On a class of homographies of the hyperbolic plane on its spherical model. (English) Zbl 0868.51020
The paper deals with the real hyperbolic plane and compiles facts that can be found in text-books on this subject. See e.g. W. Killing, ‘Grundlagen der Geometrie’ (F. Schöningh) (1893); D. Hilbert, ‘Neue Begründung der Bolyai-Lobatschefskyschen Geometrie’, Math. Ann. 57, 137-150 (1903); F. Bachmann, ‘Aufbau der Geometrie aus dem Spiegelungsbegriff (Springer) (1959; Zbl 0085.14502); R. S. Millman and G. D. Parker, ‘Geometry, A Metric Approach with Models (Springer) (1981; Zbl 0484.51019); and E. M. Schröder, Vorlesungen über Geometrie (Band 1, BI) (1991; Zbl 0754.51002).
Some statements are wrong, e.g. “the rotations, the horocyclic motions and the translations are abelian groups” (page 4).
Some statements are wrong, e.g. “the rotations, the horocyclic motions and the translations are abelian groups” (page 4).
Reviewer: F.Knüppel (Kiel)
MSC:
| 51M10 | Hyperbolic and elliptic geometries (general) and generalizations |
Keywords:
real hyperbolic planeReferences:
| [1] | Pasquali-Coluzzi D., Sulle simmetrie assiali del piano iperbolico (1978) · Zbl 0418.51006 |
| [2] | Pasquali-Coluzzi D., On some properties of the hyperbolic plane directly on its spherical model · Zbl 0880.51009 |
| [3] | Pasquali-Coluzzi D., La trigonometria e le simmetrie del piano iperbolico (1979) · Zbl 0427.51009 |
| [4] | Discuonzo V., Annali di Matematica pura ed applicata (IV) pp 93– (1974) · Zbl 0281.50014 · doi:10.1007/BF02414015 |
| [5] | Guggenheimer H. W., Plane Geometry and its groups · Zbl 0147.38801 |
| [6] | Coxeter H. S. M., Mathematical Exposition 2, in: Non-Euclidean Geometry |
| [7] | Kerekjarto B., Les fondaments de la géometrie (1966) |
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