When dealing with quantities modulo $2\pi$, one naturally wants to know if the concept of 'arithmetic mean' can be extended. One way of doing this would be to project each 'angle' to the unit circle to get a complex number, take the arithmetic mean of the set of complex numbers, and then project back onto the unit circle to get an 'average angle'. https://en.wikipedia.org/wiki/Mean_of_circular_quantities

However this value does not always correspond to the intuitive notion of average; for instance the 'circular average' of $\{0,0,\pi /2\} = \{0^{\circ},0^{\circ},90^{\circ}\}$ is $$\arctan\left( \frac{Im(2+i)}{Re(2+i)}\right)=\arctan\left(\frac{1}{2}\right)=0.463 = 26.6^{\circ}$$ Which is not quite the $30^{\circ}$ we were expecting. My question is; is it possible to define a meaningful 'circular average' on the unit circle which corresponds to our natural understanding of the arithmetic mean in simple cases?

I was thinking, in line with this discussion How to average cyclic quantities?, that the best definition would be the one which minimises the mean squared error of the sampling points. However that still leaves one asking; how can we calculated the point which minimises mean squared error on the unit circle without using brute force?

Note: I recognise that due to the symmetry of the circle, there are cases where there will be multiple points which can both serve as a suitable average. E.g. when the points are distributed as an equilateral polygon on the unit circle. I am not interested in these pathological cases for the time being.


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