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Regular octagons in Poincare model of Lobachevsky geometry

Oleksandr Pryshliak

Abstract

To investigate the topological structure of Morse flows on the 2-disk we use the planar graphs as destinguished graph of the flow. We assume, that the flow is transversal to the boundary of the 2-disk. We give a list of all planar graph with at least 3 edges and describe all planar graphs with 4 edges. We use a list of spherical graph with at least 4 edges.

Regular octagons in Poincare model of Lobachevsky geometry

Abstract

To investigate the topological structure of Morse flows on the 2-disk we use the planar graphs as destinguished graph of the flow. We assume, that the flow is transversal to the boundary of the 2-disk. We give a list of all planar graph with at least 3 edges and describe all planar graphs with 4 edges. We use a list of spherical graph with at least 4 edges.
Paper Structure (14 sections, 21 equations, 6 figures)

This paper contains 14 sections, 21 equations, 6 figures.

Table of Contents

  1. Relevance of the topic
  2. Literature review
  3. The purpose of the paper
  4. History of emergence and basic concepts of Lobachevsky's geometry
  5. Emergence History
  6. Basic Concepts
  7. Poincaré model of Lobachevsky geometry
  8. Trigonometric relations in Lobachevsky's geometry
  9. Properties of triangles in the Lobachevsky plane
  10. Theorems of sines and cosines
  11. Right triangles
  12. Regular octagons in the Poincaré model
  13. Coordinates of vertices
  14. Visualization of octagons

Figures (6)

  • Figure 1: Right triangle in the Poincaré model
  • Figure 2: Denotation of points on a regular octagon
  • Figure 3: A regular octagon with interior angle $2\alpha = 2\pi/3$
  • Figure 4: Regular octagon with interior angle $2\alpha = \pi/2$
  • Figure 5: Regular octagon with interior angle $2\alpha = \pi/3$
  • ...and 1 more figures