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Given two points on the arc of a circle, $A$ and $B$, how do I approximate the shortest circle arc between them using $N$ connected line segments where each line segment has length 2? (It would be nice to solve this for other pre-determined lengths as well.)

The center of the circle is known (as well as its radius).

The sum of the length of the N segments will be larger than the $A$ to $B$ chord.

Approximating an arc with a fixed number of line segments. seems relevant as well as the answer there of "Approximation of Curves by Line Segments" by Henry Stone, but I'm not capable enough to adapt that to my problem.

(I'm pretty sure this is a subset of what I need to solve in order to solve Approximating a circle with an irregular polygon, as this would allow for the broken line segments that need to be circle/grid-aligned to intersect with the circle on both sides and the remaining line segments to follow the circle arc as best as they can per the answer to this question.)

Anne
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    Do you know the center? – Thomas Andrews Jun 29 '22 at 16:14
  • Yes, that's known. Edited that in. Thanks for pointing that out! – Anne Jun 29 '22 at 16:21
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    What have you tried? What do you mean by "broken?" – Thomas Andrews Jun 29 '22 at 16:23
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    If the word "broken" means what I'd guess, I get $$2Nr\sin\frac{\arcsin\left(\frac{AB}{2r}\right)}{N}$$ where $r$ is the radius of the circle. – Thomas Andrews Jun 29 '22 at 16:26
  • I used the word broken as that is what the paper from Henry Stone used. Perhaps the proper term is polygonal chain? If so I can edit the question accordingly.

    As to what I've tried, this is where I got stuck. I can solve this for certain values essentially by plotting circles (as seen in https://www.eurobricks.com/forum/index.php?/forums/topic/163513-ldd-erroneous-gap-using-hinges-to-create-360-circle/ for instance), but not generally.

     – Anne Jun 29 '22 at 16:41
  • Hey Thomas, after some searching I think I understand your answer and realized that something was unclear in my question. Both the number and length of the line segments is pre-determined and not flexible. I have attempted to clarify the question further. – Anne Jun 29 '22 at 20:24
  • Must A be the begin of the first segment and B the end of the last segment? What is goal function for approximation? – Ivan Kaznacheyeu Jul 04 '22 at 12:57
  • Hey Ivan, thank you for asking. Yeah, the idea is that A is the begin and B the end. I don't know what the goal function should be, exactly. One idea I had (but I don't know how to execute) is that you try to maximize the area of the polygon when also connect A and B with a straight line. – Anne Jul 07 '22 at 14:28

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