Plücker, Hesse, Higher Plane Curves, and the Resolution of the Duality Paradox

Part of the Springer Undergraduate Mathematics Series book series (SUMS)


An algebraic plane curve is, by definition, the locus in the plane corresponding to a polynomial equation of some degree, n: f (x,y) = 0. For example x 3 y + y3 + x = 0 represents a quartic. The equation may be written in homogeneous coordinates [x,y,z] by setting x = x′/z′, y = y′/z′ and multiplying through by the lowest power of z′ that produces a polynomial equation (if you wish you may then remove the primes) when the curve is considered to lie in the projective plane. The example above becomes x3y + y3z + z3x = 0 in homogeneous form.


Double Point Analytic Geometry External Point Quartic Curve Dual Curve 
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© Springer-Verlag London Limited 2007

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