Reiss relation
{{Short description|Algebraic geometry theorem}}
In algebraic geometry, the Reiss relation, introduced by {{harvs|txt|last=Reiss|authorlink=Michel Reiss|year=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
:
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 {{harv|Griffiths|Harris|1994|loc=p. 675}}.
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 {{harv|Griffiths|Harris|1994|loc=p. 677}}.
References
- {{Citation | last1=Griffiths | first1=Phillip | author1-link=Phillip Griffiths | last2=Harris | first2=Joseph | author2-link=Joe Harris (mathematician) | title=Principles of algebraic geometry | publisher=John Wiley & Sons | location=New York | series=Wiley Classics Library | isbn=978-0-471-05059-9 |mr=1288523 | year=1994}}
- {{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).