# What does the covariance of a quaternion *mean*?

If I have a set of Euler angles (representing the orientation of an object) and I find the covariance of those angles then I have some intuition that $\sigma^2$ is in units of $\text{rad}^2$ and I can visualize what a normal distribution of angles with a given mean is.

If I represent that same set of orientations with quaternions, I have no idea what the resulting variance $\sigma^2$ means. How can I visualize it or gain an intuitive sense of how to interpret it?

EDIT: So my actual answer is that it is difficult to visualize quaternions (and you should not feel alone in that), thus the conversion to Euler angles. The thesis is there in case you wanted the covariance laws, and to add to the rationale above.

Here is some information from a thesis (the full thesis PDF link is at the bottom)

You may also try locating this book: Vanicek, P. and E.J. Krakiwsky (1986): Geodesy: The Concepts, North-Holland, Amsterdam.   source: www.ucalgary.ca/engo_webdocs/GL/96.20096.JSchleppe.pdf

• This answer has more than earned the bounty, because on pp 73-76 it derives the discrete time process error covariance $Q_k$ for a Kalman Filter whose state contains a quaternion orientation and an angular velocity $\omega$. – Ben Jackson Oct 20 '14 at 4:49
• Glad I could be of use @Ben. – I Heart Beats Oct 20 '14 at 13:07