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.

Hiroshi Maeharais an emeritus professor of mathematics at Ryukyu University in Japan. He obtained his Ph.D. in mathematics from Tokyo University in 1986. He was a guest professor at the Institute of Statistical Mathematics in Tokyo from 1994 to 2005. He also held professor position at the Research Institute of Education Development of Tokai University in Tokyo from 2009 to 2012. His main research interests lie in discrete geometry and combinatorics.

Horst Martiniis a distinguished retired professor of mathematics from the Chemnitz University of Technology in Germany. He obtained his Ph.D. in mathematics from the University of Dresden 1988, followed by his Habilitation from the University of Jena in 1993. His research interests include convex geometry, discrete geometry, functional analysis, and classical subfields of geometry, as well as applications in optimization and related fields. His scholarly pursuits also extend to certain fields in history of mathematics. He previously served as editor in chief of the Springer journal Contributions to Algebra and Geometry. In 2015, he received an honorary professorship from the Harbin University for Science and Technology in China. His passions include travelling to interesting places of geographical and historical significance and studying different genres of good music.