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Mathematical Problems in Engineering
Volume 2012 (2012), Article ID 618148, 15 pages
http://dx.doi.org/10.1155/2012/618148
Research Article

Rational Generalized Offsets of Rational Surfaces

Faculty of Mathematics and Informatics, Konstantin Preslavsky University, 9712 Shumen, Bulgaria

Received 20 May 2011; Revised 1 September 2011; Accepted 1 November 2011

Academic Editor: Kui Fu Chen

Copyright © 2012 Georgi Hristov Georgiev. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. H. Pottmann, “Rational curves and surfaces with rational offsets,” Computer Aided Geometric Design, vol. 12, no. 2, pp. 175–192, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. M. Peternell and H. Pottmann, “A Laguerre geometric approach to rational offsets,” Computer Aided Geometric Design, vol. 15, no. 3, pp. 223–249, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. R. Krasauskas, “Branching blend of natural quadrics based on surfaces with rational offsets,” Computer Aided Geometric Design, vol. 25, no. 4-5, pp. 332–341, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. R. Krasauskas and M. Peternell, “Rational offset surfaces and their modeling applications,” in Nonlinear Computational Geometry, vol. 151 of IMA Vol. Math. Appl., pp. 109–135, Springer, New York, NY, USA, 2010. View at Google Scholar
  5. B. Jüttler, “Triangular Bézier surface patches with a linear normal vector field,” in The Mathematics of Surfaces VIII, pp. 431–446, Information Geometers, Winchester, UK, 1998. View at Google Scholar
  6. B. Jüttler and M. L. Sampoli, “Hermite interpolation by piecewise polynomial surfaces with rational offsets,” Computer Aided Geometric Design, vol. 17, no. 4, pp. 361–385, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. M. L. Sampoli, M. Peternell, and B. Jüttler, “Rational surfaces with linear normals and their convolutions with rational surfaces,” Computer Aided Geometric Design, vol. 23, no. 2, pp. 179–192, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. M. Peternell and B. Odehnal, “On generalized LN-surfaces in 4-space,” pp. 223–230, ACM, New York, NY, USA.
  9. B. Bastl, B. Jüttler, J. Kosinka, and M. Lávička, “Computing exact rational offsets of quadratic triangular Bézier surface patches,” Computer Aided Design, vol. 40, no. 2, pp. 197–209, 2008. View at Publisher · View at Google Scholar
  10. T. Maekawa, “Self-intersections of offsets of quadratic surfaces: part I, Explicit surfaces,” Engineering with Computers, vol. 14, no. 1, pp. 1–13, 1998. View at Google Scholar
  11. N. M. Patrikalakis and T. Maekawa, Shape Interrogation for Computer Aided Design and Manufacturing, Springer, Berlin, Germany, 2002.
  12. M. Aigner, B. Jüttler, L. Gonzalez-Vega, and J. Schicho, “Parameterizing surfaces with certain special support functions, including offsets of quadrics and rationally supported surfaces,” Journal of Symbolic Computation, vol. 44, no. 2, pp. 180–191, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. B. Bastl, B. Jüttler, J. Kosinka, and M. Lávička, “Volumes with piecewise quadratic medial surface transforms: computation of boundaries and trimmed offsets,” Computer Aided Design, vol. 42, no. 6, pp. 571–579, 2010. View at Publisher · View at Google Scholar
  14. L. A. Piegl and W. Tiller, “Computing offsets of NURBS curves and surfaces,” Computer Aided Design, vol. 31, no. 2, pp. 147–156, 1999. View at Publisher · View at Google Scholar
  15. G. V. V. Ravi Kumar, K. G. Shastry, and B. G. Prakash, “Computing non-self-intersecting offsets of NURBS surfaces,” Computer Aided Design, vol. 34, no. 3, pp. 209–228, 2002. View at Publisher · View at Google Scholar
  16. Y. F. Sun, A. Y. C. Nee, and K. S. Lee, “Modifying free-formed NURBS curves and surfaces for offsetting without local self-intersection,” Computer Aided Design, vol. 36, no. 12, pp. 1161–1169, 2004. View at Publisher · View at Google Scholar
  17. E. Arrondo, J. Sendra, and J. R. Sendra, “Parametric generalized offsets to hypersurfaces,” Journal of Symbolic Computation, vol. 23, no. 2-3, pp. 267–285, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. E. Arrondo, J. Sendra, and J. R. Sendra, “Genus formula for generalized offset curves,” Journal of Pure and Applied Algebra, vol. 136, no. 3, pp. 199–209, 1999. View at Google Scholar
  19. J. R. Sendra and J. Sendra, “Algebraic analysis of offsets to hypersurfaces,” Mathematische Zeitschrift, vol. 234, no. 4, pp. 697–719, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. H. Hagen, S. Hahmann, and T. Schreiber, “Visualization and computation of curvature behaviour of free-form curves and surfaces,” Computer-Aided Design, vol. 27, no. 7, pp. 545–552, 1995. View at Google Scholar
  21. S. Hahmann, A. Belyaev, L. Busé, G. Elber, B. Mourrain, and C. Rössl, “Shape interrogation,” in Shape Analysis and Structuring, Math. Vis., pp. 1–51, Springer, Berlin, Germany, 2008. View at Google Scholar
  22. H. P. Moon, “Equivolumetric offset surfaces,” Computer Aided Geometric Design, vol. 26, no. 1, pp. 17–36, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  23. D. G. Jang, H. Park, and K. Kim, “Surface offsetting using distance volumes,” International Journal of Advanced Manufacturing Technology, vol. 26, no. 1-2, pp. 102–108, 2005. View at Publisher · View at Google Scholar
  24. H. Shen, J. Fu, Z. Chen, and Y. Fan, “Generation of offset surface for tool path in NC machining through level set methods,” International Journal of Advanced Manufacturing Technology, vol. 46, no. 9–12, pp. 1043–1047, 2010. View at Publisher · View at Google Scholar
  25. R. P. Encheva and G. H. Georgiev, “Similar Frenet curves,” Results in Mathematics, vol. 55, no. 3-4, pp. 359–372, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  26. A. Fujioka and J.-I. Inoguchi, “Deformations of surfaces preserving conformal or similarity invariants,” in From Geometry to Quantum Mechanics, vol. 252 of Progr. Math., pp. 53–67, Birkhäuser, Boston, Mass, USA, 2007. View at Google Scholar
  27. M. Lávička and B. Bastl, “PN surfaces and their convolutions with rational surfaces,” Computer Aided Geometric Design, vol. 25, no. 9, pp. 763–774, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  28. A. Coffman, A. J. Schwartz, and C. Stanton, “The algebra and geometry of Steiner and other quadratically parametrizable surfaces,” Computer Aided Geometric Design, vol. 13, no. 3, pp. 257–286, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  29. A. Gray, E. Abbena, and S. Salamon, Modern Differential Geometry of Curves and Surfaces with Mathematica, Studies in Advanced Mathematics, Chapman & Hall/CRC, Boca Raton, Fla, USA, 3rd edition, 2006.
  30. T. Rando and J. A. Roulier, “Designing faired parametric surfaces,” Computer-Aided Design, vol. 23, no. 7, pp. 492–497, 1991. View at Google Scholar