1st Edition

Topological Circle Planes and Topological Quadrangles

By Andreas E Schroth Copyright 1995
    166 Pages
    by Chapman & Hall

    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.

    Features

    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

    Biography

    Andreas E Schroth

    "This book is a must read for anyone interested in incidence geometry and especially anybody interested in topological incidence geometry."
    -Mathematical Reviews, Issue 97b