This is my first question, so I apologize in advance if I am not clear. I need to calculate the great-circle distance between two points on a sphere (of known radius). Only the latitudes and longitudes of these points are known. I am aware of the law of cosines approach, but am thinking of solving the problem by parametrizing the sphere and using the length of a curve formula. That is, $$\gamma(t) = \sigma(u(t), v(t)) = \langle r\cos{u}\cos{v}, r\cos{u}\sin{v}, r\sin{u} \rangle$$ and so since arclength $s$ is given by $$ s = \int_{t_1}^{t_2} |\gamma ' (t)| dt,$$ finding the time derivative and so on I found $$s = r \int_{t_1}^{t_2} \left[ (\frac{du}{dt})^2 + \cos{u}(\frac{dv}{dt})^2 \right]^{1/2} dt.$$ I tried simplifying, i.e. $$s = r \int_{t_1}^{t_2} \left[ (du)^2 + \cos{u}(dv)^2 \right]^{1/2}, $$ $$ s = r \left[ \int_{u(t_1)}^{u(t_2)}du + \int_{v(t_1)}^{v(t_2)}dv\sqrt{\cos{u}} \right],$$ leaving $$ s = r \left[ (u_2-u_1) + \sqrt{\cos{u}}(v_2-v_1) \right]. $$ Since the values $u_{1/2}$ and $v_{1/2}$ by definition represent the latitude and longitude, and are therefore known, the majority of the expression is taken care of. However, this leaves an unwieldy $\cos{u}$ and so I cannot compute $s$. Is my working out incorrect? Is there an approach that better fits the problem (aside from spherical cosines)? Is there some step I am missing that cleanly finishes the problem?
Thanks :)