Издательство Elsevier, 2002, -388 pp.
About forty years ago, the development of computer-aided design and manufacturing created the strong need for new ways to mathematically represent curves and surfaces" the new representations should possess enough flexibility to describe almost arbitrary geometric shapes; be compatible with efficient algorithms; and be readily accessible to designers who could manipulate them simply and intuitively. Although these new requirements presented a difficult challenge, the search for appropriate mathematical tools has been very successful within a relatively short period of time. Curves and surfaces with a piecewise polynomial or rational parametric representation have become the favorites, in particular if they are represented in so-called B6zier or B-spline form. A new field, called computer-aided geometric design (CAGD) emerged. Deeply rooted in approximation theory and numerical analysis, CAGD greatly benefited from results in the classical geometric disciplines such as differential, projective, and algebraic geometry.
Today, CAGD is a mature field that branches into various areas of mathematics, computer science, and engineering. Its boundaries have become less defined, but its keel still consists of algorithms for interpolation and approximation with piecewise
polynomial or rational curves and surfaces.
Pyramid Algorithms presents this keel in a unique way. A few celebrated examples of pyramid algorithms are known to many people: I think of the de Casteljau algorithm and de Boor's algorithm for evaluation and subdividing a B6zier or Bo spline curve, respectively. However, as Ron Goldman tells us in this fascinating book, pyramid algorithms occur almost everywhere in CAGD: they are used for polynomial interpolation, approximation, and change of basis procedures; they are even dualizable. Dr. Goldman discusses pyramid algorithms for polynomial curves, piecewise polynomial curves, tensor product surfaces, and triangular and multisided surface patches.
Though the book focuses mainly on topics well known in CAGD, there are many parts with unconventional approaches, interesting new views, and new insights. Surprises already appear in Chapter 1 on foundations: I had always thought that projective geometry was the ideal framework for a deeper study of rational curves and surfaces, but I must admit that Ron Goldman's preference of Grassmann space over projective space has its distinct advantages for the topics discussed in this book. Surprises continue to occur throughout the book and culminate in the last chapter on multisided patches. Here we find a brand new exposition of very recent results, which eloquently connects the well-established theory of CAGD to ongoing research in the field.
I am convinced that reading this book will be a pleasure for everyone interested in the mathematical and algorithmic aspects of CAGD. Ron Goldman is a leading expert who knows the fundamental concepts and their interconnectedness as well as the small details. He skillfully guides the reader through subtle subjects without getting lost in pure formalism. The elegance of the writing and of the methods used to present the material allows us to get a deep understanding of the central concepts of CAGD. The presentation is clear and precise but never stiff or too abstract. This is a mathematically substantial book that lets the reader enjoy the beauty of the subject. It achieves its goal even without illustrating the creative shape potential of free-form curves and surfaces and without espousing the many important applications this field has in numerous branches of science and technology. In its simplicity and pure beauty, the theory indeed resembles the pyramids.
Introduction: Foundations
Lagrange Interpolation and Neville's Algorithm
Hermite Interpolation and the Extended Neville Algorithm
Newton Interpolation and Difference Triangles
Bezier Approximation and Pascal's Triangle
Blossoming
B-Spline Approximation and the de Boor Algorithm
Pyramid Algorithms for Multisided Bezier Patches
About forty years ago, the development of computer-aided design and manufacturing created the strong need for new ways to mathematically represent curves and surfaces" the new representations should possess enough flexibility to describe almost arbitrary geometric shapes; be compatible with efficient algorithms; and be readily accessible to designers who could manipulate them simply and intuitively. Although these new requirements presented a difficult challenge, the search for appropriate mathematical tools has been very successful within a relatively short period of time. Curves and surfaces with a piecewise polynomial or rational parametric representation have become the favorites, in particular if they are represented in so-called B6zier or B-spline form. A new field, called computer-aided geometric design (CAGD) emerged. Deeply rooted in approximation theory and numerical analysis, CAGD greatly benefited from results in the classical geometric disciplines such as differential, projective, and algebraic geometry.
Today, CAGD is a mature field that branches into various areas of mathematics, computer science, and engineering. Its boundaries have become less defined, but its keel still consists of algorithms for interpolation and approximation with piecewise
polynomial or rational curves and surfaces.
Pyramid Algorithms presents this keel in a unique way. A few celebrated examples of pyramid algorithms are known to many people: I think of the de Casteljau algorithm and de Boor's algorithm for evaluation and subdividing a B6zier or Bo spline curve, respectively. However, as Ron Goldman tells us in this fascinating book, pyramid algorithms occur almost everywhere in CAGD: they are used for polynomial interpolation, approximation, and change of basis procedures; they are even dualizable. Dr. Goldman discusses pyramid algorithms for polynomial curves, piecewise polynomial curves, tensor product surfaces, and triangular and multisided surface patches.
Though the book focuses mainly on topics well known in CAGD, there are many parts with unconventional approaches, interesting new views, and new insights. Surprises already appear in Chapter 1 on foundations: I had always thought that projective geometry was the ideal framework for a deeper study of rational curves and surfaces, but I must admit that Ron Goldman's preference of Grassmann space over projective space has its distinct advantages for the topics discussed in this book. Surprises continue to occur throughout the book and culminate in the last chapter on multisided patches. Here we find a brand new exposition of very recent results, which eloquently connects the well-established theory of CAGD to ongoing research in the field.
I am convinced that reading this book will be a pleasure for everyone interested in the mathematical and algorithmic aspects of CAGD. Ron Goldman is a leading expert who knows the fundamental concepts and their interconnectedness as well as the small details. He skillfully guides the reader through subtle subjects without getting lost in pure formalism. The elegance of the writing and of the methods used to present the material allows us to get a deep understanding of the central concepts of CAGD. The presentation is clear and precise but never stiff or too abstract. This is a mathematically substantial book that lets the reader enjoy the beauty of the subject. It achieves its goal even without illustrating the creative shape potential of free-form curves and surfaces and without espousing the many important applications this field has in numerous branches of science and technology. In its simplicity and pure beauty, the theory indeed resembles the pyramids.
Introduction: Foundations
Lagrange Interpolation and Neville's Algorithm
Hermite Interpolation and the Extended Neville Algorithm
Newton Interpolation and Difference Triangles
Bezier Approximation and Pascal's Triangle
Blossoming
B-Spline Approximation and the de Boor Algorithm
Pyramid Algorithms for Multisided Bezier Patches