Worlds Out of Nothing pp 155-164 | Cite as

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

Chapter

## Abstract

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* + *y*^{3} + *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 *x*^{3}*y* + *y*^{3}*z* + *z*^{3}*x* = 0 in homogeneous form.

## Keywords

Double Point Analytic Geometry External Point Quartic Curve Dual Curve
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Springer-Verlag London Limited 2007