Re: puzzling mathworld formulas

*To*: mathgroup at smc.vnet.net*Subject*: [mg66228] Re: puzzling mathworld formulas*From*: Paul Abbott <paul at physics.uwa.edu.au>*Date*: Thu, 4 May 2006 05:21:45 -0400 (EDT)*Organization*: The University of Western Australia*References*: <e36vt6$nao$1@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

In article <e36vt6$nao$1 at smc.vnet.net>, Jones Douglas <bush.dubya at yahoo.com> wrote: > at http://mathworld.wolfram.com/Ellipse.html, there are a set of equations > called (22), (23). > > which is the source used to verify, and is it possible to check their truth > with mathematica? Did you download the Mathematica Notebook at this link? The derivation that you are after is in there, along with a lot of other material. The general quadratic curve can be written {x, y} . {{a, b}, {b, c}} . {x, y} + 2 {d, f} . {x, y} + g == 0 Linear translation, {x -> x - (b f - c d)/(b^2 - a c), y -> y - (b d - a f)/(b^2 - a c)} eliminates the linear term, 2 {d, f} . {x, y}, and modifies the constant term g to cons = (g b^2 - 2 d f b + c d^2 + a f^2 - a c g)/(b^2 - a c) which you will recognize in (22) and (23). Diagonalizing the symmetric matrix, {{a, b}, {b, c}}, to reduce the ellipse equation to standard form, requires its eigenvalues, ev[1] = (1/2) (a + c - Sqrt[a^2 - 2 c a + 4 b^2 + c^2]) and ev[2] = (1/2) (a + c + Sqrt[a^2 - 2 c a + 4 b^2 + c^2]) The semi-axes lengths (squared) are just a^2 == -cons / ev[1] and b^2 == -cons / ev[2] These formula are rather ugly and I suspect that for any implementation, it would be better to work with Root objects rather than explicit radicals. > but notice that no similar analagous formulas at > http://mathworld.wolfram.com/Hyperbola.html . > > are the ellipses formulas just lucky guesses or what? Follow the above process and see. > does these also hold for ellipses that are called type imaginary ellipses? Definition? As an aside, you can load the gifs of these equations into Mathematica using Show@Import[ "http://mathworld.wolfram.com/images/equations/Ellipse/inline68.gif";] and Show@Import[ "http://mathworld.wolfram.com/images/equations/Ellipse/inline71.gif";] where the inline figure number can be found by clicking on the graphic of the equation at http://mathworld.wolfram.com/Ellipse.html. Cheers, Paul _______________________________________________________________________ Paul Abbott Phone: 61 8 6488 2734 School of Physics, M013 Fax: +61 8 6488 1014 The University of Western Australia (CRICOS Provider No 00126G) AUSTRALIA http://physics.uwa.edu.au/~paul

**Follow-Ups**:**Command completion maxes out CPU***From:*gardyloo <gardyloo@mail.wsu.edu>