Circles, Spheres and Spherical Geometry
Series: Birkhäuser Advanced Texts Basler Lehrbücher;
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Product details:
- Edition number 2024
- Publisher Springer Nature Switzerland
- Date of Publication 10 August 2024
- Number of Volumes 1 pieces, Book
- ISBN 9783031627750
- Binding Hardback
- No. of pages336 pages
- Size 235x155 mm
- Language English
- Illustrations XIV, 336 p. 85 illus., 1 illus. in color. Illustrations, black & white 704
Categories
Long description:
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.
MoreTable of Contents:
- 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.
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