Least Squared Best Double Fit

Least Squares Best Fit Methods are well known for a point, a straight
line, a circle, a conic section, in general: for a _simple_ figure.
I've programmed several of these methods myself:

http://hdebruijn.soo.dto.tudelft.nl/www/programs/delphi.htm#BFC

But that's not really cool. I'd rather have a least squares best fit
method with _complicated_ figures, for example: two points, crossing
or parallel lines, a bunch of circles, the image of a character 'A',
my ultimate dream being the least squares best fit with an idealized
treble clef :-)

So far so good about dreams. But one has to start somewhere ..

Let's consider the most simple case in two dimensions: a least squares
best fit with TWO points instead of one. Writing the minimum principle
for this case is not too difficult.

Let the function  M  to be minimized be defined by  M(a,b,p,q) =

Sum_k [ (x_k - a)^2 + (y_k - b)^2 ].[ (x_k - p)^2 + (y_k - q)^2 ]

Here the (x_k,y_k) are the fixed points of a cloud in the plane and we
are satisfied if we can find some fairly unique points (a,b) and (p,q).

A procedure to accomplish this is rather straigtforward. Differentiate
the above Sum_k to (a,b,p,q) and put the four outcomes of this partial
differentiations equal to zero. Then employ a four dimensional Newton-
Rhapson iteration procedure in order to find the zeroes. It turns out
that, given some neatly chosen conditions - one always can arrange this
with a snippet of _purified_ applied mathematics - iterations converge
very quickly to the desired result. The whole thing is published as a
couple of web items at:

http://hdebruijn.soo.dto.tudelft.nl/jaar2008/index.htm

Any fresh ideas to accomplish more of this are quite welcome !

Han de Bruijn