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?

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

>   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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