Arclength between two points on a circle not knowing theta

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.

• you can use the formula for length of an arc $l=\int_{a}^{b}\sqrt{<v(t),v(t)>}dt$ – Nebo Alex Jan 1 '16 at 10:37
• 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. – bubba Jan 1 '16 at 10:37
• I meant arc length – Sangam Jan 1 '16 at 10:41
• @Boris, I am not good at integration. Please can you explain how to use the formula. – Sangam Jan 1 '16 at 10:43
• 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 $\theta$ – 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).