So I know that all the geodesics on the sphere ($\mathbb{S}^2\subseteq\mathbb{R}^3$) lie on great circles. However, I have been having a bit of trouble coming up with a time parameterization of these great circle arcs. Specifically, if I have 2 points $(\theta_1,\phi_1)$ and $(\theta_2,\phi_2)$ that lie on $\mathbb{S}^2$ what is the function $\gamma:[0,1]\to\mathbb{S}^2$ such that $\gamma([0,1])$ is the geodesic that connects these two points?
This doesn't seem like it should be terribly difficult; however, I've been getting stuck. The reason that I care about this parameterization is that I am trying to get some visualization working in Mathematica.
Use $$ \mathbf{x} = (\cos\theta \sin \phi, \sin\theta \sin \phi, \cos\phi ) $$
to get vector forms $v_1$, $v_2$ for your two points.
Let $$ w = v_2 - (v_2 \cdot v_1) v_1 \\ u = \frac{1}{\|w\|} w $$
(This is basically just Gram-Schmidt on the basis $\{v_1, v_2 \}$.)
Now let $$ \alpha(t) = \cos(t) v_1 + \sin(t) u $$ As $t$ goes from $0$ to $2\pi$, you'll traverse the great circle containing $v_1$ and $v_2$, starting from $v_1$, passing through $v_2$ before you get to angle $\pi$, and continuing on back to $v_1$.
If you want to stop at $v_2$, just let $t$ run from $0$ to $c$, where $$ c = \cos^{-1} (v_2 \cdot v_1). $$