Skip to main content

A Computational Approach to the Unwrappings of the Developable Surfaces

  • Chapter
  • First Online:
Contributions in Mathematics and Engineering
  • 726 Accesses

Abstract

We present a closed-formula description for the unwrappings of the developable surfaces based on information obtained mainly by plane intersection curves on them. Among other applications, this description is suitable for computational implementation in CAD systems, for the construction of visual pictures of unwrappings onto planes, and for the computer graphic modeling of the approximations of real-object surfaces via the developable ones.

In Honor of Constantin Carathéodory

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover 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

References

  1. Aumann, G.: Interpolation with developable Bézier patches. Comput. Aided Geom. Des. 8, 409–420 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bennis, C., Vezien, J.M., Iglesias, G.: Piecewise flattening for non-distorted texture mapping. In: Proceedings of ACM SIGGRAPH ’91, pp. 237–246 (1991)

    Google Scholar 

  3. Bodduluri, R.M.C., Ravani, B.: Design of developable surfaces using duality between plane and point geometries. Comput. Aided Des. 25, 621–632 (1993)

    Article  MATH  Google Scholar 

  4. Do Carmo, M.P.: Differential Geometry of Curves and Surfaces. Prentice-Hall, Englewood Cliffs (1976)

    MATH  Google Scholar 

  5. Faux, I.D., Pratt, M.J.: Computational Geometry for Design and Manufacture. Ellis Horwood, Chichester (1980)

    MATH  Google Scholar 

  6. Kergosien, Y.L., Gotoda, H., Kunii, T.L.: Bending and creasing virtual paper. IEEE Comput. Graph. Appl. 14, 40–48 (1994)

    Article  Google Scholar 

  7. Kodokostas, D.: Centers of curvature for unwrappings of plane intersections of tame developable surfaces. Int. Electron. J. Geom. 6, 112–128 (2013)

    MathSciNet  MATH  Google Scholar 

  8. Kuhnel, W.: Differential Geometry, Curves-Surfaces-Manifolds. American Mathematical Society, Providence, RI (2006)

    MATH  Google Scholar 

  9. Lipschutz, M.M.: Differential Geometry. Schaum’s Outlines Series of Theory and Problems. McGraw-Hill, New York (1974)

    MATH  Google Scholar 

  10. Maillot, J., Yahia, J., Verroust, A.: Interactive texture mapping. In: Proceedings of ACM SIGGRAPH ’93, pp. 27–34 (1993)

    Google Scholar 

  11. Nutbourne, A., McLellan, P., Kensit, R.: Curvature profiles for plane curves. Comput. Aided Des. 24, 176–184 (1992)

    Google Scholar 

  12. Pottmann, H., Farin, G.: Developable rational Bézier and B-spline surfaces. Comput. Aided Geom. Des. 12, 513–531 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  13. Redont, P.: Representation and deformation of developable surfaces. Comput. Aided Des. 21, 13–20 (1989)

    Article  MATH  Google Scholar 

  14. Sun, M., Fiume, E.: A technique for constructing developable surfaces. In: Graphic Interface, pp. 176–185 (1996)

    Google Scholar 

  15. Weiss, G., Furtner, P.: Computer-aided treatment of developable surfaces. Comput. Graph. 12, 39–51 (1988)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Dimitrios Kodokostas .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Kodokostas, D. (2016). A Computational Approach to the Unwrappings of the Developable Surfaces. In: Pardalos, P., Rassias, T. (eds) Contributions in Mathematics and Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-31317-7_17

Download citation

Publish with us

Policies and ethics