Parameterization for Clairaut's cubic cone? Higher order "conic sections"?

I'm trying to find parametric equations for Clairaut's cubic cone,

z y^2 = a x^3 + b x^2 z + c x z^2 + d z^3 ,

which, as I understand it, was used in his proof of Newton's assertion that all third degree equations could be reduced to four standard forms by projective transformations (if I've got that right).

In Robert Bix's "Conics and Cubics" (Springer, 1998, beginning of Chapter 3, "Cubics") he talks about how

"The Weierstrass's P-function x = P(u) parametrizes the nonsingular, irreducible complex cubic y^2 = g(x) in the following sense: Equation (9) [i.e., x = P(u) ] and the equation y = P'(u) match up the complex numbers on and inside a parallelogram in the complex plane with the points (x,y) of the complex cubic (10) [i.e., y^2 = 4 x^3 + ct + d ], except that any two complex numbers u in corresponding positions on opposite sides of the parallelogram map to the same point (x,y). The function P(u) can be written in the form

P(u) = (1/u^2) + a_2 u^2 + a_4 u^4 + . . .

for complex numbers a_2, a_4, . . . ."

It goes on to say:

"We obtain the Weierstrass P-function in x = P(u) by inverting the elliptic integral u = Integral of g(x)^-(1/2) dx and considering x as a function of u."

"The idea of parameterizing the complex cubic in

[ y^2 = g(x) where g(t) = 4 t^3 + ct + d ]

by inverting the elliptic integral

u = Integral of g(x)^-(1/2) dx

arose by drawing analogies with the following familiar facts: the unit circle y^2 = 1 - x^2 is parameterized by setting x = sin(u) and y = sin'(u) = cos(u), where the relationg give by x = sin(u) is the inverse of the relation

u = arcsin(x) = Integral from 0 to x of (1 - t^2) ^(-1/2) dt .

Drawing parallels between the cubic g(t) = 4 t^3 + ct + d and the quadratic 1 - t^2 and between the Weierstras P-function x = P(u) and the sine functopm x = sin(u) creates analogies between

x = P(u) and x = sin(u) ,
y^2 = 4 x^3 + ct + d and y^2 = 1 - x^2 ,
y = P'(u) and y = sin'(u) = cos(u), and

u = Integral of (4 x^3 + ct + d) ^(-1/2) dx and
u = arcsin(x) = Integral from 0 to x of (1 - t^2) ^(-1/2) dt."

This all sounds very interesting and promising, but I can't find any actual parameterization formulas anywhere in the book, or anywhere else, for that matter.

Is it just a matter of finding values of a standard elliptic integral, with a few adjustments, possibly?

Any help, clues, tips where to look, etc. would by hugely appreciated. Will by putting together some nice graphs with Graphing Calculator which I will post on my website and post links to. The ultimate aim is to try to find parallels to the Dandelin sphere construction involving 2nd-degree conic sections. What, if any, are the analogs to the foci and directrices of ellipses, parabolas, and hyperbolas?

Chris Young

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