ABSTRACT
The properties of conformal mappings can be illustrated by considering the simplest transformations, namely, the isometric, linear, and bilinear. A conformal mapping preserves the modulus and direction of angles; if it is also isometric, that is, preserves distances, then it leaves all figures unchanged; the isometric, conformal mapping consists of rotations and translations (Section 35.1). Adding homotheties, that is, projections from the origin, leads to the linear mapping that is the most general transformation without critical points, that is, conformal everywhere (Section 35.2). The ratio of two linear transformation specifies the bilinear, homographic, or Mobius mapping (Section 35.4) that has critical points; it consists of the preceding transformation plus the inversion; it is the most general mapping univalent on the whole complex plane, that is, it maps distinct points on to distinct points, and its domain and image are the whole complex plane (Section 35.5). The bilinear mappings form a group, of that the linear and isometric mappings are subgroups. A point that is mapped onto itself is called (Section 35.1) a fixed point of the transformation: (i) the isometric mapping has one fixed point at infinity (Section 35.1); (ii) the linear mapping has one fixed point at finite distance (Section 35.2); (iii) the bilinear mapping has two fixed points, distinct or coincident, at finite distance or at infinity (Section 35.6). The repeated application of these mappings makes all points tend to (or away from) the limit points that are then called attractive (repulsive); the third possibility, is that of indifferent fixed points, when successive mappings do not change the distance, that is, move the points around a circle. The linear (bilinear) mapping [Section 35.3 (35.6)] have the three types of limit points. The linear (bilinear) mapping leaves invariant a three (four) point cross-ratio [Section 35.3 (35.6)]. This is an instance of the general property of the bilinear transformation of mapping circles and straight lines into circles or straight lines (Sections 35.8 and 35.9); it relates to the inversion (Section 35.7) with regard to the straight line (circle), that is, the image (circle) methods in potential fields with plane walls (Chapters 16, 24, and 26) [cylindrical bodies (Chapters 24, 26, 28, 34, and 36)]. Besides being the most general univalent mapping, and mapping between straight lines and circles, the bilinear transformation appears in a variety of other problems, for example, the composition of spatial rotations.
