Construction of Fair Surfaces Over Irregular Meshes

This paper describes the process of constructing a fair, open or closed $C1$ surface over a given irregular curve mesh. The input to the surface construction consists of point and/or curve data which are individually marked to be interpolated or approximated and are arranged according to an arbitrary irregular curve mesh topology (Fig. 1). The surface constructed from these data will minimize flexibly chosen fairness criteria. The set of available fairness criteria is able to measure surface characteristics related to curvature, variation of curvature, and higher order surface derivatives based on integral functionals of quadratic form derived from the second, third and higher order parametric derivatives of the surface. The choice is based on the desired shape character. The construction of the surface begins with a midpoint refinement decomposition of the irregular mesh into aggregates of patch complexes in which the only remaining type of building block is the quadrilateral Be´zier patch of degrees 4 by 4. The fairing process may be applied regionally or to the entire surface. The fair surface is built up either in a single global step or iteratively in a three stage local process, successively accounting for vertex, edge curve and patch interior continuity and fairness requirements. This surface fairing process will be illustrated by two main examples, a benchmark test performed on a topological cube, resulting in many varieties of fair shapes for a closed body, and a practical application to a ship hull surface for a modern container ship, which is subdivided into several local fairing regions with suitable transition pieces. The examples will demonstrate the capability of the fairing approach of contending with irregular mesh topologies, dealing with multiple regions, applying global and local fairing processes and will illustrate the influence of the choice of criteria upon the character of the resulting shapes.