Geometry of Seeing: series consistency + accuracy pass

_posts/2026-04-25-geometry-of-seeing-visual-cortex-se2.md

@@ -119,7 +119,8 @@ Mapped over a patch of cortex, the preferred orientations rotate continuously 
 completing a full $\pi$-rotation over a distance of about 1 mm.
 The resulting pattern of orientation preferences is called an *orientation map*,
 and it exhibits a characteristic structure of *pinwheels*: point singularities
-around which the preferred orientation rotates by $\pm\pi/2$.
+of topological charge $\pm\tfrac12$ — encircle one and the preferred orientation
+sweeps through a full half-turn, $\pm\pi$.
 
 <aside>
 The pinwheel structure was first measured in the cat striate cortex using
@@ -312,7 +313,7 @@ Each leg is small (length $\varepsilon$). The net effect is to leave the
 $\varepsilon^2$ in the **perpendicular** direction:
 
 $$\Phi^{X_2}_{-\varepsilon} \!\circ\! \Phi^{X_1}_{-\varepsilon} \!\circ\! \Phi^{X_2}_{\varepsilon} \!\circ\! \Phi^{X_1}_{\varepsilon}\;(g_0)
-\;=\; g_0 \,+\, \varepsilon^2\, X_3 \,+\, O(\varepsilon^3).$$
+\;=\; g_0 \,-\, \varepsilon^2\, X_3 \,+\, O(\varepsilon^3).$$
 
 <aside id="note-lie-bracket">
 The <strong>Lie bracket</strong> $[X, Y]$ measures the failure of two
@@ -324,9 +325,13 @@ side-by-side derivation.
 </aside>
 
 The infinitesimal limit of this loop is exactly the <span class="annotated-term" data-note="note-lie-bracket">Lie bracket</span> of $X_1$ and
-$X_2$:
+$X_2$. With $X_3$ fixed as above, a direct computation gives
 
-$$[X_1, X_2] \;=\; X_3 \;=\; -\sin\theta\,\partial_x + \cos\theta\,\partial_y.$$
+$$[X_1, X_2] \;=\; -X_3 \;=\; \sin\theta\,\partial_x - \cos\theta\,\partial_y.$$
+
+The sign is a convention artefact — $[X_2, X_1] = X_3$ — and changes nothing
+that follows: what matters is that the bracket points along $X_3$, the
+direction the two horizontal moves cannot reach on their own.
 
 </div><!-- /.l-body -->
 
@@ -441,7 +446,7 @@ $$[X_1, X_2] \;=\; X_3 \;=\; -\sin\theta\,\partial_x + \cos\theta\,\partial_y.$$
 
 ### Hörmander condition ⇒ V1 is a contact manifold
 
-Because $X_3 = [X_1, X_2]$ is *not* in $\mathrm{span}\{X_1, X_2\}$ but the
+Because $[X_1, X_2] = -X_3$ is *not* in $\mathrm{span}\{X_1, X_2\}$ but the
 three together $\{X_1,\, X_2,\, [X_1, X_2]\}$ span the full tangent space
 $T_g\,\mathrm{SE}(2) \cong \mathbb{R}^3$ at every $g$, the 2-plane field
 
@@ -452,13 +457,13 @@ The <strong>Hörmander condition</strong> (1967): the vector fields and all thei
 Lie brackets together span the full tangent space at every point.
 It is the key criterion for hypoellipticity and, via Chow–Rashevskii, for the
 existence of horizontal paths between any two points.
-For SE(2): $\{X_1, X_2, [X_1,X_2]\} = \{X_1, X_2, X_3\}$ already spans
-$\mathbb{R}^3$, so depth-1 brackets suffice.
+For SE(2): $\{X_1, X_2, [X_1,X_2]\}$ spans the same space as $\{X_1, X_2, X_3\}$ —
+all of $\mathbb{R}^3$ — so depth-1 brackets suffice.
 See Appendix A2 for the full statement.
 </aside>
 
 satisfies the <span class="annotated-term" data-note="note-hormander"><strong>Hörmander (bracket-generating) condition</strong></span>. By the
-**Chow–Rashevskii theorem** (1938, 1938) any two configurations in
+**Chow–Rashevskii theorem** (Rashevskii 1938, Chow 1939) any two configurations in
 $\mathrm{SE}(2)$ can therefore be joined by a *horizontal* path — a curve
 whose velocity lies in $\mathcal{H}$ at every point. Geometrically,
 $\mathcal{H}$ is a **contact structure** on $\mathrm{SE}(2)$, and Petitot's
@@ -496,11 +501,12 @@ The spatial projection $(x(t), y(t))$ of the solution is the **perceptually comp
 contour** that the visual system infers between two oriented line elements
 $(x_0,\theta_0)$ and $(x_1,\theta_1)$.
 
-The simplification $u_1 = 1$ (unit forward speed) reduces the problem to minimising
-$$\int_0^L \kappa^2(s)\,ds$$
-where $L$ is arc length and $\kappa = u_2/u_1 = \dot\theta$ is the signed curvature.
-This is the **Euler elastica functional**: the total squared bending energy of the
-projected curve.
+With the normalisation $u_1 = 1$ (unit forward speed), $s$ becomes arc length and
+$\kappa = u_2/u_1 = \dot\theta$ is the signed curvature. The sub-Riemannian extremals
+then project onto the critical curves of the **Euler elastica functional**
+$$\int_0^L \kappa^2(s)\,ds,$$
+the total squared bending energy of the projected curve — the classical bridge,
+made precise in Part&nbsp;2, between geodesics on SE(2) and Euler's elastica.
 
 </div><!-- /.l-body -->
 
@@ -527,9 +533,9 @@ projected curve.
     $k \in (0, 1)$ — <strong>inflectional</strong>, $\kappa(s) = 2k\,\mathrm{cn}(s\mid k^{2})$;
     $k = 1$ — the <strong>Euler / Cornu spiral</strong>, the separatrix
     $\kappa(s) = 2\,\mathrm{sech}\,s$;
-    $k > 1$ — <strong>non-inflectional</strong>, parametrised by
-    $m = 2 - k \in (0, 1)$ with $\kappa(s) = 2\,\mathrm{dn}(s\mid m)$
-    (closed-loop curves with one-signed curvature).
+    $k > 1$ — <strong>non-inflectional</strong>, $\kappa(s) = 2\,\mathrm{dn}(s\mid m)$
+    with its own modulus $m \in (0, 1)$ (closed-loop curves with one-signed
+    curvature); the slider just sweeps $m$ downward as $k$ runs past&nbsp;1.
     Drag the slider to set the maximum $k$ rendered; the vertical bar on the
     right is the colour scale, with the red tick marking the separatrix
     $k = 1$. As $k \to 1$ the period $4K(k^{2})$ diverges and the inflectional
@@ -583,8 +589,10 @@ $L_{\mathrm{SR}}$:
   equal or smaller $L_{\mathrm{SR}}$. At a Maxwell point, two distinct
   globally-optimal geodesics meet with *exactly* the same SR length.
 
-For a sub-Riemannian manifold as symmetric as SE(2), the cut and Maxwell loci coincide
-(this is part of what Sachkov and I proved in arXiv:0807.4731).
+For a sub-Riemannian manifold as symmetric as SE(2), the cut and Maxwell loci are
+tightly linked: characterising the Maxwell strata is the main result of
+arXiv:0807.4731 (Sachkov and I), and pinning the cut locus down exactly is the
+subject of arXiv:0903.0727.
 The first point on the cut locus along a given geodesic is the **cut time**
 $t_\mathrm{cut}$, and it equals the first time the exponential map is no longer
 injective.

_posts/2026-04-28-geometry-of-seeing-elastica-jacobi.md

@@ -85,9 +85,13 @@ The **Hamiltonian** for the PMP maximisation condition is
 
 $$H = h_1 u_1 + h_2 u_2 - \nu \sqrt{u_1^2 + u_2^2},$$
 
-where $\nu \in \{0, \tfrac{1}{2}\}$ is the abnormality constant.
-For **normal extremals** ($\nu = \tfrac{1}{2}$), maximising over $u_1, u_2$
-gives the **normal Hamiltonian**
+where $\nu \in \{0, 1\}$ is the abnormality constant ($\nu = 1$ for
+**normal** extremals, $\nu = 0$ for abnormal; Appendix A3 carries out the
+maximisation in full).
+Minimising length is equivalent to minimising the energy
+$\tfrac12\!\int(u_1^2 + u_2^2)\,dt$ over a fixed interval (Cauchy–Schwarz),
+and for normal extremals the PMP maximisation then
+delivers the **normal (sub-Riemannian) Hamiltonian**
 
 $$\mathcal{H}_n = \tfrac{1}{2}\bigl(h_1^2 + h_2^2\bigr).$$
 
@@ -191,15 +195,14 @@ delivered for the SR-on-SE(2) costate phase.
 
 The pendulum equation lives on the costate, but the figures below plot the
 **curvature $\kappa(s)$ of the projected plane curve in its own Euclidean
-arc length $s$**. These two are tied together by a quick computation.
+arc length $s$** — and $s$ is not the costate time $t$: the two are tied
+together by $ds = |h_1|\,dt$.
 
-In unit-speed parametrisation $h_1^{2}+h_2^{2}=1$, the projected speed is
-$|h_1| = |\sin(\varphi/2)|$ and the heading derivative is $\dot\theta = h_2 = \cos(\varphi/2)$.
-Reparametrising by Euclidean arc length $s$ with $ds = |h_1|\,dt$, and using
-$\dot\varphi = \pm 2 h_3$, a short calculation gives an explicit closed-form
-$\kappa(s)$ in each regime — the three Jacobi-elliptic curvature profiles
-below. They satisfy the **elastica curvature ODE** (the Duffing form
-equivalent to the pendulum)
+Pushing the pendulum solution through that reparametrisation is the technical
+heart of the reduction, and it is carried out in full in Appendix A3. The
+outcome is what matters here: in each regime $\kappa(s)$ comes out in closed
+form as one of the three Jacobi-elliptic profiles below, and each one solves
+the **elastica curvature ODE** — the Duffing form equivalent to the pendulum,
 
 $$\kappa''(s) + \tfrac{1}{2}\kappa(s)^{3} - \mu\,\kappa(s) \;=\; 0,$$
 
@@ -374,8 +377,9 @@ The browser figures on this page use the same AGM algorithm implemented in
 For the Jacobi functions themselves:
 
 ```python
-from elliptic import ellipj
+from elliptic import ellipj, ellipticK
 
+K = ellipticK(k**2)            # quarter-period of sn, cn
 s = np.linspace(-2*K, 2*K, 800)
 sn, cn, dn = ellipj(s, k**2)
 kappa = 2 * k * cn             # curvature of inflectional elastica
@@ -417,9 +421,9 @@ In the `elliptic` package:
 ```python
 from elliptic import elliptic12
 
-phi = np.arcsin(k * sn)
-E_vals, F_vals = elliptic12(phi, k**2)   # E(φ|k²) and F(φ|k²)
-x = 2 * (E_vals - F_vals / 2)            # exact formula from Sachkov (2011)
+am = np.arcsin(sn)                       # Jacobi amplitude am(s | k²)
+F_vals, E_vals = elliptic12(am, k**2)    # F(am | k²) = s  and  E(am | k²)
+x = 2 * E_vals - F_vals                  # x(s) = 2 E(am(s)|k²) − s  (Sachkov 2011)
 ```
 
 The full closed-form expressions — due to Sachkov (2011) — express every
@@ -435,8 +439,9 @@ A few landmarks worth noting:
 - **$k = 0.1$** (inflectional): nearly straight, very gentle curvature oscillation.
   The curve barely bends before straightening again.
 
-- **$k \approx 0.71$** (inflectional): the "figure-eight" lemniscate — the curve
-  crosses itself once per period and the endpoints of one period coincide.
+- **$k \approx 0.909$** (inflectional): the "figure-eight" lemniscate — at the
+  modulus where $2E(k^2) = K(k^2)$, the curve crosses itself once per period and
+  the endpoints of one period coincide.
   This is the **Maxwell stratum** for the symmetric geodesics (Part 3).
 
 - **$k \to 1^-$** (inflectional → Euler spiral): the period $4K(k^2)$ diverges and