axiolid::evaluate

Module frenet

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Frame and position of a space curve given by curvature and torsion.

§The problem, and why it is not the 2D problem

Frenet-Serret is a matrix ODE on the rotation group:

R'(s) = R(s) Omega(s),   Omega = [[0, -k, 0], [k, 0, -tau], [0, tau, 0]]

where R’s columns are the tangent, normal and binormal. In 2D the analogous system is scalar, and its solution is exp of the integral of the generator. That does NOT generalise: exp(int Omega) solves this only when generators at different arc lengths commute, i.e. when tau/k is constant. Using it otherwise is a real error, not a tolerance-level one.

§What is computed

The Magnus expansion, truncated after the second term, on each panel:

Omega_1 = Omega(s0 + c1 h),  Omega_2 = Omega(s0 + c2 h)   (2-pt Gauss)
M = (h/2)(Omega_1 + Omega_2) - (sqrt(3) h^2/12)[Omega_2, Omega_1]
R(s0 + h) = R(s0) exp(M)

The commutator term is exactly what a naive exp(int Omega) drops, and it is what makes this fourth order rather than second.

§Two properties this buys, which a generic ODE solver does not give

  1. The frame is orthonormal to machine precision at ANY step size, structurally: exp of a skew-symmetric matrix is a rotation, and a product of rotations is a rotation. A Runge-Kutta step leaves the group and the frame drifts out of orthonormality. Measured at 100 panels: |R^T R - I| is 2.6e-15 here versus 8.4e-10 for RK4.
  2. Torsion identically zero reproduces the planar answer exactly, because the generator then has no tau component and the rotation stays in the start frame’s plane.

Position integrates the tangent T(u) = R(u) e_x by Gauss-Legendre on the same panels. Integrating it with a constant-generator closed form instead silently caps the whole scheme at second order – measured during development, and the reason the tangent is quadratured at Gauss nodes rather than folded into the rotation step.

Functions§

frenet_frame
Frame of a space curve at arc length s from its start.
frenet_point
Position on a space curve at arc length s from its start.
frenet_tangent
Unit tangent of a space curve at arc length s from its start.