×

zbMATH — the first resource for mathematics

On elementary circle-geometry in Cayley-Klein planes. (English) Zbl 1236.51002
Summary: This contribution can be seen as an addendum to a recently paper published by H. Martini and M. Spirova on circle-geometries in Cayley-Klein planes [Publ. Math. 72, No. 3–4, 371–383 (2008; Zbl 1174.51005)], as it deals with further generalisations and extensions of the results therein to circle-geometries in all Cayley-Klein planes. The main methods in this paper are the interpretation of planar figures in space and dualizing according to the duality principle of projective spaces. There are, in principle, only three types of \(\mathbb R^{2}\)-ring structures and, thus, only three types of corresponding circle-geometries [W. Benz, Vorlesungen über Geometrie der Algebren. Geometrien von Möbius, Laguerre-Lie, Minkowski in einheitlicher und grundlagengeometrischer Behandlung. Berlin-Heidelberg-New York: Springer-Verlag (1973; Zbl 0258.50024)]. Therefore, each generalisation to non-Euclidean planes must turn out to be just another representation of the classical Euclidean cases. This point of view gives more insight into why some elementary geometric theorems remain valid when changing the place of action from the Euclidean plane to non-Euclidean circle planes and makes explicit proofs of such elementary geometric theorems in non-Euclidean circle planes superfluous. We believe that even the Euclidean cases of circle-geometries comprise, in principle, already all non-Euclidean cases. Representations of such non-Euclidean circle-geometries might also be of interest in their own. For example, among the planar Cayley-Klein geometries the quasi-elliptic and quasi-hyperbolic geometry usually are neglected. They can be treated, similar to the isotropic Möbius geometry, by suitable projections of the Blaschke cylinder.

MSC:
51A25 Algebraization in linear incidence geometry
51M09 Elementary problems in hyperbolic and elliptic geometries
51M15 Geometric constructions in real or complex geometry
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Benz W.: Geometrie der Algebren. Springer, Heidelberg (1973a) · Zbl 0253.50017
[2] Benz W.: Zum Büschelsatz in der Geometrie der Algebren. Monatshefte f. Math. 77, 1–9 (1973b) · Zbl 0253.50017 · doi:10.1007/BF01300522
[3] Brauner H.: Geometrie projektiver Räume I. Bibliographisches Institut, Mannheim (1976a) · Zbl 0332.50010
[4] Brauner H.: Geometrie projektiver Räume II. Bibliographisches Institut, Mannheim (1976b) · Zbl 0336.50002
[5] Brauner H.: Kreisgeometrie in der isotropen Ebene. Monatshefte f. Math. 69(2), 105–128 (1965) · Zbl 0141.19002 · doi:10.1007/BF01298316
[6] Giering O.: Vorlesungen über höhere Geometrie. Vieweg, Braunschweig-Wiesbaden (1987) · Zbl 0493.51001
[7] Herzer A.: Büschelsätze zur Charakterisierung projektiv darstellbarer Zykelebenen. Math. Z. 164, 215–238 (1979) · Zbl 0384.51005 · doi:10.1007/BF01182268
[8] Hirano, K.: On some centre-circles and their relations. Sugaku 8, 210–211 (1956/57)
[9] Hoffman A.J.: Chains in the projective line. J. Duke Math. 18, 827–830 (1951) · Zbl 0044.35201 · doi:10.1215/S0012-7094-51-01877-7
[10] Kantor S.: Über das vollständige Fünfseit. Sitzungsberichte der Kgl. Akademie der Wissenschaften in Wien 78, 165–174 (1878) · JFM 10.0385.02
[11] Kovačević, N., Jurkin, E.: Some remarks on Clifford’s chain of theorems in affine CK-planes (submitted, 2010)
[12] Martini H., Spirova M.: Circle-geometry in affine Cayley–Klein planes. Periodica Mathematica Hungarica 57, 197–206 (2008a) · Zbl 1199.51013 · doi:10.1007/s10998-008-8197-5
[13] Martini H., Spirova M.: Clifford’s chain of theorems in strictly convex Minkowski planes. Publicationes Mathematicae Debrecen 72, 371–383 (2008b) · Zbl 1174.51005
[14] Morley F.: Extensions of Clifford’s chain-theorem. Am. J. Math. 51, 465–472 (1929) · JFM 55.0362.02 · doi:10.2307/2370734
[15] Müller E., Krames L.: Die Zyklographie. Vorlesungen über Darstellende Geometrie. Deuticke, Leipzig (1929)
[16] Pottmann H., Wallner J.: Computational Line Geometry. Springer, Berlin (2001) · Zbl 1006.51015
[17] Samaga, H.-J.: A Unified Approach to Miquel’s Theorem and its Degenerations. Lecture Note in Math. vol. 792, pp. 132–142. Springer, Berlin (1980) · Zbl 0433.51005
[18] Šimić, M.: Power of the point and the line with respect to the circle in the isotropic plane. Conference Contribution. Conference on Geometry: Theory and Application, Plsen (2009)
[19] Wagner R.: Interne Kennzeichnung projektiver Abbildungen auf Quadriken. Math. Z. 77(1), 94–100 (1961) · Zbl 0102.15703 · doi:10.1007/BF01180165
[20] Weiss G.: Einige ebene Schließungssätze und mögliche Erweiterungen derselben. Wissenschaftliche Zeitschrift der TU Dresden 45(Heft 6), 18–22 (1996)
[21] Weiss G., Nestler K., Meinl G.: Some Moebius-geometric theorems connected to Euclidean kinematics. J. Geom. Grap. 3(2), 183–191 (1999) · Zbl 0979.53016
[22] Zeitler H.: Kreisgeometrie in Schule und Wissenschaft oder: Klassische und moderne Kreisgeometrie. Did. d. Math. 11(3), 169–201 (1983)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.