The geometry behind the library
Concepts
These pages follow the same path as the implementation: from a Finsler metric to the geometry derived from it, and then from observed paths and arrival times back to an unknown metric.
Riemann’s Habilitationsvortrag of 1854 introduced the -dimensional case with a line element coming from a positive definite quadratic form, and in the same lecture he named the next case — a length that is not the square root of one — and set it aside, judging that it would demand considerable work and throw comparatively little new light. Finsler’s 1918 thesis under Carathéodory developed the general case from the calculus of variations. The quadratic form is no longer required, but positivity, positive homogeneity and strong convexity remain.
Much of the Riemannian vocabulary remains: geodesics, parallel translation, curvature and distance. The objects themselves are less rigid. Distance may depend on the order of its arguments, a reversed geodesic need not remain geodesic, and canonical parallel translation is generally homogeneous rather than linear. Its holonomy acts by diffeomorphisms of the indicatrix instead of orthogonal maps of a tangent space; in the generic case that group is infinite-dimensional.
Each page begins with the geometric question, gives the definition needed to answer it, and then identifies the corresponding classes or solvers. Formal details are kept close to the discussion rather than replacing it.
- Finsler metricsA smooth, positively homogeneous and strongly convex norm on each tangent space. Its unit sphere is the indicatrix.1
- Randers metrics and Zermelo navigationThe family generated by a Riemannian medium and a drift field, the navigation problem it answers, and the bound it must respect.2
- Geodesics and the sprayThe spray, initial- and boundary-value formulations, and why the way back may follow a different curve.3
- Arrival time and frontsOne Hamilton–Jacobi solve giving the cost to every point at once, and the fronts its level sets trace.4
- Parallel transport and holonomyHorizontal translation along a curve, and the holonomy group generated by closed curves.5
- CurvatureThe obstruction to the horizontal distribution being integrable, and the flag curvature it produces — a plane together with a direction in it.6
- Learned metricsFitting F to paths, timings and fronts, and the drift components that path shapes alone cannot identify.7
Conventions are fixed in the package’s spec/MATH_SPEC.md, and these pages follow it rather than restating it: coordinate indices are superscripts, the fundamental tensor carries the factor , and the sign of the drift is set so that a headwind increases cost. Where a source in the literature differs, the implementation is what is described here.