Overview

As the perspicacious reader will have already noticed in the presentation of the main governed mesh generation methods (Chapters 5 to 7) and as will also be seen in the chapters devoted to curve and surface meshing (Chapters 14 and 15), as well as in the sections dealing with h, p and hp-methods (Chapters 21 and 22), lengths, distances and other metric-like relations play an important role and are key features in numerous mesh generation and evaluation algorithms.

From a mathematical point of view, the definition of the length of a given vector (resp. a segment) or, similarly, of the distance between two points is based on an adequate definition of the dot product. Algebraic results indicate that this product is related to a quadratic form (associated with a bilinear form). Depending on the objectives, various definitions of these notions can be exhibited, therefore leading to various definitions of a metric.

This chapter begins with some elementary reviews of quadratic forms (which can be found in textbooks), then the notion of length and metric are introduced and explained. The definition of the unit length is introduced and discussed in detail as a simple way to measure the length of a given item (segment, vector, etc.) with respect to a given metric. Examples of metrics are given to emphasize the different types of control that can be applied based on the previous notions.

Different metric-related operators are then suggested. They allow us to apply the various metric...