Arclength between two points on a circle not knowing theta

Question

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.

Answer 1

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 other answers seem to do).

Of course, the problem only makes sense if $d \le 2r$, for otherwise there can be no circle passing through the two given points.

As mentioned in a comment, on any given circle passing through the two points, there is a shorter arc and a longer arc. The formula above gives the arclength $s$ of the shorter arc. The arclength of the longer one is simply $2\pi r - s$.