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Question
Hello Sir!

1) I am looking at images with curves. All of the curves resemble closely to a half circle. I selected 5 (x,y) points on the curve (including the two tips of the arc and the general midpoint). From these 5 points, I'd like to calculate the circularity. If these 5 points resemble closely an arc - it would return a value of 1, if it resembles an ellipse or something else, it would return a lower value.

2) I am interested in this topic for a graduate research project. I am not sure if it will prove what I am looking for, but it is worth a shot.

3) I can use anything. I tried to use an ImageJ plugin earlier (http://rsbweb.nih.gov/ij/plugins/circularity.html), but I couldn't get the file to work. All of the numbers 5 (x,y) coordinates are now already in excel, so I think I need to do it within excel if possible.

4) No text. These are the resources I've looked into.
http://www.geometer.org/mathcircles/fourpoints.pdf
http://en.wikipedia.org/wiki/Eccentricity_%28mathematics%29
along with others.

5) Not for a course.

I'm still a little stuck. My instinct was to generate the equation of the arc and then compare the points to the arc and calculate an Rsquared value by comparing the equation with the points. However, I don't know how to generate an equation from these 5 points.

Do you have any insight to share? Thanks!

Katie

Best-Fit circle
This question is hard to answer, in the sense that it is open-ended. There is no particular answer that is correct or standard.

However, I will give you some input.

If you have five points (say p,q,r,s,t) in the plane, you want to know how close they are to a circle. Like you said, you can't just "make" a circle out of them because three points define a circle and you have five.

Here are two approaches I recommend as starting places. Throughout, p,q,r,s,t are fixed points.

1. Fit the best possible circle.

Let d( (x,y) ,  = √(x²+y²), the standard distance in the plane.

Consider O to be a point O=(a,b) the define a function D(O,r) as:

D(O,r) = √[ d(O,p)² + d(O,q)² + d(O,r)² + d(O,s)² + d(O,t)² )

This is basically the distance from the points {p,q,r,s,t} to the circle defined by O and r.

You can use calculus or numerical approximation to find the minimum value of D for all choices of O and r. This is the "least squares" sort of thing applied in one way.

2. Fit all 5-choose-3 = 10 circles through 3 of {p,q,r,s,t}

This is like above, but only consider circles that go through three of your five points. Obviously, if there is a circle through four of them, it gets tested repeatedly, but that would likely be a good fit. In other words, define:

O(x,y,z) = center of circle going through x,y,z
r(x,y,z) = radius of circle going through x,y,z

Test the following:

D(O(a,b,c),r(a,b,c))
D(O(a,b,d),r(a,b,d))
D(O(a,b,e),r(a,b,e))
D(O(a,c,d),r(a,c,d))
...etc

Whichever has the lowest value of D you could consider this the "best fit" -- this will always be worse (or equal) to method 1, since method 1 allows for all 10 of these circles as well as infinitely many more! However, if you want the "candidate" circle to go through some of your points, this is how to do it. This also eliminates calculus/numerical stuff and makes this a short computation.

Either one of these can be easily implemented in computation environments.

Update: I have attached a rudimentary implementation of method #1 using the software Mathematica. Note that using this also shows you that if the points are not arranged in a circle-like formation, the computer may not be able to find a best-fit circle at all. Of course, you said (and so we assume) that the points are roughly circular.

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