Fairing-PIA: Progressive iterative approximation for fairing curve and surface generation

TL;DR

This paper introduces a novel progressive-iterative approximation method called fairing-PIA for generating and improving fairing curves and surfaces with adjustable parameters, proving convergence and demonstrating effectiveness.

Contribution

The paper develops a new iterative fairing method that allows local and global shape optimization with convergence proof, extending beyond traditional energy minimization models.

Findings

Effective and efficient shape fairing demonstrated through numerical examples.

Convergence of the proposed fairing-PIA method is mathematically proven.

Traditional energy minimization fairing is a special case of fairing-PIA.

Abstract

The fairing curves and surfaces are used extensively in geometric design, modeling, and industrial manufacturing. However, the majority of conventional fairing approaches, which lack sufficient parameters to improve fairness, are based on energy minimization problems. In this study, we develop a novel progressive-iterative approximation method for fairing curve and surface generation (fairing-PIA). Fairing-PIA is an iteration method that can generate a series of curves (surfaces) by adjusting the control points of B-spline curves (surfaces). In fairing-PIA, each control point is endowed with an individual weight. Thus, the fairing-PIA has many parameters to optimize the shapes of curves and surfaces. Not only a fairing curve (surface) can be generated globally through fairing-PIA, but also the curve (surface) can be improved locally. Moreover, we prove the convergence of the developed fairing-PIA and show that the conventional energy minimization fairing model is a special case of fairing-PIA. Finally, numerical examples indicate that the proposed method is effective and efficient.