In a few days I will be giving a talk to (smart) high-school students on a topic which includes a brief overview on the notions of curvature and of gedesic lines. As an example, I will discuss flight paths and, more in general, shortest length arcs on the 2-dimensional sphere.
Here is my question: let $p,q$ be distinct (and, for simplicity, non-antipodal) points on the sphere $S^2$. Is there an elementary proof of the fact that the unique shortest path between $p$ and $q$ is the shortest arc on the unique great circle containing $p$ and $q$?
By ''elementary'' I mean a (possibly not completely formal, but somehow convinving) argument that could be understood by a high-school student. For example, did ancient Greeks know that the geodesics of the sphere are great circles, and, if so, how did they “prove'' this fact?
Here is the best explanation that came into my mind so far. Let $\gamma\colon [0,1]\to S^2$ be any path, and let $\gamma(t_0)=a$. The best planar approximation of $\gamma$ around $a$ is given by the orthogonal projection of $\gamma$ on the tangent space $T_a S^2$. This projection is a straight line (that is, a length-miminizing path on $T_a S^2$) only if $\gamma$ is supported in a great circle containing $a$. This sounds quite convincing, and with a bit of differential geometry it is not difficult to turn this argument into a proof. However, I was wondering if I could do better…
Best Answer
Imagine that the sphere is the surface of the earth, and that an earthquake happens at the south pole, point $p$. A seismic wave goes out from the point $p$, so that the wave front is a circle of latitude moving at a constant speed $c$. At some point this circle/wave will meet the point $q$ (let's say that $q$ lies within the continent of Antarctica, so the seismic wave will reach it). For any path from $p$ to $q$, flying along this path at the speed $c$, starting at the moment of the earthquake, will never overtake the wave, and will keep apace of it only if it coincides with the line of longitude going through $q$.
Underlying this physical explanation is Gauss's lemma, from which a rigorous calculus argument can be extracted. But maybe this intuition is enough to convince a high schooler, or at least make them think about why it might be true?