Ebook: Eighteen essays in Non-Euclidean geometry

Foreword / Vincent Alberge and Athanase Papadopoulos -- Introduction / Vincent Alberge and Athanase Papadopoulos -- Prologue: An excerpt from "Sur les hypothèses fondamentales de la géométrie" / Henri Poincaré (translated by Athanase Papadopoulos) -- Part I. Spherical and hyperbolic geometries -- Chapter 1. Area in non-Euclidean geometry / Norbert A'Campo and Athanase Papadopoulos -- Chapter 2. The area formula for hyperbolic triangles / Elena Frenkel and Weixu Su -- Chapter 3. On a problem of Schubert in hyperbolic geometry / Vincent Alberge and Elena Frenkel -- Chapter 4. On a theorem of Lambert: medians in spherical and hyperbolic geometries / Himalaya Senapati -- Chapter 5. Inscribing a triangle in a circle in spherical geometry / Himalaya Senapati -- Chapter 6. Monotonicity in spherical and hyperbolic triangles / Himalaya Senapati -- Chapter 7. De Tilly's mechanical view on hyperbolic and spherical geometries / Dmitriy Slutskiy -- Chapter 8. The Gauss-Bonnet theorem and the geometry of surfaces / Son Lam Ho -- Chapter 9. On the non-existence of a perfect map from the 2-sphere to the Euclidean plane / Charalampos Charitos and Ioannis Papadoperakis -- Chapter 10. Area preserving maps from the sphere to the Euclidean plane / Charalampos Charitos -- Chapter 11. Area and volume in non-Euclidean geometry / Nikolay Abrosimov and Alexander Mednykh -- Chapter 12. Statics and kinematics of frameworks in Euclidean and non-Euclidean geometry / Ivan Izmestiev -- Part II. Projective geometries -- Chapter 13. Contributions to non-Euclidean geometry I / Eduard Study (translated by Annette A'Campo-Neuen) -- Chapter 14. Notes on Eduard Study's paper "Contributions to non-Euclidean geometry I" / Annette A'Campo-Neuen and Athanase Papadopoulos -- Chapter 15. Spherical and hyperbolic conics / Ivan Izmestiev -- Chapter 16. Spherical, hyperbolic, and other projective geometries: convexity, duality, transitions / François Fillastre and Andrea Seppi -- Part III. Projective geometries -- Chapter 17. Hermitian trigonometry / Boumediene Et-Taoui -- Chapter 18. A theorem on equiareal triangles with a fixed base / Victor Pambuccian.;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.