Local Least Squares Fitting for Surface Mesh Fairing in Automobile Panel Design

Abstract

Three-dimensional laser scanning equipment is being used more frequently to convert clay model automobile designs to large, detailed meshes for computer-aided design of outer-body panels. The panels are generally composed of large, constant curvature patches with small local features, called character lines, superposed to give the car a distinctive look. Although modern laser scanners are very accurate and precise, their tolerances nevertheless admit meshes with geometric flaws that destroy the constant curvature of patches and make character lines nearly invisible in a reflection simulation. Thus, we require an algorithm to fair the mesh by restoring the intended curvature while minimizing the vertex displacemtents. Existing approaches such as Laplacian and curvature flow operators are not suitable because they tend to shrink the mesh and introduce a bias toward planar geometries. Our approach aims to solve both of these problems by fitting a least squares surface to a set of vertices adjacent to the target vertex and moving the target vertex vertically onto the least squares surface in a local coordinate system. This algorithm has linear time complexity in the number of vertices and makes convergence likely while eliminating the planar bias of other operators. We show the effectiveness of our operator with both geometric and real-world mesh examples.