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## Demonstratio Mathematica

Tytuł artykułu

### The invariants of a pair of directions in geometry [En1] and their interpretation

Autorzy Misiak, A.  Stasiak, E.  Szmuksta-Zawadzka, M.
Treść / Zawartość http://www.degruyter.com/view/j/dema
Warianty tytułu
Języki publikacji EN
Abstrakty
 EN Solving a certain functional equation, we find all invariants of a pair of directions in n-dimensional pseudo-Euclidean geometry of index one [En1]. In (n-1)-dimensional space we construct a model for these directions by means of concepts characteristic of Euclidean geometry. Because it is a pseudo-orthogonal group, not orthogonal, that operates in this model, the distance between two points and the measure of an angle are not invariants. Using these changeable quantities we construct invariant quantities.
Słowa kluczowe
 PL geometria pseudoeuklidesowa   geometria nieeuklidesowa   skalar EN pseudo-Euclidean geometry   scalar   invariant mapping
Wydawca De Gruyter
Czasopismo Demonstratio Mathematica
Rocznik 2013
Tom Vol. 46, nr 2
Strony 361--371
Opis fizyczny Bibliogr. 9 poz.
Twórcy
 autor Misiak, A. School of Mathematics, West Pomeranian University of Technology, Al. Piastów 17, 70-310 Szczecin, Poland, misiak@ps.pl autor Stasiak, E. autor Szmuksta-Zawadzka, M. School of Mathematics, West Pomeranian University of Technology, Al. Piastów 17, 70-310 Szczecin, Poland, mszmuksta@zut.edu.pl
Bibliografia
[1] J. Aczél, S. Gołąb, Functionalgleichungen der Theorie der geometrischen Objekte, P.W.N. Warszawa 1960. Zbl 0100.32901
[2] L. Bieszk, E. Stasiak, Sur deux formes équivalents de la notion de (r,s)-orientation de la géometrié de Klein, Publ. Math. Debrecen 35 (1988), 43–50.
[3] B. Glanc, A. Misiak, Z. Stępień, Equivariant mappings from vector product into G-space of vectors and ε-vectors with G = O(n, 1, R), Math. Bohem. 130 (2005), 265–275.
[4] B. Glanc, A. Misiak, M. Szmuksta-Zawadzka, Equivariant mappings from vector product into G-spaces of φ-scalars with G = O(n – 1,1), Math. Bohem. 132 (2007), 325–332.
[5] M. Kucharzewski, Über die Grundlagen der Kleinschen Geometrie, Period. Math. Hungar. 8(1) (1977), 83–89. Zbl 0335.50001
[6] A. Misiak, E. Stasiak, Equivariant maps between certain G-spaces with G = O(n – 1,1), Math. Bohem. 126 (2001), 555–560. Zbl 1031.53031
[7] A. Misiak, E. Stasiak, G-space of isotropic directions and G-spaces of φ-scalars with G = O(n – 1,1), Math. Bohem. 133 (2008), 289–298.
[8] E. Stasiak, On a certain action of the pseudoorthogonal group with index one O(n, 1, R) on the sphere Sn2 ( Polish ), Prace Naukowe P. S. 485 (1993).
[9] E. Stasiak, Scalar concomitants of a system of vectors in pseudo-Euclidean geometry of index 1, Publ. Math. Debrecen 57 (2000), 55–69. Zbl 0966.53012
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