Hyperbolic Geometry: The All in One Textbook

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 - Poincar?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 M鐽ius 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.