Topic
Möbius plane
About: Möbius plane is a research topic. Over the lifetime, 14 publications have been published within this topic receiving 97 citations.
Papers
Book•
3 Nov 1995
TL;DR: In this paper, the authors define and notation models for classical circle planes based on the Lie Geometry of a Mobius Plane and Antiregular Quadrangles, as well as their corresponding structures.
Abstract: 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 Mobius Planes and Antiregular Quadrangles Introduction The Lie Geometry of a Mobius Plane The Lifted Lie Geometry of a Flat Mobius Plane Constructing Topologies on the Lifted Lie Geometry Characterizing Quadrangles Obtained from Flat Mobius 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 Mobius Planes Sisters of Minkowski Planes The Problem of Apollonius Introduction The Problem of Apollonius in Laguerre Planes The Problem of Apollonius in Mobius 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
34 citations
TL;DR: Lower bounds are given for the number of lines blocked by a set of q + 2 points in a projective plane of order q and implications are discussed to the theory of blocking sets.
22 citations
1 Jan 1994
TL;DR: Polster and Steinke as mentioned in this paper proved that a Laguerre plane or Minkowski plane with a given topology on the point set is a 2-dimensional plane if and only if each derived affine plane at points of at least one parallel class is 2-dimensions with respect to the induced topology.
Abstract: We give some easy to use criteria for deciding whether a Mobius plane, Laguerre plane or Minkowski plane, given in some normal form, is 2-dimensional. As an application of our results we prove that a Laguerre plane or Minkowski plane with a given topology on the point set is a (topological) 2-dimensional plane if and only if each derived affine plane at points of at least one parallel class is 2-dimensional with respect to the induced topology. To appear in Beitreige fur Algebra and Geometrie. CRITERIA FOR TWO-DIMENSIONAL CIRCLE PLANES B. POLSTER AND G. F. STEINKE Department of Mathematics University of Canterbury Christchurch, New Zealand ABSTRACT. We give some easy to use criteria for deciding whether a Mobius plane, Laguerre We give some easy to use criteria for deciding whether a Mobius plane, Laguerre plane or Minkowski plane, given in some normal form, is 2-dimensional. As an application of our results we prove that a Laguerre plane or Minkowski plane with a given topology on the point set is a (topological) 2-dimensional plane if and only if each derived affine plane at points of at least one parallel class is 2-dimensional with respect to the induced topology.
17 citations
TL;DR: In this article, it was shown that Miquel's theorem for circles of equal radii is true in every normed plane, without the assumptions of strict convexity and smoothness.
Abstract: Asplund and Grunbaum proved that Miquel’s six-circles theorem holds in strictly convex, smooth normed planes if the considered circles have equal radii. We extend this result in two directions. First we prove that Miquel’s theorem for circles of equal radii (more precisely, a generalized version of it) is true in every normed plane, without the assumptions of strict convexity and smoothness, and give also some properties of the circle configuration related to this theorem. Second we clarify the situation if the circles of the corresponding configuration do not necessarily have equal radii.
8 citations
TL;DR: In this article, a family of locally compact 2D topological Mobius planes is introduced, which can be obtained by pasting together two halves of the classical real Mobius plane suitably.
Abstract: A new rather large family of locally compact 2-dimensional topological Mobius planes is introduced here. This family consists exactly of those Mobius planes which can be obtained by pasting together two halves of the classical real Mobius plane suitably. Isomorphism classes, automorphisms, and the Hering type of these planes are determined.
5 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2021 | 2 |
| 2020 | 1 |
| 2010 | 1 |
| 2008 | 1 |
| 1999 | 1 |
| 1995 | 1 |