The main aim of this thesis is the classification of the circular curves of the 3rd class in the quasi-hyperbolic plane and the analysis of the possibilities of their construction obtained by the line inversion, pedal transformation and projective mapping. The theory of the curves in the Euclidean plane has been given in the first chapter. In the second chapter the quasi-hyperbolic plane has been studied i.e. the basics, the types of the 2nd class curves, and notions of the circular curves of class n in the quasi-hyperbolic plane are given. The circular curves of the 3rd class have been classified according to the degree of circularity into 1-circular, 2-circular or entirely circular curves and, within these types, according to the type of circularity and the type of the isotropic lines of the curve into 18 subtypes. In each of the remaining chapters one mapping is observed - line inversion, pedal transformation and projective mapping. First, the properties of every mapping are investigated and then the conditions of obtaining the circular curves of the 3rd class by these mappings are observed. Furthermore, the connection between line inversion and pedal transformation is given and the construction of the curve dual to the curve obtained by these two mappings is explained. The synthetic and analytical methods have been used in the studies. It is shown that every subtype of the circular curves of the 3rd class can be obtained by projective mapping and five subtypes can be obtained by pedal transformation, while only one subtype of the circular curves of the 3rd class can not be obtained by line inversion.