The aim of the work is to present construction and some characteristics of Möbius planes. We begin with introducing their concept through the Euclidean plane, their classification into a wider class of circle planes and their connection with affine planes. Continuing with generalised construction over any chosen field and a geometric theorem, that characterizes said contruction, will lead to a space model of Möbius planes. With it, an even wider class of examples will be given as plane sections of ovoids. Final chapter deals with a characteristic mapping circle inversion and its properties, some geometric constructions and a metric for measuring distance between circles.
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