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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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