Sunday, April 05, 2009

Rational parameterization conic sections result

It's worth pulling out into a separate post a wonderful unifying result for 3 of the 4 conic sections that follows from the parameterization of Pell's Equation:

Given x2 - Dy2 = 1, in rationals:

y = 2t/(D - t2)


x = (D + t2)/(D - t2)

showing it more traditionally versus the way that results from my own derivation.

You get hyperbolas with D>0, the circle with D=-1, and ellipses with D<0.

You can see the D=-1 case from a well-known mainstream source at the following link:
See: eqns. 16 & 17

I actually feel very honored to have my own derivation leading to such a beautiful result, which allows categorization of ellipses and hyperbolas by a single number, which then is probably connected to the eccentricity.

So what mathematicians call "Pell's Equation" is in other areas a one-stop equation for generating 3 of the 4 conic sections in rationals just by fiddling with one variable--D.

James Harris
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