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MathGroup Archive 2006

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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


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