Skip to main content
SpringerLink
Log in

Star-Unfolding Polygons

  • Conference paper
  • First Online:
Book cover Automated Deduction in Geometry (ADG 2014)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 9201))

Included in the following conference series:

Abstract

In this paper we initiate the study of geodesic star unfoldings. They are a generalization of shortest-path star unfoldings of 3D convex polyhedra and have a very simple characterization. We also address several problems concerning the existence of shortest-path star unfoldings on specified source point sets, and of reconstructing shortest-path star unfoldings with given ridge tree combinatorics.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 34.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 44.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    The 1-skeleton is the graph of polyhedral edges.

  2. 2.

    The modifier “abstract” is used to emphasize that, a priori, such polygons are not guaranteed to arise from star unfoldings of 3D convex polyhedra.

References

  1. Aggarwal, A., Guibas, L.J., Saxe, J., Shor, P.W.: A linear-time algorithm for computing the Voronoi diagram of a convex polygon. Discret. Comput. Geom. 4(1), 591–604 (1989)

    Article  MathSciNet  Google Scholar 

  2. Agarwal, P.K., Aronov, B., O’Rourke, J., Schevon, C.A.: Star unfolding of a polytope with applications. SIAM J. Comput. 26, 1689–1713 (1997)

    Article  MathSciNet  Google Scholar 

  3. Alexandrov, A.D.: Convex Polyhedra. Springer Monographs in Mathematics. Springer Verlag, Berlin Heidelberg (2005). English Translation of Russian edition, Gosudarstv. Izdat. Tekhn.-Teor.Lit, Moscow-Leningrad (1950)

    Google Scholar 

  4. Aronov, B., O’Rourke, J.: Non-overlap of the star unfolding. Discret. Comput. Geom. 8, 219–250 (1992)

    Article  MathSciNet  Google Scholar 

  5. Demaine, E.D., O’Rourke, J.: Geometric Folding Algorithms: Linkages, Origami, and Polyhedra. Cambridge University Press, Cambridge (2007)

    Book  Google Scholar 

  6. Dillencourt, M.B.: Realizability of Delaunay triangulations. Inf. Process. Lett. 33(6), 283–287 (1990)

    Article  MathSciNet  Google Scholar 

  7. Lambert, T.: An optimal algorithm for realizing a Delaunay triangulation. Inf. Process. Lett. 62(5), 245–250 (1997)

    Article  Google Scholar 

  8. Mount, D.: On finding shortest paths on convex polyhedra. Technical report 1495, University of Maryland (1985)

    Google Scholar 

  9. Sharir, M., Schorr, A.: On shortest paths in polyhedral spaces. SIAM J. Comput. 15(1), 193–215 (1986)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgement

This research was supported by the NSF grants CCF-1016988 and CCF-1319366.

Author information

Authors and Affiliations

  1. Computer Science Department, University of Massachusetts Amherst, Amherst, MA, 01003, USA

    Md. Ashraful Alam & Ileana Streinu

  2. Computer Science Department, Smith College, Northampton, MA, 01063, USA

    Ileana Streinu

Authors
  1. Md. Ashraful Alam
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ileana Streinu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ileana Streinu .

Editor information

Editors and Affiliations

  1. University of Vigo, Pontevedra, Spain

    Francisco Botana

  2. University of Coimbra, Coimbra, Portugal

    Pedro Quaresma

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this paper

Cite this paper

Alam, M.A., Streinu, I. (2015). Star-Unfolding Polygons. In: Botana, F., Quaresma, P. (eds) Automated Deduction in Geometry. ADG 2014. Lecture Notes in Computer Science(), vol 9201. Springer, Cham. https://doi.org/10.1007/978-3-319-21362-0_1

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-21362-0_1

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-21361-3

  • Online ISBN: 978-3-319-21362-0

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation