Topological Circle Planes and Topological Quadrangles: 1st Edition (Hardback) book cover

Topological Circle Planes and Topological Quadrangles

1st Edition

By Andreas E Schroth

Chapman and Hall/CRC

168 pages

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Hardback: 9780582288119
pub: 1995-11-03
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Description

This research note presents a complete treatment of the connection between topological circle planes and topological generalized quadrangles. The author uses this connection to provide a better understanding of the relationships between different types of circle planes and to solve a topological version of the problem of Apollonius.

Topological Circle Planes and Topological Quadrangles begins with a foundation in classical circle planes and the real symmetric generalized quadrangle and the connection between them. This provides a solid base from which the author offers a more generalized exploration of the topological case. He also compares this treatment to the finite case.

Subsequent chapters examine Laguerre, Möbius, and Minkowski planes and their respective relationships to antiregular quadrangles. The author addresses the Lie geometry of each and discuss the relationships of circle planes-the "sisters" of Möbius, Laguerre, and Minkowski planes - and concludes by solving a topological version of the problem of Apollonius in Laguerre, Möbius, and Minkowski planes.

The treatment offered in this volume offers complete coverage of the topic. The first part of the text is accessible to anyone with a background in analytic geometry, while the second part requires basic knowledge in general and algebraic topology. Researchers interested in geometry-particularly in topological geometry-will find this volume intriguing and informative. Most of the results presented are new and can be applied to various problems in the field of topological circle planes.

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Reviews

"This book is a must read for anyone interested in incidence geometry and especially anybody interested in topological incidence geometry."

-Mathematical Reviews, Issue 97b

Table of Contents

Introduction

Circle Planes

Introduction

Definitions and Notation

Models for Classical Circle Planes

Derived Structures

Antiregular Quadrangles

Introduction

Generalized Quadrangles

Square Projections

The Twisting Number

Antiregular Quadrangles

Characterization of Antiregular Quadrangles

Laguerre Planes and Antiregular Quadrangles

Introduction

Laguerre Planes Constructed from Antiregular Quadrangles

Antiregular Quadrangles Constructed from Laguerre Planes

Constructing Topologies on the Lie Geometry

Möbius Planes and Antiregular Quadrangles

Introduction

The Lie Geometry of a Möbius Plane

The Lifted Lie Geometry of a Flat Möbius Plane

Constructing Topologies on the Lifted Lie Geometry

Characterizing Quadrangles Obtained from Flat Möbius Planes

Minkowski Planes and Antiregular Quadrangles

Introduction

The Point Space and Parallel Classes

The Circle Space

The Other Spaces

The Derivation of a Minkowski Plane

The Lie Geometry of a Minkowski Plane

The Lifted Lie Geometry of a Minkowski Plane

The Topology on the Lifted Lie Geometry

Characterizing Quadrangles Obtained from Minkowski Planes

Relationship of Circle Planes

Introduction

Sisters of Laguerre Planes

Sisters of Möbius Planes

Sisters of Minkowski Planes

The Problem of Apollonius

Introduction

The Problem of Apollonius in Laguerre Planes

The Problem of Apollonius in Möbius Planes

One Point and Two Circles

Three Circles

The Problem of Apollonius in Minkowski Planes

Two Points and One Circle

One Point and Two circles

Three Circles

Index

Glossary

References

About the Series

Chapman & Hall/CRC Research Notes in Mathematics Series

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Subject Categories

BISAC Subject Codes/Headings:
MAT012000
MATHEMATICS / Geometry / General