I've searched everywhere for something to help me with this problem, but I can't find anything. What I want to calculate is the midpoint between two locations (latitude and longitude) on a sphere. The midpoint must lie on the shortest path between them. And for this, I need the equation of the great circle on this sphere that passes through these two points.
What I tried to do is first start with an arbitrary great circle given by the following parametric equation:
${x=0}$
${y=cos\space \theta}$
${z=sin\space \theta}$
Or:
$\left( \begin{array}{} 0 \\ cos\space \theta \\ sin\space \theta \end{array} \right)$
Then if we multiply this vector by three rotation matrices $R_x, R_y,$ and $ R_z$ about angles $\alpha, \beta, $ and $\gamma$, and then solve the equations for the rotation angles using our known values of the radius and the coordinates of the two points, we should get the exact equations of this circle in 3D space, but the trigonometric equations are extremely messy and I'm not sure if my assumption is correct.
There must be a statement of the equation somewhere but I just can't find it. What is it?
Perhaps a simpler method would be to calculate the midpoint in geographic coordinates and then convert them to cartesian, but I'm not sure how geometry works in geographic coordinates.