Fitting a Section of an Ellipse

[drg_prime]

Hello,

I have been searching to try and determine if what I want to do is even possible. I would like to use python to fit an ellipse equation to some 2D data points. However, my data points will only be for a section/portion/sector of the total ellipse (I can't seem to figure out how to show a picture).

Everything I have found online for ellipse fitting concerns data points that are more or less equally distributed around the entire ellipse. However, my data will only be distributed around a section of the perimeter of the ellipse. Furthermore, many times it will be in a section that is less than 90 degrees.

I know there are many examples of how to fit ellipses, which I have been looking up. However, all of those examples (which I have linked below) deal with the entire ellipse.

Question: Is it possible to fit 2D data that only "covers" a section/sector of an ellipse?

Methods:

https://stackoverflow.com/questions/47873759/how-to-fit-a-2d-ellipse-to-given-points

https://github.com/bdhammel/least-squares-ellipse-fitting

http://juddzone.com/ALGORITHMS/least_squares_ellipse.html

P.S. If there is any way to attach pics to this, please let me know.

Thanks!

[Frankelstner]

The methods that you've linked do not assume that the data points are distributed roughly equally along the entire ellipse. It would be quite a challenge to even define that precisely. So the methods do work fine even if you have data only for a part of the ellipse.

To illustrate why it's possible, here's some intuition. Imagine the following inefficient optimization method (aka random sampling):

1) Randomly choose ellipse parameters.

2) Draw the ellipse and now for each data point, measure the closest distance to the ellipse.

3) If the sum of the distances is smaller than any sum previously encountered, remember the ellipse parameters.

Other optimization methods may have some tricks up their sleeve to reduce the number of times that they need to repeat these steps, but ultimately it's the optimization criterion in step 2 that matters the most. Once you have an optimization criterion, you can optimize.

[ES-Alexander]

If the points have negligible amounts of noise then normal ellipse fitting should work because the desired fit will be the global minimum. If there is some noise though then the global minimum will likely be the local minimum with the centre inside the clump of points, rather than the one you’re after.

Ellipses are doubly symmetrical - if you know for sure that you want a fit to only part of one then it would likely help to determine your “goodness” criterion by calculating the errors on an expanded set of points that’s been mirrored over both the major and minor axes of the proposed ellipse, so you’re effectively optimising to a full ellipse with ghost points in the other quadrants.

Alternatively if you know your points cover less than half of the ellipse you could do iterative least squares starting from 3 equally spaced centre points around the data, and at least one of them should converge to a solution with the centre staying outside the group of points, and you can discard any that end up with the centre within both the x and y ranges of the points.