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From: robison@uiucdcsb.CS.UIUC.EDU
Newsgroups: net.math
Subject: Re: Question came up at a party
Message-ID: <9700044@uiucdcsb>
Date: Mon, 3-Feb-86 22:48:00 EST
Article-I.D.: uiucdcsb.9700044
Posted: Mon Feb  3 22:48:00 1986
Date-Received: Thu, 6-Feb-86 21:28:04 EST
References: <532@well.UUCP>
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Nf-ID: #R:well.UUCP:532:uiucdcsb:9700044:000:1303
Nf-From: uiucdcsb.CS.UIUC.EDU!robison    Feb  3 21:48:00 1986


>
> Does anyone know (or know where to find) expressions for the coefficients of
> the (oft quoted) general second order equation in terms of the radii
> ($r_{major},r_{minor}$), center ($x_c,y_c$) and rotation ($theta$) of a
> circle or ellipse...

One simple approach is use matrix transformations.  The equation

          2          2
	Ax + Bxy + Cy + Dx + Ey + F = 0

can be expressed with the matrix equation:

         t        
(1)     V M V = 0
 
                                    t
where V is the column vector [x,y,1]  and M is a 3x3 matrix.  (Note that M
has 9 coefficients, we could define M as being symmetric to remove the extra
3 degrees of freedom without losing generality).

Any translation, scaling, or rotation of (x,y) to (x',y') can be expressed
with a matrix multiplication (as shown in most computer graphics texts).

(2)     U = T V

                                    t
where U is the column vector [x',y',1]

We can solve (2) for V and plug back into (1), i.e.

             -1
        V = T  U

          -1  t   -1
        (T  U) M T  U = 0

which simplifies to the equation:

         t   -1t   -1
        U  (T   M T  ) U = 0

from which you can dig out the coefficients for the conic section equation.

- Arch D. Robison
  University of Illinois at Urbana-Champaign