Jump to ContentJump to Main Navigation
Show Summary Details

Advances in Geometry

Managing Editor: Grundhöfer, Theo / Joswig, Michael

Editorial Board Member: Bannai, Eiichi / Coskun, Izzet / Duzaar, Frank / Eberlein, Patrick / Gentili, Graziano / Henk, Martin / Kantor, William M. / Korchmaros, Gabor / Kreuzer, Alexander / Lagarias, Jeffrey C. / Leistner, Thomas / Löwen, Rainer / Ono, Kaoru / Pasini, Antonio / Ratcliffe, John G. / Scharlau, Rudolf / Scheiderer, Claus / Van Maldeghem, Hendrik / Weintraub, Steven H. / Weiss, Richard

4 Issues per year

IMPACT FACTOR increased in 2015: 0.578

SCImago Journal Rank (SJR) 2015: 0.489
Source Normalized Impact per Paper (SNIP) 2015: 0.864
Impact per Publication (IPP) 2015: 0.530

Mathematical Citation Quotient (MCQ) 2015: 0.43

See all formats and pricing
Volume 6, Issue 2 (Mar 2006)


Geodesics in non-positively curved plane tessellations

Oliver Baues
  • Math. Institut II, Universität Karlsruhe, 76128 Karlsruhe, Germany.
  • Email:
/ Norbert Peyerimhoff
  • Department of Math. Sciences, Durham University, Durham DH1 2LE, UK.
  • Email:
Published Online: 2006-05-17 | DOI: https://doi.org/10.1515/ADVGEOM.2006.014


We introduce a natural combinatorial curvature function on the corners of plane tessellations and relate it to the global metric geometry of their corresponding edge and dual graphs. If the combinatorial curvature in the corners is non-positive then we prove that any geodesic path in such a graph may be extended to infinity. Moreover, if the combinatorial curvature is negative we show that every pair of geodesic segments with the same end points does not enclose any vertices. We apply these results to establish an estimate for the growth of distance balls, Gromov hyperbolicity, and four-colourability of certain classes of plane tessellations.

About the article

Received: 2004-04-27

Revised: 2005-04-27

Published Online: 2006-05-17

Published in Print: 2006-03-24

Citation Information: Advances in Geometry, ISSN (Online) 1615-7168, ISSN (Print) 1615-715X, DOI: https://doi.org/10.1515/ADVGEOM.2006.014. Export Citation

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

Alessandro Maddaloni and Carol T. Zamfirescu
Discrete Mathematics, 2016, Volume 339, Number 1, Page 354
International Journal of Computational Geometry & Applications, 2014, Volume 24, Number 01, Page 1
Byung-Geun Oh
Discrete & Computational Geometry, 2014, Volume 51, Number 4, Page 859
Frank Bauer, Bobo Hua, and Matthias Keller
Advances in Mathematics, 2013, Volume 248, Page 717
Matthias Keller, Daniel Lenz, and Radosław K. Wojciechowski
Mathematische Zeitschrift, 2013, Volume 274, Number 3-4, Page 905
Matthias Keller
Discrete & Computational Geometry, 2011, Volume 46, Number 3, Page 500
Matthias Keller and Norbert Peyerimhoff
Mathematische Zeitschrift, 2011, Volume 268, Number 3-4, Page 871
Matthias Keller
Mathematische Annalen, 2010, Volume 346, Number 1, Page 51
Mohammed Mostefa Mesmoudi, Leila De Floriani, and Umberto Port
Computer Graphics Forum, 2008, Volume 27, Number 5, Page 1333
Sven Thatje, Olaf Heilmayer, and Jürgen Laudien
Helgoland Marine Research, 2008, Volume 62, Number S1, Page 5
Jürgen Laudien, Manuel E. Rojo, Marcelo E. Oliva, Wolf E. Arntz, and Sven Thatje
Helgoland Marine Research, 2007, Volume 61, Number 2, Page 103

Comments (0)

Please log in or register to comment.
Log in