Overview

Mesh generation of curves and surfaces ^[1] is an operation known to be tedious to carry out, in a robust fashion, in the context of numerical simulations based on a finite element method as well as in other types of applications. The accuracy of the results in finite element numerical simulations is partly related to the quality of the geometric approximation (i.e., the mesh). Therefore, the mesh of the boundary of a two- or three-dimensional arbitrary domain must have certain properties that are directly related to the geometry it represents.

The construction of a mesh of a curve (resp. surface) requires, in particular, knowledge of the local intrinsic characteristics of the curve (resp surface) such as the curvature, the normal(s), the tangent (the tangent plane), etc. at any point of this geometric support. These geometric characteristics have a significance that will be made precise in this chapter. This analysis is based in practice on a limited expansion of the function, ? or ?, which gives a local approximation of the corresponding curve or surface. The local behavior and the main features of the function can be deduced from this approximation. Thus, in a mesh generation context, the analysis of curves and surfaces can be deduced from ? or ? as well as from their successive derivatives.

Differential geometry was introduced in the early 18th century and then established in the 19th century as a way of defining a general theoretical framework for the local...