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**On the use of infinite control points in CAGD.**
*(English)*
Zbl 0622.65142

The author points out that it is a waste of computer resources to use polynomial Béziers and B-splines in computer graphics and CAD. Just as rational functions give numerical approximations superior to those by polynomials, so rational Béziers splines turn out to be very versatile even for low degrees. In addition, rational splines allow one to work in a projective setting and therefore use points at infinity (i.e., directions) as Beźiers. Explicit formulas are given for torus patches and surfaces of revolution and it is shown by impressive examples that these satisfy for a great many modeling problems.

Reviewer: H.Guggenheimer

### MSC:

65S05 | Graphical methods in numerical analysis |

41A20 | Approximation by rational functions |

65D07 | Numerical computation using splines |

53A04 | Curves in Euclidean and related spaces |

51N05 | Descriptive geometry |

51N15 | Projective analytic geometry |

53A05 | Surfaces in Euclidean and related spaces |

### Keywords:

computer aided geometric design; projective geometry; B-splines; computer graphics; rational Béziers splines; torus patches; surfaces of revolution; modeling
Full Text:
DOI

### References:

[1] | Bézier, P., Numerical control, mathematics and applications, (1972), Wiley New York · Zbl 0251.93002 |

[2] | Boehm, W., On cubics: A survey, Computer graphics and image processing, 19, 201-226, (1982) · Zbl 0534.65095 |

[3] | Boehm, W.; Farin, G.; Kahmann, J., A survey of curve and surface methods in CAGD, Computer aided geometric design, 1, 1-60, (1984) · Zbl 0604.65005 |

[4] | Coons, S.A., Surfaces for computer aided design of space forms, MIT project MAC-TR-41, (1967) |

[5] | Faux, I.D.; Pratt, M.J., Computational geometry for design and manufacture, (1981), Ellis Horwood Chichester · Zbl 0601.51001 |

[6] | Forrest, A.R., Curves and surfaces for computer aided design, () · Zbl 0727.65010 |

[7] | Forrest, A.R., The twisted cubic curve: A CAGD approach, Computer aided design, 12, 165-172, (1980) |

[8] | Lee, E.T.Y., The rational Bézier representation for conics, () |

[9] | Mullineux, G., Surface Fitting using boundary data, (), 137-147 |

[10] | Piegl, L., Defining C1 curves containing conic segments, Computers & graphics, 8, 177-182, (1984) |

[11] | Piegl, L., A geometric investigation of the rational Bézier scheme of computer aided design, Computers in industry, 7, 401-410, (1986) |

[12] | Piegl, L., Representation of rational Bézier curves and surfaces by recursive algorithms, Computer aided design, 18, 361-366, (1986) |

[13] | Piegl, L., The sphere as a rational Bézier surface, Computer aided geometric design, 3, 45-52, (1986) · Zbl 0631.65013 |

[14] | Riesenfeld, R.F., Applications of B-spline approximation to geometric problems of computer aided design, () · Zbl 0675.41026 |

[15] | Roberts, L.G., Homogeneous matrix representation and manipulation of N-dimensional constructs, (1965), MIT Lincoln Lab Cambridge, MA, MS-1405 |

[16] | Rowin, M.S., Conic, cubic and T-conic segments, (1964), The Boeing Company Seattle, Document No. D2-23252 |

[17] | Tiller, W., Rational B-splines for curve and surface representation, IEEE computer graphics and applications, 3, 61-69, (1983) |

[18] | Tiller, W., Geometric modeling using non-uniform rational B-splines: mathematical techniques, SIGGRAPH86 tutorial notes, (1986) |

[19] | Versprille, K.J., Computer aided design applications of the rational B-spline approximation form, () |

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