On the sphere, what is the shortest smooth curve passing through three points? I think it is an arc, but I do n’t know how to prove it to be an arc.
It is not necessarily a geodesic, because these three points are not necessarily on a great circle.
On the sphere, what is the shortest smooth curve passing through three points
geometry
Best Answer
I guess, often the shortest piecewise smooth curve would be a V-shaped curve that goes directly from one of the points to another one, then bends sharply there and goes directly to the last point. May have to permutate the three points to find the shortest "V".
Once you have the shortest piecewise smooth curve, let us say with length $L$, then for any tiny positive $\epsilon$, you can smooth the bend at the middle point to get a smooth curve with length $L + \epsilon$. None of these is "the shortest" one. So the shortest may not exist.
And if you mean the shortest closed curve, again you start with piecewise smooth curves, and get a spherical triangle (each edge shall have a length of at most half the circumference of the sphere). If the perimeter of the triangle is $L$, then smooth curves exist for all lengths $L + \epsilon$ (but not for length $L$).
Then only case where you can find a shortest smooth curve, is if the three points are on the same great circle.