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Plücker's Equations

Relationships between the number of Singularities of plane algebraic curves. Given a Plane Curve,

$\displaystyle m$ $\textstyle =$ $\displaystyle n(n-1)-2\delta-3\kappa$ (1)
$\displaystyle n$ $\textstyle =$ $\displaystyle m(m-1)-2\tau-3\iota$ (2)
$\displaystyle \iota$ $\textstyle =$ $\displaystyle 3n(n-2)-6\delta-8\kappa$ (3)
$\displaystyle \kappa$ $\textstyle =$ $\displaystyle 3m(m-2)-6\tau-8\iota,$ (4)

where $m$ is the Class, $n$ the Order, $\delta$ the number of Nodes, $\kappa$ the number of Cusps, $\iota$ the number of Stationary Tangents (Inflection Points), and $\tau$ the number of Bitangents. Only three of these equations are Linearly Independent.

See also Algebraic Curve, Bioche's Theorem, Bitangent, Cusp, Genus (Surface), Inflection Point, Klein's Equation, Node (Algebraic Curve), Stationary Tangent


Boyer, C. B. A History of Mathematics. New York: Wiley, pp. 581-582, 1968.

Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, pp. 99-118, 1959.

© 1996-9 Eric W. Weisstein