Abstract
We prove that Euler’s ratio-sum formula is valid in a projective-metric space if and only if it is either elliptic, hyperbolic, or Minkowskian.
Similar content being viewed by others
References
Busemann, H.: The geometry of geodesics. Academic Press, New York (1955)
Busemann, H., Kelly, P.J.: Projective geometries and projective metrics. Academic Press, New York (1953)
Euler, L.: Geometrica et sphaerica quaedam, Memoires de lAcademie des Sciences de Saint-Petersbourg, 5 (1815), 96–114; Opera Omnia Series 1, vol. XXVI, 344–358; Original: http://eulerarchive.maa.org/docs/originals/E749.pdf; English translation: http://eulerarchive.maa.org/Estudies/E749t.pdf. Accessed 24 Sept 2018
Grünbaum, B., Klamkin, M.S.: Euler’s Ratio-Sum Theorem and Generalizations, Mathematics Magazine, 79:2, 122–130, (2006) http://www.jstor.org/stable/27642919. Accessed 24 Sept 2018
Grünbaum, B.: Cyclic ratio sums and products, Crux Mathematicorum, 24(1), 20–25, (1998) https://cms.math.ca/crux/v24/n1/page20-25.pdf. Accessed 24 Sept 2018
Kozma, J., Kurusa, Á.: Hyperbolic is the only Hilbert geometry having circumcenter or orthocenter generally. Beiträge zur Algebra und Geometrie 57(1), 243–258 (2016). https://doi.org/10.1007/s13366-014-0233-3
Kurusa, Á., Kozma, J.: Euler’s ratio-sum theorem revisited (submitted)
Kurusa, Á.: Support theorems for totally geodesic radon transforms on constant curvature spaces. Proc. Am. Math. Soc. 122(2), 429–435 (1994). https://doi.org/10.2307/2161033
Lexell, A.J.: Solutio problematis geometrici ex doctrina sphaericorum, Acta academiae scientarum imperialis Petropolitinae, 5(1), 112–126, (1784) http://www.17centurymaths.com/contents/euler/lexellone.pdf. Accessed 24 Sept 2018
Montejano, L., Morales, E.: Characterization of ellipsoids and polarity in convex sets. Mathematika 50, 63–72 (2003). https://doi.org/10.1112/S0025579300014790
Papadopoulos, A., Su, W.: On hyperbolic analogues of some classical theorems in spherical geometry, (2015) arXiv arXiv:1409.4742
Sandifer, C.E.: 19th century Triangle Geometry (May 2006), How Euler did it, Math. Assoc. Am., (2007), pp. 19–27 http://eulerarchive.maa.org/hedi/HEDI-2006-05.pdf. Accessed 24 Sept 2018
Shephard, G.C.: Euler’s Triangle Theorem, Crux Mathematicorum, 25(3), 148–153, (1999) https://cms.math.ca/crux/v25/n3/page148-153.pdf. Accessed 24 Sept 2018
Szabó, Z.I.: Hilbert’s fourth problem. I. Adv. Math. 59(3), 185–301 (1986). https://doi.org/10.1016/0001-8708(86)90056-3
Author information
Authors and Affiliations
Corresponding author
Additional information
This research was supported by NFSR of Hungary (NKFIH) under grant numbers K 116451 and KH_18 129630.
Rights and permissions
About this article
Cite this article
Kurusa, Á., Kozma, J. Euler’s ratio-sum formula in projective-metric spaces. Beitr Algebra Geom 60, 379–390 (2019). https://doi.org/10.1007/s13366-018-0422-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13366-018-0422-6