The Series
| Part | Title | Key objects | Status |
|---|---|---|---|
| 1 | The Visual Cortex as a Contact Manifold | SE(2), contact structure, Kanizsa | PUBLISHED |
| 2 | Euler's Elastica and Jacobi Elliptic Functions | sn, cn, dn, K(k²), elastica | PUBLISHED |
| 3 | Maxwell Strata: When Optimal Paths Fork | discrete symmetries, first Maxwell time | PUBLISHED |
| 4 | The Exact Cut Time on SE(2) — and the Open Problem Beyond It | cut locus, conjugate time, conjecture | PUBLISHED |
Appendices — Theory Background
The five appendices below build, from first principles, the mathematics the main parts use without proof. Each is self-contained, with derivations and 2–3 interactive figures, and re-uses code from the moiseevigor/elliptic package. Read in the order A1 → A2 → A3 → A4 → A5 — or as needed from the main parts.
| Part | Title | Builds toward | Status |
|---|---|---|---|
| A1 | Lie Groups, Lie Algebras, and the Exponential Map of SE(2) | $\mathfrak{se}(2)$, brackets, $\exp$ | PUBLISHED |
| A2 | Distributions, Frobenius, and Contact Geometry | Chow, contact, V1 horizontality | PUBLISHED |
| A3 | Calculus of Variations and the Pontryagin Maximum Principle | Lie–Poisson on $\mathfrak{se}(2)^*$ | PUBLISHED |
| A4 | Jacobi Elliptic Functions, Elliptic Integrals, and the AGM | $\mathrm{sn}, \mathrm{cn}, \mathrm{dn}, K(m)$ | PUBLISHED |
| A5 | The Sub-Riemannian Exponential Map of SE(2) | conjugate / cut / Maxwell | PUBLISHED |
Mathematical Setting
The model rests on three ingredients.
The state space is an orientation lift. An idealised orientation-selective cortical state records retinal position $(x,y)$ and a preferred line orientation $[\theta]$, with $\theta\sim\theta+\pi$. The biologically natural space is therefore $\mathbb{R}^2\times\mathbb{P}^1$. For calculation the series uses its oriented double cover $\mathbb{R}^2\times S^1\cong\mathrm{SE}(2)$, where heading is remembered modulo $2\pi$; endpoint statements modulo $\pi$ are explicitly projected back to the line bundle.
The metric is sub-Riemannian. Not all directions in $\mathrm{SE}(2)$ are allowed at unit cost. An idealised cortical state at orientation $\theta$ can move cheaply along its preferred direction $(\cos\theta, \sin\theta)$ and rotate cheaply by $d\theta$, but moving transversally is forbidden outright. This defines a rank-2 distribution (a contact structure) with the Pontryagin Hamiltonian
\[H = \frac{1}{2}(p_x \cos\theta + p_y \sin\theta)^2 + \frac{1}{2} p_\theta^2.\]One pendulum, two curve families. Pinning the forward speed turns the problem into Euler’s elastica — the same curves Euler studied in 1744 when minimising the integral of squared curvature, with $\kappa(s) = 2k\,\mathrm{cn}(s \mid k^2)$ and spatial period $T = 4K(k^2)$. The free SE(2) geodesics share the pendulum but project to cuspidal curves; their cut time is $2K(k^2)$ — half the elastica clock (Sachkov 2010–2011).
Key Results Covered
- Petitot’s contact model (Part 1): how a neurogeometric model lifts orientation data to a contact bundle and formulates one class of completion problems variationally.
- Complete parametrisation (Part 2): all three families — the inflectional family, the borderline (separatrix) elastica, and the non-inflectional family — written in closed form using $\mathrm{sn}, \mathrm{cn}, \mathrm{dn}$.
- Maxwell strata (Part 3): the pendulum’s reflection group \((\mathbb{Z}_2)^3\), the mirror-pair tie at one curvature period \(4K(k^2)\) on the elastica family, and the strata bounding the free problem’s cut time.
- The theorem and the open problem (Part 4): Sachkov’s exact cut time \(t_{\mathrm{cut}} = 2K(k^2)\) on the oscillating-pendulum family (\(C_1\)) — with no conjugate points anywhere along it — and the general Maxwell-equals-cut question that remains open beyond SE(2).
Code and Data
The interactive figures use the elliptic library — Jacobi elliptic functions and complete/incomplete integrals implemented without Maple calls, accepting tensors as input.
- GitHub: moiseevigor/elliptic
- arXiv: 0807.4731 — Moiseev & Sachkov (2010)
- arXiv: 0903.0727 — Sachkov (2010, 2011)
Core References
- J. Petitot (2003). "The neurogeometry of pinwheels as a sub-Riemannian contact structure." J. Physiology–Paris 97(2–3): 265–309.
- I. Moiseev & Yu. L. Sachkov (2010). "Maxwell strata in sub-Riemannian problem on the group of motions of a plane." ESAIM: COCV 16(2): 380–399. arXiv:0807.4731
- Yu. L. Sachkov (2011). "Cut locus and optimal synthesis in the sub-Riemannian problem on the group of motions of a plane." ESAIM: COCV 17(2): 293–321. arXiv:0903.0727
- G. Citti & A. Sarti (2006). "A cortical based model of perceptual completion in the roto-translation space." J. Math. Imaging Vision 24(3): 307–326.