Question

The great circle / small circle / half circle / full circle route between the two points represent the shortest line between the two points​

Answer 1

Answer:

Tilt your head as necessary to consider the first point the North Pole and the second point to lie somewhere on the Prime Meridian. A great circle path between the two will be the one which just plods straightforwardly south along the Prime Meridian.

Why is this shorter than any other? Well, on small scales, the Earth is approximately flat, so we can decompose any small movement into a north-south movement plus an east-west movement, with the former's length given from the latter two by the Pythagorean Theorem.

In particular, the overall movement is at least as long as the amount it moves southwards, and strictly longer just in case the overall movement is not due south. (In algebraic terms, S2+W2−−−−−−−√≥S , with the inequality being strict unless W=0 and S≥0 )

Since every small movement is at least as long as the amount it moves south, and strictly longer if it is not itself purely southward, the same is true of every path whatsoever (a path being just a bunch of small movements strung together), and so, between our two points, a due south path is shorter than any other.

Answer 2

Explanation:

The great-circle distance, orthodromic distance, or spherical distance is the shortest distance between two points on the surface of a sphere, measured along the surface of the sphere (as opposed to a straight line through the sphere's interior). The distance between two points in Euclidean space is the length of a straight line between them, but on the sphere there are no straight lines. In spaces with curvature, straight lines are replaced by geodesics. Geodesics on the sphere are circles on the sphere whose centers coincide with the center of the sphere, and are called great circles.