Schwarz-Christoffel transformation

This geometry can be transformed into a half-plane (fig. 3) by means of a Schwarz-Christoffel transformation. 

The conformal transformation is an approach to calculate complex potential fields by means of transmapping two holomorphic functions such as z and w, as described earlier. The potential field on the z-plane can be obtained if a function w = f z) is known. When the potential field is not described by a known function, it can be analyzed by the Schwarz-Christoffel transformation, which is outlined here. 

Conformal mapping (the Schwarz-Christoffel transformation) was employed to calculate conductivity from experimentally measured biofilm resistance. The details of the conformal mapping procedure are presented elsewhere [53, 58], Variables are defined in Figure 7.3a. In the limiting cases a [Pg.231]

Figure 10.5 Schematic representation of band electrodes in real space and after Schwarz-Christoffel transformations. Double-band assembly in real space (A) and conformal space (B, C) for steady-state (B) and non-steady-state (C) conditions. Interdigitated array of band electrodes in real...

The Schwarz-Christoffel transformation from the w-plane to the Z-plane is... 

The upper half of the r-plane is transformed in the z-plane conformally by the Schwarz-Christoffel mapping as follows. 

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