Numerical reparametrization of periodic planar curves via 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.