International Journal of CAD/CAM

Society for Computational Design and Engineering (한국CDE학회)
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Construction of Cubic Triangular Patches with $C^1$ Continuity around a Corner

  • Zhang, Renjiang (Department of Mathematics, College of Science, China Jiliang University) ;
  • Liu, Ligang (Department of Mathematics, Zhejiang University) ;
  • Wang, Guojin (Department of Mathematics, Zhejiang University) ;
  • Ma, Weiyin (Department of Manufacturing Engr. and Engr. Management, City University of Hong Kong)
  • Published : 2006.12.31
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Abstract

This paper presents a novel approach for constructing a piecewise triangular cubic polynomial surface with $C^1$ continuity around a common corner vertex. A $C^1$ continuity condition between two cubic triangular patches is first derived using mixed directional derivatives. An approach for constructing a surface with $C^1$ continuity around a corner is then developed. Our approach is easy and fast with the virtue of cubic reproduction, local shape controllability, $C^2$ continuous at the corner vertex. Some experimental results are presented to show the applicability and flexibility of the approach.

Keywords

References

  1. Farin, G. (1982), A construction for visual $C^1$ continuity of polynomial surface patches, Computer Graphics and Image Processing, 20(7), 272-282 https://doi.org/10.1016/0146-664X(82)90085-5
  2. Farin, G. (1986), Triangular Bernstein-Bezr Patches, Computer Aided Geometric Design, 3(2), 83-127 https://doi.org/10.1016/0167-8396(86)90016-6
  3. Gregory, J. A. (1986), N-sided surface patches, in: J.A. Gregory, ed., The Mathematics of Surfaces, Clarendon Press, Oxford, 217-232
  4. Gregory J.A. and Yuen P.K.(1992), An arbitrary mesh network scheme using rational splines, in: T. Lyche and L.L. Schumaker, eds., Mathematical Methods in CAGD II, Academic Press, New York, 321-329
  5. Hahmann S. and Bonneau G.-P. (2000), Triangular $G^1$ interpolation by 4-splitting domain triangles, Computer Aided Geometric Design, 17, 731-757 https://doi.org/10.1016/S0167-8396(00)00021-2
  6. Herron G. (1985), Smooth closed surfaces with drete triangular interpolants, Computer Aided Geometric Design, 2(3), 297-306 https://doi.org/10.1016/S0167-8396(85)80004-2
  7. Hermann T. (1996), $G^2$ interpolation of free form curve networks by biquintic Gregory patches, Computer Aided Geometric Design, 13, 873-893 https://doi.org/10.1016/S0167-8396(96)00013-1
  8. Hall R.and Mullineux G. (1999), Continuity between Gregory-like patches', Computer Aided Geometric Design, 16, 197-216 https://doi.org/10.1016/S0167-8396(98)00044-2
  9. Loop C.(1994), A $G^1$ triangular spline surface of arbitrary topological type, Computer Aided Geometric Design, 11, 303-330 https://doi.org/10.1016/0167-8396(94)90005-1
  10. Mann S., Loop C., Lonsbery M., Meyers D., Painter J., DeRose T. and Sloan K.(1992), A survey of parametric scattered data fitting using triangular interpolants, in: H. Hagen, ed., Curve and Surface Design, SIAM, 145-172
  11. Peters J. (1990), Local smooth surface interpolation: a classification, Computer Aided Geometric Design, 7, 191-195 https://doi.org/10.1016/0167-8396(90)90030-U
  12. Peters J.(1991), Smooth interpolation of a mesh of curves, Constructive Approximation, 7, 221-246 https://doi.org/10.1007/BF01888155
  13. Piper B.R.(1987), Visually smooth interpolation with triangular Bezier pathces, in: G. Farin, ed., Geometric Modeling:Algorithms and New Trends, SIAM, 221-233
  14. Sarraga R.F.(1987), $GC^2$ Interpolation of generally unstricted cubic Bézier curves, Computer Aided Geometric Design, 4, 23-39 https://doi.org/10.1016/0167-8396(87)90022-7
  15. Shichtel M.(1993), $GC^2$ blend surfaces and filling of n-sided holes, IEEE Computer Graphics and Its Applications, September, 68-73
  16. Shirman L.A. and Sequin C.H. (1987), Local surface interpolation with Bézier patches, Computer Aided Geometric Design, 4, 279-295 https://doi.org/10.1016/0167-8396(87)90003-3
  17. Van Wijk J.J.(1986), Bicubic patches for approximating non-rectangular control meshes, Computer Aided Geometric Design , 3, 1-13 https://doi.org/10.1016/0167-8396(86)90021-X
  18. Watkins M.A.(1988), Problems in geometric continuity, Computer-Aided Design, 20, 499-502 https://doi.org/10.1016/0010-4485(88)90012-7
  19. Walton D.J. and Meek D.S.(1996), A triangular $G^1$ patch from boundary curves, Computer-Aided Design, 28, 113-123 https://doi.org/10.1016/0010-4485(95)00046-1