Similar problems arise in computer vision and robotics applications when there is a need to average multiple estimates of an angle. Clearly, it is incorrect to just average the angles together, i.e.
because of the problems with wraparound at 2 Pi. However, you could get a reasonable answer by treating each angle as a unit vector, adding the vectors together, then taking the angle associated with the vector resultant using arctangent, i.e.
Trying to mathematically formalize the above "trick", and to extend it to multiple dimensions leads one directly into the study of directional statistics, which concerns questions about how to generalize ordinary statistical analyses to handle point observations on circles, spheres and hyperspheres.
K.V.Mardia, Statistics of Directional Data, Academic Press, New York, 1972.
G.S.Watson, Statistics on Spheres, John Wiley and Sons, New York, 1983.
P.E.Jupp and K.V.Mardia, "A Unified View of the Theory of Directional Statistics : 1975--1988," International Statistical Review, Vol.57, pp.261-294.
R.V.Lenth, "Robust Measures of Location for Directional Data," Technometrics, Vol.23, 1981, pp.77-81.