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

by Guillermo Gallego, Anthony Yezzi

Abstract:

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.

Reference:

Guillermo Gallego, Anthony Yezzi, "A Compact Formula for the Derivative of a 3-D Rotation in Exponential Coordinates", In Journal of Mathematical Imaging and Vision, Springer US, vol. 51, no. 3, pp. 378-384, 2015.

Bibtex Entry:

@ARTICLE{2015-03-MIV2014_Gallego, year={2015}, month=mar, issn={0924-9907}, journal={Journal of Mathematical Imaging and Vision}, volume={51}, number={3}, doi={10.1007/s10851-014-0528-x}, title={A Compact Formula for the Derivative of a 3-D Rotation in Exponential Coordinates}, publisher={Springer US}, keywords={Rotation; Lie group; Exponential map; Derivative of rotation; Cross-product matrix; Rodrigues parameters; Rotation vector}, author={Gallego, Guillermo and Yezzi, Anthony}, pages={378-384}, language={English} url = {http://arxiv.org/pdf/1312.0788v2.pdf}, abstract={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. } }