Overview

In this chapter, the significance of the Beltrami equation for mesh enhancement will be discussed. Specifically, the Beltrami equation in the real domain will be derived in the context of the isothermal coordinates problem. The equation will then be re-derived in the complex form, which is more useful to formulate its solution. There is a link between the Beltrami coefficient and Teichm?ller's theory, used extensively in modern physics. A generalization of isothermal coordinates to higher-dimensional spaces leads naturally to the harmonic coordinate system.

The significance of quasi-conformal mapping has long been recognized in grid enhancement problems [Belinskii et al., 1975; Mastin and Thompson, 1984]. To the extent that conformal transformations have been used to generate orthogonal curvilinear difference meshes, quasi-conformal transformations often provide a better distribution of mesh lines or simplify the grid equations to be solved in the canonical domain [Mastin and Thompson, 1978]. The relationship between the Beltrami equation and the concept of a quasi-conformal mapping involves several logical steps. First, given a two-dimensional surface whose metric properties in curvilinear coordinates ( u, v) are described by the metric tensor g _??( u, v), one may derive new coordinates ( ?, ?) in which the metric tensor is proportional to a Euclidean metric. In these coordinates, termed isothermal coordinates, the line element becomes ds ² = ? ( ?, ?)( d ? ² + d ? ²). Each of the new coordinates, viewed as a function...