Euclidean Arc-Length Waveforms on Anisotropic Quartic Lamé Curves: Fourier Spectra and Thin-Axis Limits

This preprint studies the coordinate waveforms generated by Euclidean unit-speed motion along anisotropic quartic Lamé curves, x⁴ + (ky)⁴ = 1, with k ≥ 1. After amplitude normalization, the two waveforms are related by a phase shift if and only if k = 1. As k tends to infinity, their period-normalized forms converge uniformly, with explicit error bounds, to a triangular horizontal profile and a vertical profile with fourth-root zero crossings. Integral formulas are derived for the Fourier coefficients. At each finite k, the waveforms are analytic and their coefficients decay exponentially. The limiting horizontal and vertical coefficients instead obey the algebraic laws n⁻² and n⁻⁵/⁴, with explicit leading constants. The corresponding rescaled thin-axis and large-harmonic limits do not commute. The supplementary MATLAB software generates standalone Fourier-based coordinate functions. Reported numerical tolerances concern sampled position errors and are not certified uniform error bounds.