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Mathematical aspects of geometric modeling. (English) Zbl 0864.65008

CBMS-NSF Regional Conference Series in Applied Mathematics. 65. Philadelphia, PA: SIAM, Society for Industrial and Applied Mathematics. ix, 256 p. (1995).
This CBMS-NSF monograph is based on a series of 10 lectures presented at Kent State University in December 1990, under the title Curves and surfaces: an algorithmic viewpoint. An excellent description of the contents of this enjoyable well-written monograph is to be found in the author’s “A brief overview” which we therefore quote: “This monograph examines in detail certain concepts that are useful for the modelling of curves and surfaces. Our emphasis is on the mathematical theory that underlies these ideas.” Two principal themes stand out from the rest. The first one, the most traditional and well-trodden, is the use of piecewise polynomial representation. This theme appears in one form or another in all the chapters. The second theme is that of iterative refinement, also called subdivision. Here, simple iterative geometric algorithms produce, in the limit, curves with complex analytic structure. An introduction to these ideas is given in Chapters 1 and 2. “In Chapter 1, we use de Casteljau subdivision for Bernstein-Bézier curves to introduce matrix subdivision, and in Chapter 2 the Lane-Riesenfeld algorithm for computing cardinal splines is tied to stationary subdivision and ultimately leads us to the construction of pre-wavelets of compact support. In Chapter 3 we study concepts of ‘visual smoothness’ of curves and, as a result, embark on a study of certain spaces of piecewise polynomials determined by connection matrices. Chapter 4 explores the intriguing idea of generating smooth multivariate piecewise polynomials as volumes of ‘slices’ of polyhedra. The final chapter concerns evaluation of polynomials by finite recursive algorithms. Again, Bernstein-Bézier polynomials motivate us to look at polarization of polynomial identities, dual bases, and H. P. Seidel’s multivariate \(B\)-patches, which we join together with the multivariate \(B\)-spline of Chapter 4”.

MSC:

65D17 Computer-aided design (modeling of curves and surfaces)
65D18 Numerical aspects of computer graphics, image analysis, and computational geometry
68U05 Computer graphics; computational geometry (digital and algorithmic aspects)
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
65D07 Numerical computation using splines
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