Plücker, Hesse, Higher Plane Curves, and the Resolution of the Duality Paradox
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.
KeywordsDouble Point Analytic Geometry External Point Quartic Curve Dual Curve
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