Numerical reparametrization of periodic planar curves via curvature interpolation

TL;DR

This paper introduces a static algorithm for reparametrizing periodic planar curves by interpolating curvature, achieving spectral accuracy and improved spatial resolution through Fourier analysis.

Contribution

It presents a novel curvature-based reparametrization method that enhances curve representation accuracy without altering the original parametrization.

Findings

Spectral accuracy in tangential velocity computation

Faster convergence to original shape with refined sampling

Improved spatial resolution through curvature interpolation

Abstract

A novel static algorithm is proposed for numerical reparametrization of periodic planar curves. The method identifies a monitor function of the arclength variable with the true curvature of an open planar curve and considers a simple interpolation between the object and the unit circle at the curvature level. Since a convenient formula is known for tangential velocity that maintains the equidistribution rule with curvature-type monitor functions, the strategy enables to compute the correspondence between the arclength and another spatial variable by evolving the interpolated curve. With a certain normalization, velocity information in the motion is obtained with spectral accuracy while the resulting parametrization remains unchanged. Then, the algorithm extracts a refined representation of the input curve by sampling its arclength parametrization whose Fourier coefficients are directly accessed through a simple change of variables. As a validation, improvements to spatial resolution are evaluated by approximating the invariant coefficients from downsampled data and observing faster global convergence to the original shape.