Surface Covering by Fairing Progressive Iterative Approximation in Geometry Engine

Covering is a widely employed key function in CAD geometry engine, which fits a surface through one or more given boundary curves. However, it is a difficult problem to generate a desirable covering surface due to the lack of enough geometric information, and there are very few relevant works that address the problem. In this paper, we develop a method of generating one B-spline surface to cover given boundary curves by fairing progressive iterative approximation (Fairing-PIA). Differently from traditional fairing methods which have only global smoothing weight, Fairing-PIA can set different smoothing weights for each control point, leading to great flexibility. Firstly, robust methods are developed to generate stable initial surface fitting the given boundary curves. Specifically, on the one hand, if there are no more than four boundary curves, an initial surface is obtained by constructing Coons surface; on the other hand, for more than four boundary curves, a trimmed surface is generated by fitting the boundary curves as the initial surface. Then, Fairing-PIA is applied to improve the surface quality, leading to a desirable covering surface. Numerical experiments comparing the proposed method with traditional methods across different types of boundary examples demonstrate that the proposed method can fit various types of boundaries, allows for surface shape control through fairing weight adjustments, and achieves approximately twice the performance of traditional methods that directly solve matrix equations, offering both flexibility and efficiency.