This demo displays a sphere and a curve on the sphere.

The length of the curve is given in a readout in the control window.

A hotspot P on the curve determines another point Q halfway around the curve from P. A point M, which is the midpoint of P and Q on the sphere, is displayed. A red great circle is displayed with M as the north pole.  If the length of the curve is less than 2π, the hemisphere will contain the curve.

Changing alpha rotates the half of the curve from P to Q around the north pole M, thus demonstrating the proof of the lemma.