Journal of Mathematical Imaging and Vision

, Volume 35, Issue 2, pp 155–164 | Cite as

Metrics for 3D Rotations: Comparison and Analysis

  • Du Q. Huynh


3D rotations arise in many computer vision, computer graphics, and robotics problems and evaluation of the distance between two 3D rotations is often an essential task. This paper presents a detailed analysis of six functions for measuring distance between 3D rotations that have been proposed in the literature. Based on the well-developed theory behind 3D rotations, we demonstrate that five of them are bi-invariant metrics on SO(3) but that only four of them are boundedly equivalent to each other. We conclude that it is both spatially and computationally more efficient to use quaternions for 3D rotations. Lastly, by treating the two rotations as a true and an estimated rotation matrix, we illustrate the geometry associated with iso-error measures.


Matrix Lie group Lie algebra Quaternions 3D rotations Distance functions 


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Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.School of Computer Science and Software EngineeringThe University of Western AustraliaNedlandsAustralia

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