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**Surfaces defined on surfaces.**
*(English)*
Zbl 0726.65013

The authors present two methods for constructing functions, defined over convex surfaces, interpolating data values at scattered points on the surfaces. These are local methods.

The first one is basically an extension of the 3D Shepard’s method that restrict the domain to a convex surface. The second one consists in an extension of the triangular-based method over a sphere to arbitrary convex surfaces.

The two methods are compared both numerically and by means of color interactive computer graphics on an example consisting in determination of the pressure at arbitrary locations on a wing, by using given discrete pressures measured on it. This example represents an illustration of a more general problem: given data at discrete locations on a surface, determine a function that interpolates these data over the entire surface.

The first one is basically an extension of the 3D Shepard’s method that restrict the domain to a convex surface. The second one consists in an extension of the triangular-based method over a sphere to arbitrary convex surfaces.

The two methods are compared both numerically and by means of color interactive computer graphics on an example consisting in determination of the pressure at arbitrary locations on a wing, by using given discrete pressures measured on it. This example represents an illustration of a more general problem: given data at discrete locations on a surface, determine a function that interpolates these data over the entire surface.

Reviewer: D.D.Stancu (Cluj-Napoca)

### MSC:

65D17 | Computer-aided design (modeling of curves and surfaces) |

65D05 | Numerical interpolation |

65D18 | Numerical aspects of computer graphics, image analysis, and computational geometry |

68U07 | Computer science aspects of computer-aided design |

41A05 | Interpolation in approximation theory |

### Keywords:

triangular interpolation; tessellation; 3D Shepard’s method; convex surface; computer graphics
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\textit{R. E. Barnhill} and \textit{H. S. Ou}, Comput. Aided Geom. Des. 7, No. 1--4, 323--336 (1990; Zbl 0726.65013)

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### References:

[1] | Barnhill, R. E., Surfaces in computer aided geometric design: A survey with new results, Computer Aided Geometric Design, 2, 1-17 (1985) · Zbl 0597.65001 |

[2] | Barnhill, R. E.; Birkhoff, G. B.; Gordon, W. J., Smooth interpolation in triangles, J. Approx. Theory, 8, 114-128 (1973) · Zbl 0271.41002 |

[3] | Barnhill, R. E.; Piper, B. R.; Rescorla, K. L., Interpolation to arbitrary data on a surface, (Farin, G. E., Geometric Modeling (1987), SIAM: SIAM Philadelphia), 281-289 |

[4] | Foley, T. A., Interpolation to scattered data on a spherical domain, (Cox, M.; Mason, J., Algorithms for Approximation II (1990), Chapman and Hall: Chapman and Hall London), 303-310 · Zbl 0749.41003 |

[5] | Herron, G. J., Triangular and multisided patch schemes, (Ph.D. Thesis (1979), Mathematics Department, University of Utah: Mathematics Department, University of Utah Salt Lake City) |

[6] | Lawson, C. L., \(C^1\) Surface interpolation for scattered data on a sphere, Rocky Mountain J. Math., 14, 177-202 (1984) · Zbl 0579.65008 |

[7] | Nielson, G. M., The side-vertex method for interpolation in triangles, J. Approx. Theory, 25, 318-336 (1979) · Zbl 0401.41005 |

[8] | Nielson, G. M.; Ramaraj, R., Interpolation over a sphere based upon a minimum norm network, Computer Aided Geometric Design, 4, 41-57 (1987) · Zbl 0632.65010 |

[9] | Ramaraj, R., Interpolation and display of scattered data over a sphere, (Masters Thesis (1986), Computer Science Department, Arizona State University: Computer Science Department, Arizona State University Tempe, AZ) |

[10] | Renka, R. J., Interpolation of data on the surface of a sphere, ACM Trans. Math. Software, 417-436 (1984) · Zbl 0548.65001 |

[11] | Sederberg, T. W.; Parry, S. R., Free-form deformation of solid geometric models, SIGGRAPH ’86 Conf. Proc., 151-160 (1986) |

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