## [Solved] Find radius of curvature of arc

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jmac698
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### [Solved] Find radius of curvature of arc

I have an image with multiple coecentric arcs, of the same curvature. The question is to find the origin of the the circles which produce them.

Imagine drawing centered coeccentric circles at origin x,y, of random radius. Then take some random square area of that, so it contains some arcs. Then find the center of the circles which would make those arcs, relative to the area.

I looked at hough transform, but that finds the location of (some defined object). Location is easy, they are all perfect arcs and I could just pixel search and find connected pixels.

Any ideas?
Last edited by jmac698 on 2016-09-28T17:37:50-07:00, edited 1 time in total.

jmac698
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### Re: Find radius of curvature of arc

ps I could isolate just one short curved arc, finding the radius of curvature of that would help, but I was hoping to average over a few arcs for better accuracy. The ensemble has to be a best fit.

So if you want the simplest example and explanation of mh problem, just picture a short arc, and estimate the curvature. I can even manually specify the end points if needed.

Another idea might be a polar to rectangule coordinate transform, and do some kind of binary search until the result is only perfectly straight lines, I still need some measure of curvature though, either bent convex or concave or straight, to guide the binary search in curve space.

jmac698
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### Re: Find radius of curvature of arc

It seems to me I only need the start and end points of two arcs to be able to solve for radius. I'll have to spend some time working out the formula. (to within a flip or sign).

snibgo
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### Re: Find radius of curvature of arc

If you know three points that lie on a circle, you can easily find the centre of the circle. The geometric solution: join each pair of points with a straight line. Draw the three lines that are the perpendicular bisectors of those. The three bisectors will meet at a point. That point is the centre of the circle.

From the geometric solution, an algebraic solution can be found.

I don't know what "centered coeccentric circles" are. A picture might help.
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jmac698
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### Re: Find radius of curvature of arc

Thanks, the key phrase here is "3 points which define a circle". You can find two types of answers, one gives radius as a formula, another uses matrices. Code for such solutions can be found in graphics gems.