The Shortest Path When You Cannot Move Sideways
An interactive guide to the geometry of motion in the plane: from a simple constraint to the pendulum, competing routes, and the complete shortest-path solution.
Beyond the Shortest Path: Geodesics on SE(2)
One research program, three questions: when a path loses local optimality, where nearby paths focus, and how many paths reach the same position and heading. Follow the pendulum, explore the conjugate locus, and count the inverse images.
The Geometry of Forbidden Directions: A Research Program
When a physical field forbids a direction of motion — not merely slows it — the configuration space stops being ordinary and becomes sub-Riemannian. This post scopes a program built on that distinction, states the identity it is organised around (Q = d + k + 2 — a dimensional count, verified numerically), and draws the sharp line between the physics that qualifies (magnetic fields, rotation) and the physics that only looks like it does (anisotropic transport, gravity).
From Caustics to Groups: A Research Program
A caustic is where a flow of paths focuses — the bright edge in a coffee cup, the cusp in a gravitational lens, the shell-crossing wall of the cosmic web. This post scopes a falsifiable research program for the reverse map: given an observed field of caustics, infer the hidden sub-Riemannian (Lie-group) geometry that produced it. The program is built around a hard obstruction — most caustics are provably group-blind — and the three places where the discriminating information actually lives.
The Geometry of the Cosmic Web: A Research Program
Matter in the Universe collapses into a web of filaments. This post scopes a falsifiable research program: lift the cosmic web onto the position–orientation manifold $\mathbb{R}^3 \times S^2 \cong \mathrm{SE}(3)/\mathrm{SO}(2)$ — the same machinery the visual cortex uses to complete contours — and test with public simulation and survey data whether sub-Riemannian geodesics trace, and perhaps shape, the filaments.
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