Some connections between Minkowski and hyperbolic planes

• Autorzy: Kosiorek J. , Matraś A.
• Języki publikacji: EN

Abstrakty

EN

The model of the Minkowski plane in the projective plane with a fixed conic sheds a new light on the connection between the Minkowski and hyperbolic geometries. The construction of the Minkowski plane in a hyperbolic plane over a Euclidean field is given. It is also proved that the geometry in an orthogonal bundle of circles is hyperbolic in a natural way.

Słowa kluczowe

EN

symmetric Minkowski plane; hyperbolic plane; projectivity on a conic

PL

płaszczyzna symetryczna Minkowskiego

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2004
• Tom: Vol. 52, nr 2
• Strony: 133--139
• Opis fizyczny: Bibliogr. 12 poz., rys.

Twórcy

autor

Kosiorek J.

• Faculty of Mathematics and Computer Science, University of Warmia and Mazury, Żołnierska 14, 10-561 Olsztyn, Poland

autor

Matraś A.

• Faculty of Mathematics and Computer Science, University of Warmia and Mazury, Żołnierska 14, 10-561 Olsztyn, Poland

Bibliografia

• [1] F. Bachman, Aufbau der Geometrie aus dem Spiegelungsbegriff, Springer, Berlin, 1973.
• [2] H. Beck, Ein Seitenstück zur Moebius'schen Geometrie der Kreisverwandtschaften, Trans. Amer. Math. Soc. 11 (1910), 414-448.
• [3] W. Benz, Vorlesungen über Geometrie der Algebren, Springer, Berlin, 1973.
• [4] H. S. M. Coxeter, Non-Euclidean Geometry, Math. Expositions 2, Univ. of Toronto Press, Toronto, 1978.
• [5] J. Ch. Fischer, Models and theorems of the classical circle planes, Abh. Math. Sem. Univ. Hamburg 63 (1993), 245-264.
• [6] W. Heise und H. Karzel, Symmetrische Minkowski-Ebenen, J. Geometry 3 (1973), 5-20.
• [7] H. Karzel, Symmetrische Permutationsmengen, Aequationes Math. 17 (1978), 83-90.
• [8] J. Kosiorek and A. Matraś, The model of the Minkowski plane in the projective plane with a fixed conic, submitted to Algebra Geom. Appl.
• [9] M. Kordos, The Foundations of the Projective and Projective-Metric Geometry, PWN, Warszawa, 1984 (in Polish).
• [10] H. Mäurer, Laguerre- und Blaschke-Modell der ebenen Laguerre-Geometrie, Math. Ann. 164 (1966), 124-132.
• [11] O. Veblen and J. W. Young, Projective Geometry, Vol. 1, Boston, 1910.
• [12] H. Zeitler, On reections in Minkowski planes, in: Lecture Notes in Math. 792, Springer, 1980, 183-201.