Discrete & Computational Geometry

, Volume 57, Issue 2, pp 305–317 | Cite as

Quasiconformal Dilatation of Projective Transformations and Discrete Conformal Maps

  • Stefan Born
  • Ulrike Bücking
  • Boris SpringbornEmail author


We consider the quasiconformal dilatation of projective transformations of the real projective plane. For non-affine transformations, the contour lines of dilatation form a hyperbolic pencil of circles, and these are the only circles that are mapped to circles. We apply this result to analyze the dilatation of the circumcircle preserving piecewise projective interpolation between discretely conformally equivalent triangulations. We show that another interpolation scheme, angle bisector preserving piecewise projective interpolation, is in a sense optimal with respect to dilatation. These two interpolation schemes belong to a one-parameter family.


Piecewise projective Optimal quasiconformal mapping Discrete complex analysis 

Mathematics Subject Classification

30C62 52C26 



This research was supported by the DFG Collaborative Research Center TRR 109 “Discretization in Geometry and Dynamics”.


  1. 1.
    Ahlfors, L.V.: On quasiconformal mappings. J. Anal. Math. 3, pp. 1–58 (1954); correction, pp. 207–208 (1954)Google Scholar
  2. 2.
    Ahlfors, L.V.: Lectures on Quasiconformal Mappings. University Lecture Series, vol. 38, 2nd edn. American Mathematical Society, Providence, RI (2006)Google Scholar
  3. 3.
    Bobenko, A.I., Pinkall, U., Springborn, B.A.: Discrete conformal maps and ideal hyperbolic polyhedra. Geom. Topol. 19(4), 2155–2215 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Kimberling, C.: Encyclopedia of triangle centers (ETC). Accessed 9 Apr 2015
  5. 5.
    Luo, F.: Combinatorial Yamabe flow on surfaces. Commun. Contemp. Math. 6(5), 765–780 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Papadopoulos, A., Théret, G.: On Teichmüller’s metric and Thurston’s asymmetric metric on Teichmüller space. In: Papadopoulos, A. (ed.) Handbook of Teichmüller Theory, vol. I. IRMA Lectures in Mathematics and Theoretical Physics, vol. 11, pp. 111–204. European Mathematical Society, Zürich (2007)Google Scholar
  7. 7.
    Springborn, B., Schröder, P., Pinkall, U.: Conformal equivalence of triangle meshes. In: ACM SIGGRAPH 2008 Papers, SIGGRAPH ’08, pp. 77:1–77:11. ACM, New York (2008)Google Scholar
  8. 8.
    Stephenson, K.: Introduction to Circle Packing. Cambridge University Press, Cambridge (2005)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Technische Universität BerlinBerlinGermany
  2. 2.Technische Universität BerlinBerlinGermany

Personalised recommendations