# Length of arc connecting 2 points in N dimensions

Suppose two have two points that lie on a sphere with known radius in N Dimensions.Is it possible to find the length of the shortest arc connecting them?

What about the length of the arc connecting the two points and going back to the starting point (it would be the full circumference in N=2).Is the problem even possible to solve beyond the usual 3 dimensions?

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Everything is happening in a plane, so you can use your 2 dimensional geometry. If the sphere radius is $r$ and the straight line distance between the points is $d$, the straight line makes a chord of the circle. You can calculate the central angle $\theta=2 \arcsin \frac d{2r}$, then the distance on the sphere is $r \theta$. The circumference is still $2 \pi r$
Suppose you are given the two points as $N$-dimensional real vectors $\mathbf{u, v}$, each of length $r$. Then the angle between them is just given by $\cos(\theta) = \frac{\mathbf{u\cdot v}}{r^2}$, and the length of the arc is $r \theta$.