The first topic concerns a solution for ellipse fitting [Attachment: FittingEllipse.pdf Michael J. Aramini, Ellipse-Fitting, http://mysite.verizon.net/~vze2vrva/ellipse_fitting.html, May 2007].
1. The expression being minimized (left hand side looks like chi-square):
A. Is it chi-square?
B. If it is chi-square, what is the relationship to the least squares fitting.
C. If not, what is being minimized?
2. Wikipedia [http://en.wikipedia.org/wiki/Ellipse] states that the axes of the origin centered ellipse will lie along the eigenvectors of a linear map to the unit circle, and the eigenvalues are the lengths of the semimajor and semiminor axes.
A. Correctly stated? Further explanation?
B. In general, data sets will not be centered at the origin and unrotated. Is a direct fitting solution for a rotated translated ellipse unnecessary? Why?
Prose explanations sufficient.
Here are some answers to some of your questions.
As for some of the questions their purpose is not clear (2A) and for some it is impossible to address them without much more knowledge of what they relate to (2B), I can only give a limited answer.
Although it is named chi^2, it is not what people usually mean when they say "chi-square".
You can find a description of what is usually meant at web page
What it says there is that the issue is the PROBABILITY DISTRIBUTION of chi^2, whereas you have no need for probabilities. At the top of the web page you cite, it actually says (correctly) that the method is called LEAST SQUARES.
As it is not, no answer is needed.
It is minimized to find the "best fit" for the given points by some curve or surface (or whatever manifold you happen to want) to the given points.
The words "best fit" must be understood in a "relative" sense: one can only determine what is ...
A ellipse-fitting solution is explained. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.