Journal of Mathematical Imaging and Vision

, Volume 51, Issue 3, pp 378–384 | Cite as

A Compact Formula for the Derivative of a 3-D Rotation in Exponential Coordinates

  • Guillermo GallegoEmail author
  • Anthony Yezzi


We present a compact formula for the derivative of a 3-D rotation matrix with respect to its exponential coordinates. A geometric interpretation of the resulting expression is provided, as well as its agreement with other less-compact but better-known formulas. To the best of our knowledge, this simpler formula does not appear anywhere in the literature. We hope by providing this more compact expression to alleviate the common pressure to reluctantly resort to alternative representations in various computational applications simply as a means to avoid the complexity of differential analysis in exponential coordinates.


Rotation Lie group Exponential map Derivative of rotation Cross-product matrix Rodrigues parameters Rotation vector 



The authors thank the anonymous reviewers for their helpful comments and suggestions. G. Gallego is supported by the Marie Curie - COFUND Programme of the EU, as part of the Seventh Framework Programme (FP7). A. Yezzi is supported by National Science Foundation (NSF) Grants: CCF-1347191 and CMMI-1068624. This work has been partially supported by the Spanish Government under projects TEC2010-20412 (Enhanced 3DTV) and TEC2013-48453 (MR-UHDTV), and by FP7’s project 610691 (BRIDGET).


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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Grupo de Tratamiento de ImágenesUniversidad Politécnica de MadridMadridSpain
  2. 2.Robotics and Perception Group, AI LaboratoryUniversity of ZürichZürichSwitzerland
  3. 3.School of Electrical and Computer EngineeringGeorgia Institute of TechnologyAtlantaUSA

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