A Third Degree Equation

The graph of the general third degree equation in two variables has an interesting and complex behaviour that had not been completely studied until very recently. This configuration of nippe Descartes helps to study the general behaviour of the third degree equation with no linear or second degree terms:

a*x^3+3*b*x^2*y+3*c*x*y^2+d*y^2=e

This example shows that Descartes can draw the graphs of equations, even if they are not of the form  y=f(x) or x=g(y). Descartes can draw the graph of any equation f(x,y)=0 provided f is decent enough. To do this Descartes uses an adaptation of Newton's method for finding the zeroes of a function of one variable.

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	Ministerio de Educación, Cultura y Deporte. Aņo 2000