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Geometry of Seeing Part A5 67 min

Appendix A5 — The Sub-Riemannian Exponential Map of SE(2)

How initial-costate parameters $(c, \omega_0, \phi_0)$ generate every SE(2) geodesic from the origin; what conjugate, cut, and Maxwell points are; and why the first Maxwell time on an inflectional geodesic is exactly $4K(k^2)/\omega_0$ — the same period that controls the curvature. Bridges Parts 1–2 to Parts 3–4.

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.

Mathematics
sub-riemannian SE2 optimal-control jacobi-elliptic
Appendix A5 — The Sub-Riemannian Exponential Map of SE(2)
Geometry of Seeing Part A4 39 min

Appendix A4 — Jacobi Elliptic Functions, Elliptic Integrals, and the AGM

Where $\mathrm{sn}, \mathrm{cn}, \mathrm{dn}$ come from, why $K(m)$ is the pendulum's quarter-period, and how Gauss's arithmetic–geometric mean computes both to 16 digits in 6 iterations — exactly the algorithm shipped in the moiseevigor/elliptic package.

Self-contained primer on Jacobi elliptic functions and complete elliptic integrals, derived from the pendulum equation Part 2 / Appendix A3 lands on. AGM, period $4K(k^2)$, and explicit identities used in Part 2.

Mathematics
jacobi-elliptic elliptic-integrals elliptic-functions sub-riemannian
Appendix A4 — Jacobi Elliptic Functions, Elliptic Integrals, and the AGM
Geometry of Seeing Part A3 49 min

Appendix A3 — Calculus of Variations and the Pontryagin Maximum Principle

From Euler–Lagrange to the PMP, then Lie–Poisson reduction on $\mathfrak{se}(2)^{\ast}$. Why the equations $\dot h_1 = h_2 h_3$, $\dot h_2 = -h_1 h_3$, $\dot h_3 = 0$ that Part 2 §1 used as a starting point are exactly what you get when you do the optimal-control problem carefully on a Lie group.

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.

Mathematics
sub-riemannian SE2 optimal-control pmp
Appendix A3 — Calculus of Variations and the Pontryagin Maximum Principle
Geometry of Seeing Part A2 46 min

Appendix A2 — Distributions, Frobenius, and Contact Geometry

Why "rank-2 sub-bundle of $T\mathrm{SE}(2)$" and "completely non-integrable" are the right words for V1's horizontal connectivity. Frobenius gives the test for integrability; Chow–Rashevskii gives the reachability theorem; contact geometry gives the cleanest version of both.

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.

Mathematics
sub-riemannian SE2 contact-geometry optimal-control
Appendix A2 — Distributions, Frobenius, and Contact Geometry
Geometry of Seeing Part A1 86 min

Appendix A1 — Lie Groups, Lie Algebras, and the Exponential Map of SE(2)

What "left-invariant vector field" really means, why the Lie bracket of two vector fields is the same thing as the matrix commutator, and why $\exp(tX)$ generates a 1-parameter subgroup. Everything Part 1 used about SE(2), derived from scratch.

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.

Mathematics
sub-riemannian SE2 lie-groups lie-algebra
Appendix A1 — Lie Groups, Lie Algebras, and the Exponential Map of SE(2)