Rotations as a Lie Group

In the preceding chapters, several representations of three-dimensional rotations have been introduced:

• rotation matrices;
• Euler and Tait–Bryan angles;
• axis-angle parameterization;
• infinitesimal rotations;
• quaternions.

Although these representations may appear very different from one another, they all describe the same geometric object: the set of three-dimensional rotations. Lie group theory provides a unified formalism for describing these transformations and is particularly useful in optimization algorithms, calibration, visual odometry, pose estimation, and bundle adjustment.