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From: mjw@otter.hpl.hp.com (Mike Wray)
Newsgroups: comp.graphics
Subject: Re: Ellipse formulae wanted
Message-ID: <1710005@otter.hpl.hp.com>
Date: 28 Nov 90 16:43:25 GMT
References: <1990Nov20.181757.456@vixvax.mgi.com>
Organization: Hewlett-Packard Laboratories, Bristol, UK.
Lines: 74

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1) To find the centre,c, of a circle of radius 1, given distinct points
x and y lying on the circle.

We have
                       2          2                                         
                |x - c|  = |y - c| = 1

The centre c lies on the perpendicular bisector of the line
through x and y.  The equation of this line is

                x + y                                
          w =  --------   - l u                                 
                  2                                              

Where w lies on the line, l is real number, and u is a unit
vector perpendicular to x-y.
[ Given x =(r,s), a perpendicular vector is (-s, r), and we can divide
  by the length of x to obtain a unit vector.]
[ In the following x.y is dot product, i.e.
  (a,b).(u,v) = au+bv. ]

From
                       2                                            
                |x - w|  = 1

and

                    2            2                   2   2
             |x - w|  = |(y-x)/2|  + l(y-x)/2 . u + l |u|
                             2      2
                      = |x-y| /4 + l

so
                                     2
                l  = sqrt( 1  - |x-y| )
                                -----
                                  4

We thus have

                c =   x+y  - l u
                      ---
                       2

With l as above.  We have 2 solutions depending on the sign
of sqrt we choose.

=========================================================================
2) To find the centre, c, of an ellipse of semi-major axis a, semi-minor
axis b, given distinct points x and y on the ellipse.

                            -1
Let   S = ( a  0 ) and T = S
	  ( 0  b )

The x and y lie on the ellipse if and only if u=Tx and v=Ty lie on
the circle radius 1 centre z = Tc.
Solve for z from u and v using problem 1), then c = Sz.

=========================================================================
3) Solving for the centre of an ellipse rotated by angle t is similar
to 2):

let
                                           -1
       S = (cos t   -sin t)(a  0) and T = S
	   (sin t    con t)(0  b)

and solve as before.

=========================================================================

Mike