cartan
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Special Orthogonal Group SO(N)

The group of $N \times N$ rotation matrices; a compact Lie group and Riemannian manifold simultaneously.

ManifoldRetractionConnectionParallelTransportVectorTransportCurvatureGeodesicInterp.

Phase 2; In Progress

SO(N) and SE(N) implementations are landing in the next release. Follow progress in the cartan repository.

Geometry Preview

PropertyValue
Points$N \times N$ rotation matrices
Tangent space at $R$ (skew-symmetric)
MetricBi-invariant:
Dimension$N(N-1)/2$
Sectional curvature
Injectivity radius

Exponential map (matrix exponential):

Logarithmic map (matrix logarithm, defined for ):

The QR retraction provides a cheaper alternative to the matrix exponential for optimization: where extracts the orthogonal factor from the QR decomposition.

Applications

  • Robotics: SO(3) for 3D orientation, SO(2) for planar rotation
  • Computer vision: essential/fundamental matrix estimation
  • Cryo-EM: protein orientation estimation
  • Active matter: director field dynamics (volterra)