ABSTRACT
The mapping from one complex plane to another through a holomorphic function has the conformal property of preserving the modulus and direction of the angles between two curves (Section 33.1); the verification that an isogonal mapping preserves the modulus but reverses the direction of angles, is specified by a nonholomorphic function, leads to the proof that the conformal property is exclusive of holomorphic mappings (Section 33.2). The isogonal and conformal mappings share the property of preserving the modulus of angles that is equivalent to preserving the ratio of lengths (Section 33.3); the latter may vary from one point to another, but is independent of direction at a fixed point. At the poles (zeros) of the function the conformal property is broken, since the ratio of lengths is infinite (zero), and a point becomes a neighborhood (vice-versa); at such critical points of the second (first) kind (Section 33.4), the angles are (Section 33.5) divided (multiplied) by an integer, leading to the appearance of edges (reversals of paths). By selecting suitably the location and exponent of the critical points, it is possible to map conformally a region with a smooth boundary into another with edges, for example, the upper or lower half-plane, and the interior (Section 33.8) or exterior (Section 33.9) of a circle, may be mapped into the interior (Section 33.6) or exterior (Section 33.7) of a polygon. The selection of the location and angles at the vertices, allows the conformal mapping into a variety of regions (Chapters 34 and 36), including: (i) finite polygons, such as plates, triangles, rectangles; (ii) infinite obstacles, such as slits, channels, steps, flaps, ducts, and bends.
