Two-dimensional EP Geometry in Terms of Cayley-Klein
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According to classification of Cayley-Klein, EP Geometry is one of the nine two-dimensional geometries, i.e. geometry which is dual to the classical Euclidean (PE) geometry. The first chapter contains a system of axioms of this geometry, not the Hilbert's system and principle of duality. Fundamental objects (points and straight lines) and fundamental relations (of incidence, separateness of point couples and of line segment congruence) are defined by four groups of axioms (axioms of incidence, order, continuity and congruence). Based on the proposed system of axioms, some important consequence...
According to classification of Cayley-Klein, EP Geometry is one of the nine two-dimensional geometries, i.e. geometry which is dual to the classical Euclidean (PE) geometry. The first chapter contains a system of axioms of this geometry, not the Hilbert's system and principle of duality. Fundamental objects (points and straight lines) and fundamental relations (of incidence, separateness of point couples and of line segment congruence) are defined by four groups of axioms (axioms of incidence, order, continuity and congruence). Based on the proposed system of axioms, some important consequences are proved, and some more important geometric figures and their features are defined. The second chapter is about central and axial symmetry of EP plane. The third one defines transformation of congruence and its basic characteristics. Also, the relation of figure congruence is defined. Theorems are proved about the provision according to which congruence of certain geometric figures (angle, triangle, monogon) is defined. The end of the book is dedicated to the proof of logical non-contradiction of a given axiom system, through a projective model.
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