This book is based on real inner product spaces X of arbitrary (finite or infinite) dimension greater than or equal to 2. Designed as a two term graduate course, the book helps students to understand great ideas of classical geometries in a modern and general context. A real benefit is the dimension-free approach to important geometrical theories. The only prerequisites are basic linear algebra and basic 2- and 3-dimensional real geometry.

The focus of this book and its geometric notions is on real vector spaces X that are finite or infinite inner product spaces of arbitrary dimension greater than or equal to 2. It characterizes both euclidean and hyperbolic geometry with respect to natural properties of (general) translations and general distances of X. Also for these spaces X, it studies the sphere geometries of Möbius and Lie as well as geometries where Lorentz transformations play the key role. Proofs of newer theorems characterizing isometries and Lorentz transformations under mild hypotheses are included, such as for instance infinite dimensional versions of famous theorems of A.D. Alexandrov on Lorentz transformations. A real benefit is the dimension-free approach to important geometrical theories. New to this third edition is a chapter dealing with a simple and great idea of Leibniz that allows us to characterize, for these same spaces X, hyperplanes of euclidean, hyperbolic geometry, or spherical geometry, the geometries of Lorentz-Minkowski and de Sitter, and this through finite or infinite dimensions greater than 1. Another new and fundamental result in this edition concerns the representation of hyperbolic motions, their form and their transformations. Further we show that the geometry (P,G) of segments based on X is isomorphic to the hyperbolic geometry over X. Here P collects all x in X of norm less than one, G is defined to be the group of bijections of P transforming segments of P onto segments. The only prerequisites for reading this book are basic linear algebra and basic 2- and 3-dimensional real geometry. This implies that mathematicians who have not so far been especially interested in geometry could study and understand some of the great ideas of classical geometries in modern and general contexts.

This book develops the geometric intuition of the reader by examining the symmetries (or rigid motions) of the space in question. This approach introduces in turn all the classical geometries: Euclidean, affine, elliptic, projective and hyperbolic. The main focus is on the mathematically rich two-dimensional case, although some aspects of 3- or $n$-dimensional geometries are included. Basic notions of algebra and analysis are used to convey better understanding of various concepts and results. Concepts of geometry are presented in a very simple way, so that they become easily accessible: the only pre-requisites are calculus, linear algebra and basic analytic geometry.

Features the classical themes of geometry with plentiful applications in mathematics, education, engineering, and science Accessible and reader-friendly, Classical Geometry: Euclidean, Transformational, Inversive, and Projective introduces readers to a valuable discipline that is crucial to understanding bothspatial relationships and logical reasoning. Focusing on the development of geometric intuitionwhile avoiding the axiomatic method, a problem solving approach is encouraged throughout. The book is strategically divided into three sections: Part One focuses on Euclidean geometry, which provides the foundation for the rest of the material covered throughout; Part Two discusses Euclidean transformations of the plane, as well as groups and their use in studying transformations; and Part Three covers inversive and projective geometry as natural extensions of Euclidean geometry. In addition to featuring real-world applications throughout, Classical Geometry: Euclidean, Transformational, Inversive, and Projective includes: Multiple entertaining and elegant geometry problems at the end of each section for every level of study Fully worked examples with exercises to facilitate comprehension and retention Unique topical coverage, such as the theorems of Ceva and Menalaus and their applications An approach that prepares readers for the art of logical reasoning, modeling, and proofs The book is an excellent textbook for courses in introductory geometry, elementary geometry, modern geometry, and history of mathematics at the undergraduate level for mathematics majors, as well as for engineering and secondary education majors. The book is also ideal for anyone who would like to learn the various applications of elementary geometry.

This textbook offers a geometric perspective on special relativity, bridging Euclidean space, hyperbolic space, and Einstein’s spacetime in one accessible, self-contained volume. Using tools tailored to undergraduates, the author explores Euclidean and non-Euclidean geometries, gradually building from intuitive to abstract spaces. By the end, readers will have encountered a range of topics, from isometries to the Lorentz–Minkowski plane, building an understanding of how geometry can be used to model special relativity. Beginning with intuitive spaces, such as the Euclidean plane and the sphere, a structure theorem for isometries is introduced that serves as a foundation for increasingly sophisticated topics, such as the hyperbolic plane and the Lorentz–Minkowski plane. By gradually introducing tools throughout, the author offers readers an accessible pathway to visualizing increasingly abstract geometric concepts. Numerous exercises are also included with selected solutions provided. Geometry: from Isometries to Special Relativity offers a unique approach to non-Euclidean geometries, culminating in a mathematical model for special relativity. The focus on isometries offers undergraduates an accessible progression from the intuitive to abstract; instructors will appreciate the complete instructor solutions manual available online. A background in elementary calculus is assumed.