Points $A(\cos\alpha,0,\sin\alpha)$ and $B(0,\cos\beta,\sin\beta)$, $(0<\alpha$ and $\beta<\pi/2)$ are on a unit sphere and $l$ is the shortest line (geodesic) between $A$ and $B$ on the sphere. And $C$ is a point on $l$ with maximal $z$ coordinate. If $H$ is the foot of the orthogonal projection of $C$ onto the $xy$-plane and if the angle between the positive $x$-axis and line $OH$ is $\gamma$ then find $\tan(\gamma)$ in terms of $\alpha$ and $\beta$.
Can someone point out to me how to tackle this problem? I tried to refer books on geodesics but most of them are based on high-level mathematics. A hint on how to solve the problem or a reference to a book containing similar problems will really be appreciated.
Best Answer
Some hints:
The shortest curve $l$ connecting $A$ with $B$ is on a great circle $\gamma$ through $A$ and $B$. This great circle lies in a plane through $O$. This plane is spanned by the given vectors $a=(\cos\alpha,0,\sin\alpha)$ and $b=(0,\cos\beta,\sin\beta)$. The plane intersects the $(x,y)$-plane in a line $g$, and the great circle $\gamma$ intersects the $(x,y)$-plane in two points of $g$. The direction of this line indicates where the $z$-highest point $C$ of $\gamma$ lies, and therefore the direction in which the point $H$ lies.
Make a figure!