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Re: puzzling mathworld formulas

  • To: mathgroup at
  • Subject: [mg66228] Re: puzzling mathworld formulas
  • From: Paul Abbott <paul at>
  • Date: Thu, 4 May 2006 05:21:45 -0400 (EDT)
  • Organization: The University of Western Australia
  • References: <e36vt6$nao$>
  • Sender: owner-wri-mathgroup at

In article <e36vt6$nao$1 at>,
 Jones Douglas <bush.dubya at> wrote:

> at, 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])


  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]


  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 

>   but notice that no similar analagous formulas at 
> . 
>   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?


As an aside, you can load the gifs of these equations into Mathematica 




where the inline figure number can be found by clicking on the graphic 
of the equation at


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)    

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