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Is it possible to find the (best) centre of an imperfect circle with a given radius and coordinates?

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There's a lot of terms that don't have a clear meaning here. What is an imperfect circle? Are we just working with an arbitrary set of points? I'm guessing that "coordinates" means "points on the imperfect circle". What does it mean to "extend the given coordinates by its radius to get its respective centers"?

Assuming the "coordinates" are points on the shape, you probably just want to take the center of mass, whose x coordinate is the average of the points' x values, and likewise for the y (and however many others) direction. but it depends on what kind of data you're working with.
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Perhaps you can treat it as a centre of mass problem
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You can Google "circle fitting", there's a bunch of methods. I *think* it doesn't really matter that your true circle is imperfect, they should all still work.

Here's one method I've used for visual circle detection (Umbach&Jones 2000 "Modified Least Squares method"):

     X₁ = ∑x      X₂ = ∑x²      X₃ = ∑x³
     Y₁ = ∑y      Y₂ = ∑y²      Y₃ = ∑y³
    Z₁₁ = ∑xy    Z₁₂ = ∑xy²    Z₂₁ = ∑x²y

    A = N·X₂  - X₁·X₁
    B = N·Z₁₁ - X₁·Y₁
    C = N·Y₂  - Y₁·Y₁
    D = (N·Z₁₂ - X₁·Y₂ + N·X₃ - X₁·X₂)/2
    E = (N·Z₂₁ - Y₁·X₂ + N·Y₃ - Y₁·Y₂)/2
    
    xₘ = (C·D - B·E) / (A·C - B·B)
    yₘ = (A·E - B·D) / (A·C - B·B)
    r = 1/N · ∑√( (x-xₘ)² + (y-yₘ)² )        (just the average radius)

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