Tracking and the Euler rotation theorem

Suppose you are in an air traffic control tower observing a plane moving in a straight line and you want to rotate your frame of reference to align with the plane. In the new frame the plane is moving along a coordinate axis with no component of motion in the other directions.

You could do this in two steps. First, imagine the line formed by projecting the plane’s flight path onto the ground, as if the sun were directly overhead and you were watching the shadow of the plane move. The angle that this line makes with due north is the plane’s heading. You turn your head so that you’re looking horizontally along the projection of the plane’s path. Next, you look up so that you’re looking at the plane. The angle you lift your head is the elevation angle.

You’ve transformed your frame of reference by composing two rotations. Turning your head is a rotation about the vertical z-axis. Lifting your head is a rotation about the y-axis, if you label the plane’s heading the x-axis.

By Euler’s rotation theorem, the composition of two rotations can be expressed as a single rotation about an axis known as the Euler axis, often denoted e. How could you find the Euler axis and the angle of rotation about this axis that describes your change of coordinates?

You can find this calculation worked out in [1]. If the heading angle is α and the elevation angle is β then the Euler axis is

$\mathbf{e} = \left( -\sin\frac{\alpha}{2} \sin \frac{\beta}{2},\,\, \cos \frac{\alpha}{2} \sin \frac{\beta}{2},\,\, \sin\frac{\alpha}{2} \cos\frac{\beta}{2} \right)$

and the angle ϕ of rotation about e is given by

$\phi = \arccos\left(\frac{\cos\alpha\cos\beta + \cos\alpha + \cos\beta - 1}{2}\right)$

[1] Jack B. Kuipers. Quaternions and Rotation Sequences. Princeton University Press. If I remember correctly, earlier editions of this book had a fair number of errors, but I believe these have been corrected in the paperback edition from 2002.

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