The infinite regions of various geometries

A Review and Mathematical Treatment of Infinity on the Smith Chart, 3D Smith Chart and Hyperbolic Smith Chart

“…Described in References [7][8][9][10], the 3D Smith chart has its roots in the group theory of inversive geometry [11] where the point at infinity seen as a pole on a sphere plays a key role. However, the 3D Smith chart has some inconveniencies, for example it cannot be easy used without a software tool, while the idea of using a 3D embedded surface may be still discouraging for the electrical engineering community which is used to the 2D Smith chart [12].…”

“…The expression (1) maps the impedances with positive normalized resistance (belonging to the right half plane (RHP)) into the limited area of the unit circle and the impedances with negative resistance are projected outside the unit circle, as we observe in Figure 1: The transformation (1) regarded throughout the references [1][2][3][4][5][6] and [12,19] as "conformal transformation", "bilinear map" or "Möbius transformation" is from a geometrical perspective a very simple direct inversive transformation [8,9,11,15]. Direct inversive transformations are defined by (2) and together with the transformations defined by (3) (indirect inversive) they form the group of inversive transformations [8,15].…”

“…Infinity on a 2D Smith chart embedded on a planar surface means Figure 1 which makes it difficult to use when loads with negative resistance occur. Facing this problem, Maxime Bocher, a student of Felix Klein and president of the American Mathematical Society, proposed in Reference [11] the general solution for all Möbius transformations and indirect inversive transformations, so that these should always map circles into circles or lines into circles. If we extended the Möbius transformation (1) such that…”

“…The reason for seeking a 3D embedded generalization was motivated by the aspiration to have a single compact chart proper for dealing with negative resistance impedances without distorting the circular shapes of the Smith chart, thus keeping its properties. Described in References [7][8][9][10], the 3D Smith chart has its roots in the group theory of inversive geometry [11] where the point at infinity seen as a pole on a sphere plays a key role.…”

“…In this article, we present the geometrical treatment of infinity on the 2D Smith chart, 3D Smith chart and further develop the equations and properties of the intuitive hyperbolic Smith chart drawing [13,14]. This is done by using the Kleinian view of geometry, that is, infinity treatment of Möbius transformations [11,15] for the 2D and 3D Smith chart and basic elements of pure Non-Euclidean geometry [16][17][18] for the hyperbolic Smith chart.…”