What is the formula to calculate the distance (arc length) between 2 points $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)$ on the circumference of a circle of radius $r$ without knowing the angle $\theta$ between them. I found that arc length can be calculated knowing $\theta$. But I know only the $x,y,z$ co-ordinates of 2 points on the circumference of a circle. Please suggest.

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    $\begingroup$ you can use the formula for length of an arc $l=\int_{a}^{b}\sqrt{<v(t),v(t)>}dt $ $\endgroup$ – Nebo Alex Jan 1 '16 at 10:37
  • $\begingroup$ Do you mean arclength distance (measured along the circle), or straight-line distance.If the latter, then the circle is irrelevant and the answer is given by @GBeau below. $\endgroup$ – bubba Jan 1 '16 at 10:37
  • $\begingroup$ I meant arc length $\endgroup$ – Sangam Jan 1 '16 at 10:41
  • $\begingroup$ @Boris, I am not good at integration. Please can you explain how to use the formula. $\endgroup$ – Sangam Jan 1 '16 at 10:43
  • $\begingroup$ Assuming you know the radius, find the straight line distance between the points first then you have a triangle with 3 sides known. Use the cosine rule to find $\theta$ $\endgroup$ – Paul Jan 1 '16 at 10:53

Let $d$ be the (straight-line) distance between the two points. Then the arclength between them is $$ s = 2r\sin^{-1}\left( \frac{d}{2r} \right) $$ Note that this does not assume that the circle is centered at the origin (as some of the others seem to do).

  • $\begingroup$ Probably the more common formulation than mine $\endgroup$ – GPhys Jan 1 '16 at 11:04

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