On Sun, Oct 9, 2011 at 4:10 PM, Bill Gosper <billgosper@gmail.com> wrote:
Corey notes that ellipsoid girths can't be {large, tiny, tiny} and challenges us to find the "ellipsoid girth inequality" analogous to the triangle inequality.
What are the dimensions (semiaxes a,b, and c) of say, the ellipsoidoidal rock by the road near the Foothill College fire station, given the girths gab, gac, gbc; a>b>c? (Assuming you can get the damned thing off your tape measure once you've rolled it on.)
In terms of a, b, c, the girths are {gab -> 4 a EllipticE[1 - b^2/a^2], gac -> 4 a EllipticE[1 - c^2/a^2], gbc -> 4 b EllipticE[1 - c^2/b^2]}
This is a routine problem for FindRoot, but it would be nice to see how a, b, c depend on the girths in the low eccentricity case, say, by reverting the series expansions of the girths. Julian, Corey, and I were surprised by the absence of multivariate reversion from Macsyma and Mathematica, and set about building our own. After several hours of cruel hoodwinking by Mma's Series facility, we emerged having only solved the ellipsoid case. Surprisingly(?), it has at least eight (2^3) solutions, which may help to explain the shortage of multivariate reverters. To fourth order,
{a -> (gab/(2 Pi))(1 - eac + eac^2 + ebc + (eac ebc)/2 - (3 ebc^2)/2 + 1/8 (-8 eac^3 - eac^2 ebc - 9 eac ebc^2 + 18 ebc^3) + 1/16 (16 eac^4 + 2 eac^3 ebc - 3 eac^2 ebc^2 + 42 eac ebc^3 - 57 ebc^4)),
a -> (gac/(2 Pi))(1 + ebc + 3/2 (eac ebc - ebc^2) + 3/8 (eac^2 ebc - 7 eac ebc^2 + 6 ebc^3) - 3/16 (7 eac^2 ebc^2 - 26 eac ebc^3 + 19 ebc^4))}
{b -> (gab/(2 Pi))(1 + eac - (3 eac^2)/2 - ebc + (eac ebc)/2 + ebc^2 + 1/8 (18 eac^3 - 9 eac^2 ebc - eac ebc^2 - 8 ebc^3) + 1/16 (-57 eac^4 + 42 eac^3 ebc - 3 eac^2 ebc^2 + 2 eac ebc^3 + 16 ebc^4)),
b -> (gac/(2 Pi))(1 + 2 eac - eac^2/2 - ebc - (eac ebc)/2 + ebc^2 + 1/8 (6 eac^3 - 5 eac^2 ebc + 7 eac ebc^2 - 8 ebc^3) + 1/16 (-21 eac^4 + 24 eac^3 ebc - 5 eac^2 ebc^2 - 14 eac ebc^3 + 16 ebc^4))}
{c -> (gab/(2 Pi))(1 - eac + eac^2 + eac^4 - ebc - (eac ebc)/2 - (eac^3 ebc)/ 8 + ebc^2 - (5 eac^2 ebc^2)/16 - (eac ebc^3)/8 + ebc^4 + 1/8 (-8 eac^3 + eac^2 ebc + eac ebc^2 - 8 ebc^3)),
c -> (gac/(2 Pi))(1 - ebc - (3 eac ebc)/2 - (3 eac^2 ebc)/8 + ebc^2 + (9 eac ebc^2)/8 - (3 eac^2 ebc^2)/16 - ebc^3 - ( 9 eac ebc^3)/8 + ebc^4)}
where {eac -> gab/gac - 1, ebc -> gab/gbc - 1}, which are small for low eccentricity.
Actually, there are at least 3^3 solutions. The above are perturbations of either gab/2π or gac/2π, but we can multiply by (1 + ebc) gbc/gab ==1 to get perturbations of gbc/2π. E.g., for the semimajor axis: a -> (1/(2 \[Pi]))(1 - eac + 2 ebc + 1/2 (2 eac^2 - eac ebc - ebc^2) + 1/8 (-8 eac^3 + 7 eac^2 ebc - 5 eac ebc^2 + 6 ebc^3) + 1/16 (16 eac^4 - 14 eac^3 ebc - 5 eac^2 ebc^2 + 24 eac ebc^3 - 21 ebc^4)) gbc Recovering a from the girths for a,b,c ={4,3,2} (more eccentric than 5,4,3) gives a -> 3.98064, for some reason better than {a -> 3.91836, a -> 3.94558} for the gab and gac perturbations. I expected the gab one to be best, not worst. Well, actually I didn't expect multiple solutions. And there are many more if we introduce an expansion variable = gac/gbc-1 ! Presumably, these are just various forms of a single solution, artifacts of the relations eac -> gab/gac - 1, ebc -> gab/gbc - 1, but their convergence can differ substantially. In the (impossible) case of girths 2,1, and 1, the fourth order solution pairs go {{a -> 1/\[Pi], a -> 1/\[Pi]}, {b -> 1/\[Pi], b -> 1/\[Pi]}, {c -> 3/(16 \[Pi]), c -> -(17/(32 \[Pi]))}} ! --rwg
E.g., for {a -> 5., b -> 4., c -> 3.}, the girth formulæ give
{gab -> 28.3617, gac -> 25.527, gbc -> 22.1035}.
Plugging these back into the reversions: {{a -> 4.97828, a -> 4.98561}, {b -> 4.00689, b -> 4.00371}, {c -> 3.0062, c -> 3.00353}}. --rwg