A point in the plane is not enough to describe a moving vehicle. We also need its heading. Write its state as $(x,y,\theta)$: a position and an angle. Together these states form SE(2), the group of plane motions.
Now forbid sideways motion. The vehicle may drive, reverse and turn, with equal cost for the two allowed controls. Its equations are
\[\dot x=u_1\cos\theta,\qquad \dot y=u_1\sin\theta,\qquad \dot\theta=u_2, \qquad \text{length}=\int\sqrt{u_1^2+u_2^2}\,dt.\]The shortest paths for this problem are already understood through the work of Moiseev and Sachkov.12 But the equations produce more than shortest paths. Continue a geodesic after its cut time and it still solves the Hamiltonian equations. Nearby geodesics can focus, and several different sources can reach exactly the same position and heading.
This program follows that continuation. Its three articles examine the same exponential map from three directions:
- Times: locate the losses of rank along a source geodesic.
- Geometry: map those singular sources into the target space.
- Counts: hold a target fixed and recover every source in the stated class.
A geodesic, continued beyond its cut time
A pendulum behind every path
The maximum principle reduces the normal Hamiltonian equations to a mathematical pendulum. Oscillating and rotating pendulum motions produce different families of geodesics. In the standard metric considered here, only the rotating family has conjugate points.3
A rotating source is specified by a modulus $0<k<1$, an initial phase $\psi$ modulo $4K$, and a rotation sign $\varepsilon=\pm1$. The source-flow figures show $\varepsilon=1$; the inverse search considers both signs. We use the parameter $m=k^2$ in numerical elliptic functions, and the scaled time
\[p=\frac{t}{2k},\qquad \tau=\psi+p,\qquad K=K(k^2).\]These conventions matter. A statement about $p$ is a statement about physical time only after multiplying by $2k$; a complete elliptic integral written in the modulus convention has $k^2$ inside its defining integral.
There are also two different ways for a shortest path to stop being shortest. At a Maxwell point, another path reaches the same target with the same length. At a conjugate point, the differential of the exponential map loses rank: changing the source can cease to change the endpoint to first order. A global competitor and a local degeneracy are different events. The cut time occurs no later than the first conjugate time.
One crossing in each bracket
The first article starts from Sachkov’s reduced Jacobian,
\[J_2(\tau,p)=\alpha(p)\,\operatorname{sn}^2\tau+\beta(p)\,\operatorname{cn}^2\tau.\]On the relevant brackets, its zero equation becomes a crossing problem:
\[\operatorname{sn}^2(\psi+p)=r(p),\qquad r(p)=-\frac{\beta(p)}{\alpha(p)-\beta(p)}.\]The useful step is to lift $r$ back to phase. A Riccati identity and the arithmetic–geometric mean inequality give a slope bound for the two lifted branches. Each line $\tau=\psi+p$ then crosses the appropriate closed branch exactly once. The manuscript proves one conjugate time in the first bracket and one root on each of two brackets in every later Maxwell cell.
That does not always mean two distinct times. At the critical modulus $k_0\approx0.90890856$, where $K=2E$, and phase $\psi\equiv K\pmod{2K}$, the two bracket roots meet at $p=2nK$. The determinant touches zero without the usual sign change. The cells are assigned half-open endpoints, $[p_n^1,p_{n+1}^1)$, so a boundary root belongs to one cell.
Two equations, one root per bracket
The paper also derives an exact Floquet form: after an integer number of pendulum periods, the Jacobian is quadratic in that integer. Away from the exceptional phase, this gives an arithmetic-progression law for late conjugate times. It explains regular spacing without assuming the entire endpoint map is periodic: each pendulum period also produces a spatial drift.
From times to a caustic
A conjugate time is a location in source coordinates. Apply the exponential map and it becomes a point on the conjugate locus, or caustic. The second article studies this image.
The central calculation expresses the derivative of the determinant along its kernel through a polynomial $Q$. Exact rational positivity certificates establish the fold behavior of the open sheets. Separate derivative and rank conditions identify their cusp seams. At the critical point, the manuscript treats a corank-two $D_4^+$ singularity through a generating-family calculation and its versality conditions.
This is why plotting a scalar double root does not by itself classify a caustic. A singularity belongs to the full map, with its kernel, rank and higher derivatives. The article keeps those geometric tests separate from the time equation.
Where a family of geodesics focuses
Turning the map around
The third article fixes a target $q=(x,y,\theta)$ and asks which sources reach it. A direct search would vary phase, modulus and time. The scalar inverse removes two of those unknowns algebraically.
For an admissible axis angle $a$, write $z=\tan a$. The target determines a modulus $m(z)$, two endpoint amplitudes, a residual displacement $Q_0(z)$ and a positive drift per full period
\[L(m)=4\bigl(K(m)-E(m)\bigr).\]The remaining equation is a level crossing:
\[N(z)=\frac{A(z)-Q_0(z)}{L(m(z))}=n,\qquad n\in\mathbb Z_{\ge0}.\]Each admissible integer intersection reconstructs an actual rotating source. Both spatial signs must be considered, and the chart’s seam needs its own treatment. This gives the bridge between the three articles: a critical integer intersection of $N$ corresponds to a singular source, whose image lies on a caustic sheet.
Different paths, the same position and heading
Which count?
The word “count” needs a declared object. The rotating fiber is finite at strict nonzero headings in the manuscript’s stated domains. The full normal fiber can be countably infinite, because it includes oscillating geodesics in nonzero winding classes. The minimizing fiber is smaller still, and its classification uses the published optimal synthesis.
On one especially tractable half-heading ray, the manuscript gives an exact rotating count at every positive radius. Changes occur at analytically defined seam and cusp thresholds. At a threshold itself, equality matters; the count there cannot be inferred simply by drawing a step between the two neighboring values.
How the rotating fiber grows along a ray
What is proved and what remains open
The program now has three manuscript snapshots, with different proof boundaries. The table reports their author-level status; none has external journal approval.
| Layer | What the current record supports | Remaining boundary |
|---|---|---|
| Conjugate times | Analytical bracket/counting proofs and a conditional Lean development; the critical coalescence is explicit | Concrete existence of the abstract Jacobi core and finiteness of tangencies remain assumptions of the formal chain; asymptotics and full geometry are outside that chain |
| Caustic geometry | Exact fold identities and rational certificates, separate cusp/umbilic arguments and bounded independent differential checks | The global incidence/count conjecture is open; excluded endpoint domains require additional arguments |
| Interval criterion | The current audit records semantic validation of all 38,400 cell checkpoints, leaving 304 unresolved boxes in 32 cells under the original exclusions | Those residual cells are now included in the fifth exclusion set. Closure outside the five sets does not establish the predicate inside them |
| Inversion and counts | Exact scalar reconstruction and domain-specific count statements in the manuscript, with stored symbolic, interval and ODE checks | General effective onset scales, full publication review and the stated novelty checks remain open |
The declared five excluded sets occupy approximately 2.56% of the ambient endpoint-coordinate box. This is a coordinate-volume statement, not a probability, a fraction of physical targets, or a percentage of the theorem proved. The smaller 304-box residual set occupied about 0.049% before the broader fifth exclusion was declared.
The formal work has also advanced. The latest reduction constructs the 31-field Jacobi structure from a 13-hypothesis core: the differential equations, initial values and first positive zero of $\operatorname{cn}$. It derives algebraic relations, periodic and reflection identities, phase inversion and the phase congruence rule. This narrows the assumption base. It still does not formalize the entire three-paper program.
A useful negative finding survives the review: six selected targets have locally certified low-level dips in $N$. Unrestricted unimodality at all real levels is therefore false. These dips occur below one and do not settle the positive-integer-level conjecture. Several historical binary64 finite-difference signs were wrong in precisely these shallow regions; direct derivatives and higher precision were needed to distinguish them.
The next analytical work is to control the low-modulus wedge and the remaining exclusions, then complete the article-level proof and novelty reviews. Browser experiments help identify questions and explain mechanisms. They remain a different evidence layer from exact identities, interval coverage and formal proofs.
The three articles
All Conjugate Times of Rotating Geodesics
The bracket mechanism, the critical exception, the Floquet form and the reduced Lean trust base. Includes the full manuscript PDF.
The Conjugate Locus: Folds, Cusps and Incidence
The image of the singular set, geometric recognition and the exact scope of the count certificate. Includes the full manuscript PDF.
Inverting the Exponential Map and Counting Geodesics
The scalar inverse, winding classes, exact ray counts and the stated asymptotic regimes. Includes the full manuscript PDF.
All three belong to this single SE(2) program. The model is the standard left-invariant metric with $\theta$ modulo $2\pi$; extensions to other groups, metrics or quotients need their own hypotheses.
Sources and reproducibility
The primary literature is indexed in the project’s 53-record Obsidian catalogue. Its notes distinguish selected-section reading, abstract-only coverage, unread sources and edition gaps. A catalogue entry does not establish that every theorem in a source has been read or checked.
The web edition snapshots research revision 5c304fd from 27 September 2026. The source repository contains the canonical manuscripts, living notebook, formal development and audit records. The snapshot manifest binds the copied manuscript PDFs, their TeX sources and the browser math module. This publishing pass checks the figures and records current claims; it does not rerun the full interval computation.
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I. Moiseev and Y. L. Sachkov, Maxwell strata in the sub-Riemannian problem on the group of motions of a plane, ESAIM: COCV 16 (2010). DOI. Establishes the underlying parametrization and Maxwell structure used here. ↩
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Y. L. Sachkov, Cut locus and optimal synthesis in the sub-Riemannian problem on the group of motions of a plane, ESAIM: COCV 17 (2011). DOI. The complete minimizing synthesis is an established source result, distinct from post-cut counts. ↩
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Y. L. Sachkov, Conjugate and cut time in the sub-Riemannian problem on the group of motions of a plane, ESAIM: COCV 16 (2010). DOI. The known first-time bounds and rotating-stratum conjugate results are inputs to the new manuscript. ↩