On computing the average orientation of vectors and lines

Computing the "average" orientation of lines and rotations is non-trivial due to the need to account for the periodicity of rotation. For example, the average of +10 and +350 degrees is 0 (or 360) degrees, not 180. This problem arises in many disciplines, particularly those in which empirically collected data is processed or filtered. In this paper, we review a common but sub-optimal method for computing average orientation and provide a new geometric interpretation. We also propose a new method which provides additional insights into the geometry of the problem. Our new method also produces significantly more accurate results in the regime of operation usually encountered in robotics applications. We characterize this regime and provide guidance regarding which method to use, and when.