Skip to main content
SpringerLink
Log in

MOS Surfaces: Medial Surface Transforms with Rational Domain Boundaries

  • Conference paper
Mathematics of Surfaces XII (Mathematics of Surfaces 2007)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4647))

Included in the following conference series:

Abstract

We consider rational surface patches s(u,v) in the four-dimensional Minkowski space IR3,1, which describe parts of the medial surface (or medial axis) transform of spatial domains. The corresponding segments of the domain boundary are then obtained as the envelopes of the associated two-parameter family of spheres. If the Plücker coordinates of the line at infinity of the (two-dimensional) tangent plane of s satisfy a sum-of-squares condition, then the two envelope surfaces are shown to be rational surfaces. We characterize these Plücker coordinates and analyze the case, where the medial surface transform is contained in a hyperplane of the four-dimensional Minkowski space.

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 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Amenta, N., Choi, S., Kolluri, R.K.: The Power Crust, Unions of Balls, and the Medial Axis Transform. Comput. Geom. 19, 127–153 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  2. Cho, H.C., Choi, H.I., Kwon, S.-H., Lee, D.S., Wee, N.-S.: Clifford Algebra, Lorentzian Geometry and Rational Parametrization of Canal Surfaces. Computer Aided Geometric Design 21, 327–339 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  3. Choi, H.I., Choi, S.W., Moon, H.P.: Mathematical Theory of Medial Axis Transform. Pacific J. Math. 181, 57–88 (1997)

    Article  MathSciNet  Google Scholar 

  4. Choi, H.I., Han, C.Y., Moon, H.P., Roh, K.H., Wee, N.S.: Medial Axis Transform and Offset Curves by Minkowski Pythagorean Hodograph Curves. Computer-Aided Design 31, 59–72 (1999)

    Article  MATH  Google Scholar 

  5. Choi, H.I., Lee, D.S., Moon, H.P.: Clifford Algebra, Spin Representation and Rational Parameterization of Curves and Surfaces. Adv. Computational Mathematics 17, 5–48 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  6. Culver, T., Keyser, J., Manocha, D.: Exact Computation of the Medial Axis of a Polyhedron. Comput. Aided Geom. Des. 21, 65–98 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  7. Dey, T.K., Zhao, W.: Approximating the Medial Axis from the Voronoi Diagram with a Convergence Guarantee. In: Möhring, R.H., Raman, R. (eds.) ESA 2002. LNCS, vol. 2461, pp. 387–398. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  8. Dietz, R., Hoschek, J., Jüttler, B.: An Algebraic Approach to Curves and Surfaces on the Sphere and on Other Quadrics. Comput. Aided Geom. Des. 10, 211–229 (1993)

    Article  MATH  Google Scholar 

  9. Elber, G., Kim, M.–S.: The Bisector Surface of Rational Space Curves. ACM Transactions on Graphics 17, 32–49 (1998)

    Article  Google Scholar 

  10. Elber, G., Kim, M.–S.: Computing Rational Bisectors of Point/Surface and Sphere/Surface Pairs. IEEE Comput. Graph. Appl. 19, 76–81 (1999)

    Article  Google Scholar 

  11. Giblin, P., Kimia, B.B.: Transitions of the 3D Medial Axis under a One-parameter Family of Deformations. In: Heyden, A., Sparr, G., Nielsen, M., Johansen, P. (eds.) ECCV 2002. LNCS, vol. 2351, pp. 718–734. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  12. Farouki, R.T.: Pythagorean-hodograph Curves. In: Hoschek, J., Farin, G., Kim, M.-S. (eds.) Handbook of Computer Aided Geometric Design, pp. 405–427. Elsevier, North-Holland (2002)

    Chapter  Google Scholar 

  13. Farouki, R.T., al-Kandari, M., Sakkalis, T.: Hermite Interpolation by Rotation-invariant Spatial Pythagorean-hodograph Curves. Adv. Computational Mathematics 17, 369–383 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  14. Farouki, R.T., Sakkalis, T.: Pythagorean Hodographs. IBM J. Research and Development 34, 736–752 (1990)

    Article  MathSciNet  Google Scholar 

  15. Kosinka, J., Jüttler, B.: G 1 Hermite Interpolation by Minkowski Pythagorean Hodograph Cubics. Computer Aided Geometric Design 23, 401–418 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  16. Moon, H.P.: Minkowski Pythagorean Hodographs. Computer Aided Geometric Design 16, 739–753 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  17. Peternell, M., Pottmann, H.: A Laguerre Geometric Approach to Rational Offsets. Computer Aided Geometric Design 15, 223–249 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  18. Pottmann, H.: Rational Curves and Surfaces with Rational Offsets. Computer Aided Geometric Design 12, 175–192 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  19. Pottmann, H., Peternell, M.: Applications of Laguerre Geometry in CAGD. Computer Aided Geometric Design 15, 165–186 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  20. Pottmann, H., Wallner, J.: Computational Line Geometry. Springer, Berlin (2001)

    MATH  Google Scholar 

  21. Ramanathan, M., Gurumoorthy, B.: Constructing Medial Axis Transform of Extruded and Revolved 3D Objects with Free-Form Boundaries. Computer-Aided Design 37, 1370–1387 (2005)

    Article  Google Scholar 

  22. Sampl, P.: Medial Axis Construction in Three Dimensions and its Application to Mesh Generation. Eng. Comput. 17, 234–248 (2001)

    Article  MATH  Google Scholar 

  23. Walrave, J.: Curves and Surfaces in Minkowski Space. Doctoral thesis, K.U. Leuven, Fac. of Science, Leuven (1995)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Johannes Kepler University, Institute of Applied Geometry, Altenberger Str. 69, A–4040 Linz, Austria

    Jiří Kosinka & Bert Jüttler

Authors
  1. Jiří Kosinka
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bert Jüttler
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Ralph Martin Malcolm Sabin Joab Winkler

Rights and permissions

Reprints and permissions

Copyright information

© 2007 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Kosinka, J., Jüttler, B. (2007). MOS Surfaces: Medial Surface Transforms with Rational Domain Boundaries. In: Martin, R., Sabin, M., Winkler, J. (eds) Mathematics of Surfaces XII. Mathematics of Surfaces 2007. Lecture Notes in Computer Science, vol 4647. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73843-5_15

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-73843-5_15

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-73842-8

  • Online ISBN: 978-3-540-73843-5

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation