Sunday, August 15, 2010


JSH: Parameterizing conics with Pell's Equation

One result I find interesting for trying to understand the psychology of the modern mathematical community is a parameterization of conics using "Pell's Equation" which is kind of wrong you can say because it's a result over rationals! While "Pell's Equation" is traditionally considered with integers:

Given x^2 - Dy^2 = 1, in rationals:

y = 2t/(D - t^2)


x = (D + t^2)/(D - t^2)

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

Now what's fascinating to me is that the result was known to Fermat, so I've never said it was new to me, and I'm happy to note I re-derived it. It's actually kind of a funny story as I was still playing with what I call surrogate factoring, trying to solve the factoring problem, and came up with this result I terrified myself was close.

So I wouldn't finish deriving the result.

And I ended up in these arguments on the newsgroups about it's importance with me saying dire consequences for the world if it weren't acknowledged and posters arguing with me about it being this trivial thing that did NOT solve the factoring problem, and finally I relented and finished my own derivation and found this nifty thing.

So I have the pleasure of having re-derived this result, which I think is way cool.

What's fascinating to me is that you can use it to replace the Kepler favored style of equations for ellipses and consider orbits with just this D number, which then is directly related to the eccentricity, and I've pondered it for a while and the BEST explanation I think for why mathematicians don't just note this result along with so many others is that it uses "Pell's Equation".

So because to math society, x^2 - Dy^2 = 1 is a Diophantine equation, then you aren't supposed to use it with rationals is my theory, which while wacky, can give you insight into that community!!!

Math society can be very weird.

So it turns out that you can re-do stellar mechanics using "Pell's Equation" and use D instead of the eccentricity. But don't expect to read about it in a standard mathematical text!

It's a buried result. Those who wish to argue that point: cite!!!

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