Does a sphere have the property that one point can reach another point at any spherical angle?

Question

Does a sphere have the property that one point can reach another point at any spherical angle? Of course, great arcs can go from one point to another. But this is only one of the spherical angles.

A spherical angle is a particular dihedral angle; it is the angle between two intersecting arcs of circles on a sphere. It is measured by the angle between the planes containing the arcs.

Note that the definition of spherical angle here is different from that in the past. Arcs are not necessarily great arcs, and small arcs are also possible.

As shown in the figure, there are two arbitrary points $A$ and $B$ on the sphere, the red arc $AdB$ is a great arc, the green arc $AeB$ and the yellow arc $AcB$ are small arcs. Let the spherical angle of the great arc $AdB$ and the small arc (see my definition of spherical angle) be $\theta$. My question is: on the sphere, the arc from one point to another has a certain angle of theta, right?

Answer 1

Consider three points $A,\ B,\ C$ in $\mathbb{S}^2$

Case 1 : When any distance between two points is $<\pi$, then three shortest paths are unique.

Further, it forms unique geodesic triangle in a closed hemisphere, up to congruence.

When $B,\ C$ are in the boundary of given hemisphere $H$, then consider a point $A\in H$. Clearly, $\angle \ A(=\angle\ BAC) =\pi$ when $A$ is an interior point in $[BC]$

Note that $|B-C|<\pi$ so that we put $A$ s.t. $B$ is an interior point in a path $[AC]$. Hence $\angle\ A=0$. By continuity, any angle between $0$ and $\pi$ is possible for spherical angle.

Case 2 : Now consider $|B-C|=\pi$.

Here $A$ is an interior point in $[BC]$ so that only $\pi$ is possible.

Answer 2

There is a minimal and maximal angle through which a circular arc can be drawn with two points as endpoints: the minimum is the angle between the two points itself, demonstrated when the path is the short side of the great circle, and the maximum is a whole circle minus that angle, demonstrated when the path is the long side of the great circle.

Every angle between these two is feasible: there is a continuous loop of small circular arcs that brings you from the short side of the great circle to the long side.

Conveniently, for another possible interpretation of your question, this continuous loop shows paths that arrive at each point from every direction.