# A geodesic on a unit sphere

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.

• You forgot to copy the sentence defining $C$. This problem is pure elementary geometry and a bit of vector algebra. Commented May 30, 2020 at 12:32
• Thank you for the correction. Can you give me a hint?
– John
Commented May 30, 2020 at 13:12
• When tackling problems like that, you need to "grok" the geometry. In this case, A is a point on the XZ-plane, on circular arc, at angle α above the Z=0 plane. B is a point on the YZ-plane, on circular arc, at angle β above the Z=0 plane. The "great circle" between A and B is in the plane ABO, so maximal z is the largest value of Z attained on the arc AB (hint: arc is not in horizontal plane). The point along AB, where the tangent becomes horizontal (tangent vector with z-component == 0) will be the point of highest "rise" along the geodesic. Look for dz/d𝛳=0 along the circle AB. Commented Oct 25, 2023 at 19:04

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.