The derivation is long and very ugly, involving lots of nasty-looking high-order polynomials that look like they're ready to jump off the page and eat your face.
But perhaps it's worth summarizing the derivations (with a hand-wave of "omitted long algebraic derivation here") for posterity's sake. Fortunately I kept the entire derivation in a text file on my computer, so I don't have to repeat the ordeal of actually doing the derivation again. I much rather just post the resulting polynomials as I did in my earlier post, which should be enough information for anyone who's interested to run it through, say, Newton's algorithm or one of the fancy new cubic-interpolating algorithms to generate as many digits as they need for the roots.
(Having said that, though, the polynomials involved in the snub disphenoid's coordinates are pretty tame compared to the insanity that is the equations for the sphenocorona. Boy, that one really gives you a taste of just how nasty a seemingly-innocuous system of 2nd order equations can get. I have managed to get to the point where, in theory, I should be able to eliminate 4 variables and get an equation in a single variable that can be solved via polynomial root extraction. However, I say "in theory" because the substitutions involved would require squaring both sides 3-4 times, which means we're talking about a 8- to 16-fold increase in the degree of the polynomial, and the current equation is already degree 3 or 4. Each squaring multiplies the number of terms like rabbits, and the result is just so completely unwieldy that I'm not sure I have the stomach for it. How one is supposed to derive a supposedly order-4 equation from it (judging by asking Wolfram Alpha for the coordinates of the sphenocorona), eludes me. I suppose this is only tractable with a CAS like Mathematica. And mind you, this is just
one of the 5 unknowns in the system; there's no telling what will happen once we obtain the value of the first unknown and substitute it into the rest of the equations!)
(Nevertheless, the sphenocorona's system of equations is evidently still within the reach of CAS algorithms... I asked Wolfram Alpha for the coordinates of the sphenomegacorona, and it doesn't even try to give me algebraic expressions for it, just decimal expansions. My guess is that Mathematica, or whatever it is Wolfram Alpha uses in the backend, is unable to cope with the most likely utterly insane complexity of an algebraic solution, and just resorts to one of the numerical algorithms to find the solution. I have been doing some research in algorithms for solving systems of polynomial equations, and so far we haven't managed to get very far beyond the Buchberger's algorithm for computing the Gröbner basis -- this algorithm is generally exponential in time complexity, and in the case of lexicographical ordering needed for solving individual variables,
doubly-exponential (i.e., O(2^2^k)). Meaning to say, that past relatively small values of k, it is completely infeasible to compute an algebraic solution; you might as well just give up and use numerical algorithms instead. There has been various improvements to Buchberger's original algorithm, but still, they do not eliminate the double-exponential complexity, they just improve the coefficients. And a double-exponential grows so fast that even for small values of k (like, say, near 8-10 or more) it will quickly become infeasible.)
(And on that note, I used to think that since for CRFs we're really only dealing with polynomials of degree at most 2, so what's the big deal, right? Unfortunately, in the general case, degree 2 is equivalent to an arbitrarily large degree -- because you can always rewrite a system of higher-order polynomials as quadratics, by introducing new variables x and adding equations of the form x=y^2 to the system -- this way you can get as high as you want while the equations themselves only appear to be no higher than degree 2. So the fact that we're only dealing with quadratics in the starting equations doesn't really save us anything... if anything, it's probably only masking the true complexity of the system. To be able to solve these systems, you basically have to deal with polynomials of arbitrarily high degree -- and in multiple variables at that. It's an active area of research, from what I can tell, and thus far I have yet to find any research papers that have major breakthroughs in how to reduce the complexity of these things. It may well be the case that these systems are inherently complex, and are irreducible in the general case. Which, unfortunately, is bad news for us CRF hunters, because that means brute-force computer search will not be easy to implement without some major insight that gives us a feasible handle on the problem. Or, if you're an optimist, this might be construed to be good news, because the inherent complexity of these quadratic systems means that there should be plenty of places for unusual CRF crown jewels to hide in.
We'd never run out of a job, so to speak, since a lot of ingenious insights will be required to find these crown jewels.)