A geometric model of linear fractional transformations

Abstract

A model of linear fractional transformations of the complex plane in the form of points of the complex three-dimensional projective space without a linear “forbidden” quadric is presented. A model of real linear fractional transformations of the complex plane in the form of points of the real three-di­mensional projective space without a linear “forbidden” quadric is presented. A geometric separation of points corresponding to parabolic, hyperbolic and elliptic real linear fractional transformations by a “parabolic” cone touching the forbidden quadric is found. Some pro­per­ties of model points corresponding to real linear fractional transfor­ma­tions are found. Some properties of model points corresponding to fun­da­men­tal groups transformations of biconnected domains of the complex pla­ne are found.

Keywords

• linear fractional transformation
• complex plane
• three-dimensional projective space
• forbidden quadric
• parabolic cone
• fundamental group