LEADER 03741nam 22006015 450 001 9910879595403321 005 20240810131742.0 010 $a3-031-62776-8 024 7 $a10.1007/978-3-031-62776-7 035 $a(MiAaPQ)EBC31596005 035 $a(Au-PeEL)EBL31596005 035 $a(CKB)33830835900041 035 $a(DE-He213)978-3-031-62776-7 035 $a(EXLCZ)9933830835900041 100 $a20240810d2024 u| 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aCircles, Spheres and Spherical Geometry /$fby Hiroshi Maehara, Horst Martini 205 $a1st ed. 2024. 210 1$aCham :$cSpringer Nature Switzerland :$cImprint: Birkhäuser,$d2024. 215 $a1 online resource (342 pages) 225 1 $aBirkhäuser Advanced Texts Basler Lehrbücher,$x2296-4894 311 $a3-031-62775-X 327 $a- Inversion and stereographic projection -- Bend formulas -- Graphs and circle-systems -- Spherical geometry I -- Spherical geometry II -- The problem of thirteen balls -- Spherical geometry III -- Geometric probability on the sphere -- Intersection graphs of spherical caps -- Quartets on a sphere -- Higher dimensions -- The Cayley-Menger determinant -- Casey's theorem -- Solutions to the selected exercises. 330 $aThis textbook focuses on the geometry of circles, spheres, and spherical geometry. Various classical themes are used as introductory and motivating topics. The book begins very simply for the reader in the first chapter discussing the notions of inversion and stereographic projection. Here, various classical topics and theorems such as Steiner cycles, inversion, Soddy's hexlet, stereographic projection and Poncelet's porism are discussed. The book then delves into Bend formulas and the relation of radii of circles, focusing on Steiner circles, mutually tangent four circles in the plane and other related notions. Next, some fundamental concepts of graph theory are explained. The book then proceeds to explore orthogonal-cycle representation of quadrangulations, giving detailed discussions of the Brightwell-Scheinerman theorem (an extension of the Koebe-Andreev-Thurston theorem), Newton?s 13-balls-problem, Casey?s theorem (an extension of Ptolemy?s theorem) and its generalizations. The remainder of the book is devoted to spherical geometry including a chapter focusing on geometric probability on the sphere. The book also contains new results of the authors and insightful notes on the existing literature, bringing the reader closer to the research front. Each chapter concludes with related exercises of varying levels of difficulty. Solutions to selected exercises are provided. This book is suitable to be used as textbook for a geometry course or alternatively as basis for a seminar for both advanced undergraduate and graduate students alike. 410 0$aBirkhäuser Advanced Texts Basler Lehrbücher,$x2296-4894 606 $aGeometry 606 $aConvex geometry 606 $aDiscrete geometry 606 $aGraph theory 606 $aGeometry 606 $aConvex and Discrete Geometry 606 $aGraph Theory 615 0$aGeometry. 615 0$aConvex geometry. 615 0$aDiscrete geometry. 615 0$aGraph theory. 615 14$aGeometry. 615 24$aConvex and Discrete Geometry. 615 24$aGraph Theory. 676 $a516 700 $aMaehara$b Hiroshi$01765282 701 $aMartini$b Horst$060948 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910879595403321 996 $aCircles, Spheres and Spherical Geometry$94206697 997 $aUNINA