Circles Spheres And Spherical Geometry
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Circles, Spheres and Spherical Geometry
This textbook focuses on the geometry of circles, spheres, and spherical geometry. Various classic themes are used as introductory and motivating topics. The book begins very simply for the reader in the first chapter discussing the notions of inversion and stereographic projection. Here, various classical topics and theorems such as Steiner cycles, inversion, Soddy's hexlet, stereographic projection and Poncelet's porism are discussed. The book then delves into Bend formulas and the relation of radii of circles, focusing on Steiner circles, mutually tangent four circles in the plane and other related notions. Next, some fundamental concepts of graph theory are explained. The book then proceeds to explore orthogonal-cycle representation of quadrangulations, giving detailed discussions of the Brightwell-Scheinerman theorem (an extension of the Koebe-Andreev-Thurston theorem), Newton’s 13-balls-problem, Casey’s theorem (an extension of Ptolemy’s theorem) and its generalizations. The remainder of the book is devoted to spherical geometry including a chapter focusing on geometric probability on the sphere. The book also contains new results of the authors and insightful notes on the existing literature, bringing the reader closer to the research front. Each chapter concludes with related exercises of varying levels of difficulty. Solutions to selected exercises are provided. This book is suitable to be used as textbook for a geometry course or alternatively as basis for a seminar for both advanced undergraduate and graduate students alike.
A Treatise on the Circle and the Sphere
Author: Julian Lowell Coolidge
language: en
Publisher: American Mathematical Soc.
Release Date: 2004
Circles and spheres are central objects in geometry. Mappings that take circles to circles or spheres to spheres have special roles in metric and conformal geometry. An example of this is Lie's sphere geometry, whose group of transformations is precisely the conformal group. Coolidge's treatise looks at systems of circles and spheres and the geometry and groups associated to them. It was written (1916) at a time when Lie's enormous influence on the field was still widely felt. Today, there is a renewed interest in the geometry of special geometric configurations. Coolidge has examined many of the most intuitive: linear systems of circles, circles orthogonal to a given sphere, and so on. He also examines the differential and projective geometry of the space of all spheres in a given space.Through the simple vehicles of circles and spheres, Coolidge makes contact with diverse areas of mathematics: conformal transformations and analytic functions, projective and contact geometry, and Lie's theory of continuous groups, to name a few. The interested reader will be well rewarded by a study of this remarkable book.
Menelaus' ›Spherics‹
Author: Roshdi Rashed
language: en
Publisher: Walter de Gruyter GmbH & Co KG
Release Date: 2017-10-23
Despite its importance in the history of Ancient science, Menelaus’ Spherics is still by and large unknown. This treatise, which lies at the foundation of spherical geometry, is lost in Greek but has been preserved in its Arabic versions. The reader will find here, for the first time edited and translated into English, the essentials of this tradition, namely: a fragment of an early Arabic translation and the first Arabic redaction of the Spherics composed by al-Māhānī /al-Harawī, together with a historical and mathematical study of Menelaus’ treatise. With this book, a new and important part of the Greek and Arabic legacy to the history of mathematics comes to light. This book will be an indispensable acquisition for any reader interested in the history of Ancient geometry and science and, more generally, in Greek and Arabic science and culture.