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**Smooth local interpolation of surfaces using normal vectors.**
*(English)*
Zbl 1205.65042

Summary: This paper proposes a simple surface interpolation attaining tangent-plane continuity. It is a natural extension of the local quadratic \(C^{0}\) interpolator developed by the author [Comput. Aided Geom. Des. 22, No. 4, 327–347 (2005; Zbl 1205.65041)], which has already been applied successfully to diverse engineering problems. The methodology presented in this paper inherits most of the advantages possessed by the \(C^{0}\) scheme. That is, (i) The algorithm is efficient and completely local requiring only the position vectors and normals given at the nodes of a patch, and hence it is suitable for parallel processing. (ii) It converges rapidly to the given surface with the increase in the number of nodes. (iii) Singular points (apexes, sharp edges, etc.) and nonmanifolds can be treated quite easily. (iv) Because of the minimization criteria assigned to the surface coefficients, it is rather robust and amenable to computational analyses. Validity and effectiveness of the proposed technique are demonstrated through numerical examples.

### Citations:

Zbl 1205.65041
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\textit{T. Nagata}, J. Appl. Math. 2010, Article ID 952420, 24 p. (2010; Zbl 1205.65042)

### References:

[1] | T. Nagata, “Simple local interpolation of surfaces using normal vectors,” Computer Aided Geometric Design, vol. 22, no. 4, pp. 327-347, 2005. · Zbl 1205.65041 |

[2] | W. Lin, Y. Watanabe, S. Morita, Y. Uehara, and H. Ohmori, “Development of a V-CAM system and its application to manufacturing,” Journal of the Japan Society for Abrasive Technology, vol. 51, no. 5, pp. 290-295, 2007 (Japanese). |

[3] | T. Hama, M. Takamura, A. Makinouchi, C. Teodosiu, and H. Takuda, “Effect of tool-modeling accuracy on square-cup deep-drawing simulation,” Journal of the Japan Society for Abrasive Technology, vol. 48, no. 552, pp. 61-65, 2007 (Japanese). |

[4] | T. Hama, T. Nagata, C. Teodosiu, A. Makinouchi, and H. Takuda, “Finite-element simulation of springback in sheet metal forming using local interpolation for tool surfaces,” International Journal of Mechanical Sciences, vol. 50, no. 2, pp. 175-192, 2008. · Zbl 1264.74263 |

[5] | S. Morita, Y. Yamagata, and A. Makinouchi, “Ray-tracing considering form errors on the fabrication process using local interpolation for aspheric lens surface,” in Frontiers in Optics/Laser Science XXIV/Plasmonics and Metamaterials/Optical Fabrication and Testing on CD ROM, Optical Society of America, Washington, DC, USA, 2008, paper no. OWB5. |

[6] | G. Farin, “Triangular Bernstein-Bézier patches,” Computer Aided Geometric Design, vol. 3, no. 2, pp. 83-127, 1986. |

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