Ebook: Eighteen Essays in Non-Euclidean Geometry

This book consists of a series of self-contained essays in non-Euclidean geometry in a broad sense, including the classical geometries of constant curvature (spherical and hyperbolic), de Sitter, anti-de Sitter, co-Euclidean, co-Minkowski, Hermitian geometries, and some axiomatically defined geometries. Some of these essays deal with very classical questions and others address problems that are at the heart of present day research, but all of them are concerned with fundamental topics.

All the essays are self-contained and most of them can be understood by the general educated mathematician. They should be useful to researchers and to students of non-Euclidean geometry, and they are intended to be references for the various topics they present.

Keywords: Non-Euclidean geometry, spherical geometry, hyperbolic geometry, Busemann type geometry, curvature, geographical map, non-euclidean area, non-euclidean volume, Brahmagupta’s formula, Ptolemy’s theorem, Casey’s theorem, Sforza’s formula, Seidel’s problem, infinitesimal rigidity, static rigidity, Pogorelov map, Maxwell–Cremona correspondence, exterior hyperbolic geometry, de Sitter geometry, non-Euclidean conics, bifocal properties, focus-directrix properties, pencils of conics, projective geometry, convexity, duality, transition, Hermitian trigonometry, complex projective trigonometry, shape invariant, metric plane projective-metric plane