Lie groups tagged articles

• Appendix A5 — The Sub-Riemannian Exponential Map of SE(2) 05 May 2026
  The SR exponential map of SE(2): closed-form geodesic ends via Jacobi elliptic functions, conjugate points and the second variation, Maxwell pairs, and the cut locus. Self-contained capstone of the Geometry of Seeing appendices.

• Appendix A3 — Calculus of Variations and the Pontryagin Maximum Principle 03 May 2026
  Self-contained derivation of the Pontryagin Maximum Principle, applied to the sub-Riemannian length functional on $\mathrm{SE}(2)$. Lie–Poisson reduction recovers the costate equations Part 2 uses out of the box.

• Appendix A2 — Distributions, Frobenius, and Contact Geometry 02 May 2026
  Distributions, the Frobenius integrability theorem, the Chow–Rashevskii reachability theorem, and the definition of a contact structure on a 3-manifold. Worked out for $\mathrm{SE}(2)$ with the V1 distribution $\xi = \mathrm{span}\{X_1, X_2\}$. Companion to the Geometry of Seeing series.

• Appendix A1 — Lie Groups, Lie Algebras, and the Exponential Map of SE(2) 01 May 2026
  Self-contained primer on Lie groups and Lie algebras with SE(2) as the running example. Matrix exponential, left-invariant vector fields, the Lie bracket as both commutator and closing-defect, adjoint and coadjoint actions. Companion appendix to the Geometry of Seeing series.