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| 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.
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| ==Statement==
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| 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
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| :<math>\sum\frac{f_{xx}f_y^2-2f_{xy}f_xf_y+f_{yy}f_x^2}{f_y^3}=0</math>
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| where the sum is over the points of intersection of ''C'' and ''L'', and ''f''<sub>''x''</sub>, ''f''<sub>''xy''</sub> and so on stand for partial derivatives of ''f'' {{harv|Griffiths|Harris|1994|loc=p. 675}}.
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| This can also be written as
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| :<math>\sum\frac{\kappa}{\sin(\theta)^3}=0</math>
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| 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}}.
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| ==References==
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| *{{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 | id={{MathSciNet | id = 1288523}} | year=1994}}
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| *{{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 | id={{MR|0278222}} | year=1971 | volume=13}}
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| *Akivis, M. A.; Goldberg, V. V.: Projective differential geometry of submanifolds. North-Holland Mathematical Library, 49. North-Holland Publishing Co., Amsterdam, 1993 (chapter 8).
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| [[Category:Algebraic geometry]]
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