On L2 approximation by planar Pythagorean-hodograph curves

Abstract

The $L^2$ approximation of planar curves by Pythagorean-hodograph (PH) polynomial curves is addressed, based on the distance defined by a metric for planar curves represented as complex-valued functions of a real parameter. Because of the nonlinear nature of polynomial PH curves, constructing $L^2$ approximants involves solving a nonlinear optimization problem. However, a simplified method that requires only the solution of a linear system may be developed by formulating the $L^2$ approximation in the preimage space. The extension of the methodology to approximation by PH B-spline curves is also addressed, and several examples are provided to illustrate its implementation and potential.