How to check if other points are on the geodesic between 2 points on a sphere?

Question

Given any 2 points $p_1, p_2 \neq p_1$ on a sphere, how can we check if $p_3$ is on the geodesic from $p_1$ to $p_2$?

I think of checking the $||\text{arc } p_1 p_3|| + ||\text{arc } p_3 p_2|| = ||\text{arc } p_1 p_2||$. Is this condition sufficient? Any other suggestions?

Answer 1

1. Evaluate the following double integral $$ \int_{0}^{\frac{\pi}{2}} \int_{0}^{\pi}\left(x^{2} \cos y-\sin x\right) d x d y $$