Template:Harvs, 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 Template:Harv. 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 Template:Harv.

References

• Template:Citation
• Template:Citation
• Akivis, M. A.; Goldberg, V. V.: Projective differential geometry of submanifolds. North-Holland Mathematical Library, 49. North-Holland Publishing Co., Amsterdam, 1993 (chapter 8).