Discrete & Computational Geometry

, Volume 49, Issue 2, pp 157–182 | Cite as

Drawing Trees with Perfect Angular Resolution and Polynomial Area

  • Christian A. Duncan
  • David Eppstein
  • Michael T. Goodrich
  • Stephen G. Kobourov
  • Martin Nöllenburg
Article

Abstract

We study methods for drawing trees with perfect angular resolution, i.e., with angles at each node \(v\) equal to \(2\pi /d(v)\). We show:
  1. 1.

    Any unordered tree has a crossing-free straight-line drawing with perfect angular resolution and polynomial area.

     
  2. 2.

    There are ordered trees that require exponential area for any crossing-free straight-line drawing having perfect angular resolution.

     
  3. 3.

    Any ordered tree has a crossing-free Lombardi-style drawing (where each edge is represented by a circular arc) with perfect angular resolution and polynomial area.

     
Thus, our results explore what is achievable with straight-line drawings and what more is achievable with Lombardi-style drawings, with respect to drawings of trees with perfect angular resolution.

Keywords

Tree drawings Straight-line drawings Circular-arc drawings  Lombardi drawings Polynomial area Perfect angular resolution 

Notes

Acknowledgments

This research was supported in part by the National Science Foundation under grants CCF-0545743, CCF-1115971 and CCF-0830403, by the Office of Naval Research under MURI Grant N00014-08-1-1015, and by the German Research Foundation under Grant NO 899/1-1.

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Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  • Christian A. Duncan
    • 1
  • David Eppstein
    • 2
  • Michael T. Goodrich
    • 2
  • Stephen G. Kobourov
    • 3
  • Martin Nöllenburg
    • 4
  1. 1.Department of Mathematics and Computer ScienceQuinnipiac UniversityHamdenUSA
  2. 2.Department of Computer ScienceUniversity of California, IrvineIrvineUSA
  3. 3.Department of Computer ScienceUniversity of Arizona, TucsonTucsonUSA
  4. 4.Institute of Theoretical InformaticsKarlsruhe Institute of TechnologyKarlsruheGermany

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