Mathematics in action: averaging angles

This post begins a (potential) new series: mathematics in action. Let’s see how it goes.

There are many phenomena in science and engineering which one would like to describe using angles. With a pendulum, for example, one might use 0° to describe the state with the bob at the right, 90° for the bob at the centre moving left, 180° for the bob at the left, 270° for the bob at the centre moving right, and 360° = 0° for the bob at the right again:

This gets even more interesting when we have multiple pendulums (or metronomes, which are just pendulums upside down):

A more practical example of this sort of thing might be a bank of electric generators, which each generate an alternating current as they spin through their cycles:

There are even medical applications (see pages 83–84 of this thesis).

In studying this kind of thing, it is sometimes necessary to find the average of a set of angles. For example: 300°, 330°, 360°, and 30°. This is more complicated than it seems at first sight, since 330° = −30°, 300° = −60°, and so forth.

The solution is to find the average sine: in this case a = −0.2165 (the average of −0.8660, −0.5, 0, and 0.5), and the average cosine: in this case b = 0.8080 (the average of 0.5, 0.8660, 1, and 0.8660).

The average angle is the one whose sine and cosine are a and b, which in this case is: arctan(a/b) = −15° = 345° (Some care is needed in handling the case of negative or zero cosines).

We can define the extent to which the angles are the same by r = sqrt(a2 + b2). This will give r = 0 for angles evenly spaced around a circle, and r = 1 for identical angles. For the set of 300°, 330°, 360°, and 30° (which are quite similar), we get r = 0.8365. To see an example of r defined in this way being used, look at this paper of mine. Having identical or near-identical angles is also important for sets of electric generators.

For those mathematics students that have studied complex numbers, this can all be made a lot easier. Each angle θ is represented as the complex number cos(θ) + i sin(θ) = eiθ. We then average the complex numbers, giving a + b i. Conversion to polar coordinates then gives us r and the average angle φ, since a + b i = r eiφ. This helps to explain why complex numbers are a compulsory part of every engineering course – especially when studying harmonic motion, alternating current, or anything that vibrates.


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