step into a universe where straight lines bend, angles shrink, and infinite parallels redefine common sense. this definitive textbook on non-euclidean space delivers everything you need to see, measure, and command the geometry of negative curvature-whether you are a mathematics major, a graduate researcher, a theoretical physicist, or an artist chasing escher-like tessellations.what sets this book apart- complete model coverage - poincaré disk, upper half-plane, and klein projective models explained side-by-side with crystal-clear mappings.- proof-driven approach - from the independence of euclid's fifth postulate to möbius transformation matrices, every theorem is demonstrated step-by-step.- hyperbolic trigonometry toolkit - laws of sines, cosines, and right-triangle identities ready for direct calculation.- geodesics & isometries - master reflections, translations, rotations, and the elliptic-parabolic-hyperbolic classification that powers modern geometric group theory.- tessellations & tilings - build regular {p, q} mosaics, explore fundamental domains, and unlock the symmetry groups that tile the plane forever.- curvature & metrics - understand gaussian curvature, metric tensors, and why circles and areas explode exponentially with radius.- classroom to research ready - ideal for advanced undergraduates as a second course in geometry and a self-contained reference for graduate study or independent exploration.who will benefit- mathematicians seeking a rigorous yet accessible foundation.- physicists and cosmologists modeling negatively curved space-time.- computer scientists working on visualization, graphics, and network theory.- educators needing a structured curriculum packed with proofs, diagrams, and problem sets.- artists and designers inspired by hyperbolic tessellations and non-euclidean perspective.equip yourself with the knowledge to navigate curved space, calculate precise distances, and design extraordinary tilings that only hyperbolic geometry makes possible.