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**Computational geometry. Curve and surface modeling.**
*(English)*
Zbl 0679.68206

Boston etc.: Academic Press. X, 295 p. $ 39.95 (1989).

The subject of the book is Computer Aided Geometric Design (CAGD). The terminus computational geometry is used in the sense of the definition given by FORREST in 1971 as “computer-based representtion, analysis, synthesis (design) and computer-controlled manufacture of two- and three- dimensional shapes”. In this sense computational geometry is a discipline formulated from approximation theory of functions, differential geometry, algebraic geometry, computational mathematics and numerical control. In the literature computational geometry means the study of the complexity of geometric algorithms also, which is not discussed in this book.

Basic theories, numerical algorithms and geometric properties of curves and surfaces and their applications to CAD/CAM are presented from a geometric point of view. An introduction to fundamental methods of curve and surface modelling in CAGD is given. The authors include parametric splines, coons patches, Bézier and B-spline techniques. A study of the fairness of curves and networks and nonlinear spline curves has been involved also. The authors study the properties of parametric polynomial curves by using the theory of the intrinsic affine invariants for algebraic curves, the affine classification according to the number of inflection points and singularities of planar cubic Bézier curves. The practical experience of the authors in developing CAD/CAM systems at shipbuilding factories has been included. These results have been applied to CAD techniques in car, plane and mechanical industries.

The titles of the eight chapters of this book are introduction, spline functions, parametric cubic spline curves, Bézier curves and B-spline curves, spline surfaces, nonlinear splines, curve and net fairing and the intrinsic affine invariants of parametric curves in affine hyperspace.

This book is the English edition from the Chinese version from 1979 and was translated by Chang Geng-zhe, University of Science and Technology of China, He Fei Province. Many additional references to publiations after 1979 are given.

Basic theories, numerical algorithms and geometric properties of curves and surfaces and their applications to CAD/CAM are presented from a geometric point of view. An introduction to fundamental methods of curve and surface modelling in CAGD is given. The authors include parametric splines, coons patches, Bézier and B-spline techniques. A study of the fairness of curves and networks and nonlinear spline curves has been involved also. The authors study the properties of parametric polynomial curves by using the theory of the intrinsic affine invariants for algebraic curves, the affine classification according to the number of inflection points and singularities of planar cubic Bézier curves. The practical experience of the authors in developing CAD/CAM systems at shipbuilding factories has been included. These results have been applied to CAD techniques in car, plane and mechanical industries.

The titles of the eight chapters of this book are introduction, spline functions, parametric cubic spline curves, Bézier curves and B-spline curves, spline surfaces, nonlinear splines, curve and net fairing and the intrinsic affine invariants of parametric curves in affine hyperspace.

This book is the English edition from the Chinese version from 1979 and was translated by Chang Geng-zhe, University of Science and Technology of China, He Fei Province. Many additional references to publiations after 1979 are given.

Reviewer: H.D.Hecker

### MSC:

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

65-02 | Research exposition (monographs, survey articles) pertaining to numerical analysis |

68-02 | Research exposition (monographs, survey articles) pertaining to computer science |

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

51N05 | Descriptive geometry |