Plücker's Equations
Relationships between the number of singularities of plane algebraic curves. Given a plane curve,
m = n(n-1)-2delta-3kappa
(1)
n = m(m-1)-2tau-3iota
(2)
iota = 3n(n-2)-6delta-8kappa
(3)
kappa = 3m(m-2)-6tau-8iota,
(4)
where m is the class, n the curve order, delta the number of ordinary double points, kappa the number of cusps, iota the number of inflection points (inflection points), and tau the number of bitangents. Only three of these equations are linearly independent.
See also
Algebraic Curve, Bioche's Theorem, Bitangent, Curve Genus, Curve Order, Cusp, Inflection Point, Klein's Equation, Ordinary Double PointExplore with Wolfram|Alpha
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References
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.Graustein, W. C. Introduction to Higher Geometry. New York: Macmillan, pp. 220-222, 1930.Referenced on Wolfram|Alpha
Plücker's EquationsCite this as:
Weisstein, Eric W. "Plücker's Equations." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PlueckersEquations.html