Locus of points such that facing Mecca is the same as facing east

Solution 1:

I will give a simple solution using coordinates. This does not answer the question of "Is this a well-known problem that has its own name or reference?"

We have a sphere with two distinguished points: the North Pole, $N$, and Mecca, $M$. At any third point $A$, facing east means facing in the direction perpendicular to the geodesic from $A$ to $N$. Facing $M$ means facing in the direction of the geodesic $AM$. Therefore, we seek the locus of points $A$ such that the geodesics $AN$ and $AM$ are perpendicular.

Take the Earth to be a unit sphere centered at the origin $O$, with the coordinates of the points being $N=(0,0,1)$, $M=(\cos\theta,0,\sin\theta)$, and $A=(x,y,z)$. The angle between the great circles $AN$ and $AM$ is equal to the angle between the normals of the planes $AON$ and $AOM$ containing them. This gives the condition $(A\times N)\cdot(A\times M)=0$, which simplifies to $$(x^2+y^2)\sin\theta=xz\cos\theta.$$