Die euklidische Ebene und ihre Verwandten

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The text outlines a comprehensive exploration of projective and affine planes, beginning with definitions and initial results, and progressing through incidence-preserving mappings and central collineations, including Desargues' theorem. It delves into Desarguesian planes, discussing translation planes and their core structures, as well as dual planes and structural theorems. The study extends to Pappus planes, examining Hessenberg's theorem, groups of projective collineations, projectivities, and the concept of double ratios. Further, it addresses polarities and conic sections, detailing finite projective plane polarities, representations, conic generation by Steiner, and Segre's theorem on ovals. The text also covers partial ratios and orthogonality in affine planes, discussing midpoints, orthogonality relations in Pappus planes, and the angle bisector theorem. Metrical properties of conic sections are examined, including projective planes over Euclidean fields, conics in affine planes, circles, axes, foci, and algebraic descriptions of ellipses, parabolas, and hyperbolas. Finally, it addresses the real plane, exploring interrelations, arrangements, characterizations of body arrangements, and properties of Desarguesian affine planes.