This unique and richly illustrated book explores barycentric calculus, a geometric method grounded in the concept of the center of gravity. Used to elegantly determine triangle centers through weighted points, barycentric coordinates have long revealed deep insights in Euclidean geometry. Now, this book extends those insights to the fascinating realm of hyperbolic geometry, building a powerful bridge between classical and modern mathematical worlds.

In Euclidean geometry, over 3,000 triangle centers have been identified using barycentric coordinates. This book introduces readers to their hyperbolic analogs, uncovering remarkable parallels between triangle centers in Bolyai-Lobachevsky geometry and their Euclidean counterparts. The author's innovative use of Cartesian coordinates, trigonometry, and vector algebra - adapted for hyperbolic geometry - equips readers with familiar yet powerful tools to explore unfamiliar terrain.

At the heart of the book is the development of hyperbolic barycentric coordinates, or gyrobarycentric coordinates, within the framework of gyrovector spaces - a novel algebraic structure emerging from Einstein's velocity addition and Möbius addition. These gyrovectors underpin the Klein and Poincaré ball models of hyperbolic geometry, just as traditional vectors underlie analytic Euclidean geometry.

Key features of this Second Edition include three new chapters with groundbreaking results:

• Chapter 8: Derives the gyrodistance between gyropoints using gyrobarycentric coordinates and reveals hyperbolic triangle center distances that naturally reduce to classical Euclidean formulas.
• Chapter 9: Investigates the duality between classical trigonometry and gyrotrigonometry, culminating in a new hyperbolic analog of Ptolemy's Theorem.
• Chapter 10: Explores cyclic antipodal segments in both Euclidean and hyperbolic settings, offering fresh perspectives and uncovering novel hyperbolic Pythagorean identities.

Whether you are a researcher in geometry, mathematical physics, or relativity, or simply fascinated by the deep structure of space, this book offers a groundbreaking approach to analytic hyperbolic geometry through barycentric and gyrobarycentric coordinates.