Bezier and B-Spline Techniques

Book Description

This book provides a solid and uniform derivation of the various properties Bézier and B-spline representations have, and shows the beauty of the underlying rich mathematical structure. The book focuses on the core concepts of Computer Aided Geometric Design with the intension to give a clear and illustrative presentation of the basic principles, as well as a treatment of advanced material including multivariate splines, some subdivision techniques and constructions of free form surfaces with arbitrary smoothness.
The text is beautifully illustrated with many excellent figures to emphasize the geometric constructive approach of this book.

Customer Reviews:

Good reference maybe.......2006-08-09

I'm not sure what sort of person would find this book useful. If you want to LEARN about Bezier or B-Spline curves, then I don't think this book is for you. The authors' stated goal was to "provide a solid and uniform derivation" of Bezier and B-Spline properties. I believe that they have succeeded, almost every factoid I've seen in other books is included and proved rigorously. What is NOT included is any sort of meaningful explanation of the concepts. The material is presented at a breakneck pace, with everything presented exactly once, and most often in mathematical notation rather than in English. This is a good way to pack a lot of material into a few pages, but it makes for pretty difficult reading. I really don't see how a person could possibly follow the exposition in this book unless they basically already knew the material.

If you already are familiar with Bezier and B-Spline techniques, or perhaps you are an expert in a related field like approximation theory, then you might find this useful.

So if the Authors acheived their stated goal, then why only the 3-star rating? I've recently bought quite a few of these books on splines, in preperation to write a chapter in my own (introductory) 3d math book. Most of the material in this book can be found in other books. Of course, all of the books have significant overlap, since they are covering the same subject. But if you could have a book that covers most of the same material AND also has more exposition, I'd go for the one with the more exposition. I don't think there's enough unique material in this book to trade off the loss of the much more graceful exposition in the other books.

In particular, if you're interested in *learning* about Bezier curves or B-Splines, I found these two books much more accessible:

An Introduction to Splines for use Computer Graphics and Geometric Modeling by Bartels et al. has a much slower pace - paradoxially this means that if you're learning the material, you will be able to read it FASTER since you can maintain a constant pace. It focuses on B-Splines and only stops breifly to mention Bezier curves as a general case.

The other book is Curves and Surfaces for COmputer Aided Geometric Design by Farin. He introduces Bezier curves and fully develops them, at a reasonable pace, before discussing B-splines. I personally found this approach to be better from a teaching perspective, since B-Splines are more abstract and Bezier curves are easier to understand. (Be warned that I have a rather old editition, the 2nd, I believe he's on the 5th now. I don't know what all has changed since then.)

Spectral Methods: Fundamentals in Single Domains (Scientific Computation)

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Spectral Methods: Fundamentals in Single Domains (Scientific Computation)
C. Canuto , M.Y. Hussaini , A. Quarteroni , and T.A. Zang
Manufacturer: Springer
ProductGroup: Book
Binding: Hardcover

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ASIN: 3540307257

Book Description

Since the publication of "Spectral Methods in Fluid Dynamics", spectral methods, particularly in their multidomain version, have become firmly established as a mainstream tool for scientific and engineering computation. While retaining the tight integration between the theoretical and practical aspects of spectral methods that was the hallmark of the earlier book, Canuto et al. now incorporate the many improvements in the algorithms and the theory of spectral methods that have been made since 1988. The initial treatment Fundamentals in Single Domains discusses the fundamentals of the approximation of solutions to ordinary and partial differential equations on single domains by expansions in smooth, global basis functions. The first half of the book provides the algorithmic details of orthogonal expansions, transform methods, spectral discretization of differential equations plus their boundary conditions, and solution of the discretized equations by direct and iterative methods. The second half furnishes a comprehensive discussion of the mathematical theory of spectral methods on single domains, including approximation theory, stability and convergence, and illustrative applications of the theory to model boundary-value problems. Both the algorithmic and theoretical discussions cover spectral methods on tensor-product domains, triangles and tetrahedra. All chapters are enhanced with material on the Galerkin with numerical integration version of spectral methods. The discussion of direct and iterative solution methods is greatly expanded as are the set of numerical examples that illustrate the key properties of the various types of spectral approximations and the solution algorithms.

A companion book "Evolution to Complex Geometries and Applications to Fluid Dynamics" contains an extensive survey of the essential algorithmic and theoretical aspects of spectral methods for complex geometries and provides detailed discussions of spectral algorithms for fluid dynamics in simple and complex geometries.

Electronic Structure Calculations for Solids and Molecules: Theory and Computational Methods

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Electronic Structure Calculations for Solids and Molecules: Theory and Computational Methods
Jorge Kohanoff
Manufacturer: Cambridge University Press
ProductGroup: Book
Binding: Hardcover

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ASIN: 0521815916

Book Description

Electronic structure problems are studied in condensed matter physics and theoretical chemistry to provide important insights into the properties of matter. This graduate textbook describes the main theoretical approaches and computational techniques, from the simplest approximations to the most sophisticated methods. It starts with a detailed description of the various theoretical approaches to calculating the electronic structure of solids and molecules, including density-functional theory and chemical methods based on Hartree-Fock theory. The basic approximations are thoroughly discussed, and an in-depth overview of recent advances and alternative approaches in DFT is given. The second part discusses the different practical methods used to solve the electronic structure problem computationally, for both DFT and Hartree-Fock approaches. Adopting a unique and open approach, this textbook is aimed at graduate students in physics and chemistry, and is intended to improve communication between these communities. It also serves as a reference for researchers entering the field.

Finite Elements and Approximation (Dover Books on Engineering)

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Finite Elements and Approximation (Dover Books on Engineering)
O. C. Zienkiewicz , and K. Morgan
Manufacturer: Dover Publications
ProductGroup: Book
Binding: Paperback

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ASIN: 0486453014

Book Description

A powerful tool for the approximate solution of differential equations, the finite element is extensively used in industry and research. This book offers students of engineering and physics a comprehensive view of the principles involved, with numerous illustrative examples and exercises. 1983 edition.

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a small tree bearing so many fruits
A novice review
A Classic
Knuth calls this a beautiful book.

Asymptotic Methods in Analysis
N. G. de Bruijn
Manufacturer: Dover Publications
ProductGroup: Book
Binding: Paperback

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ASIN: 0486642216

Book Description

Original, effective approach teaches by explaining worked examples in detail. "Every step in the mathematical process is explained, its purpose and necessity made clear ... the reader not only has no difficulty in following the rigorous proofs, but even turns to them with eager expectation." — Nuclear Physics.

Customer Reviews:

a small tree bearing so many fruits.......2007-01-14

this book stands apart on my shelves since I used it so often that I had to replace it three times; the subject of asymptotics is both useful and fascinating; this book delves in it through many examples first showing how to deal with them in elementary fashion, then using more advanced tools and finally exhibiting heavy machinery; although a lot of complex analysis tools are present in the text, it is quite possible to get a good grasp of what is done in it since elementary methods are always stated in the first instance e.g. the study of U_n+1 = sin(U_n): using calculus, it is shown that U_n which tends to zero is not far from (3/n)^(1/2) (sorry about that for those who do not speak fluent mathematics...), then 4 terms of the asymptotic development are given followed by a more general method to get an arbitrary number or terms.
This book will prove useful to you in different fashion as the years go on
and your mathematical skill is improving.
Moreover, the price is really affordable.
As a matter of fact, I first bought this book in 1986...and I find it still really pleasurable.

A novice review.......2001-03-18

This will be an alternative review since I lack the complete knowledge foundation required by the book, yet I've purchased it knowing so in advance.

This is simply an elegantly written book in terms of language use. The book is designed with a language use; the way i prefer to write swedish text, no matter the subject at hand.

Also parsing arbitrarly there are some references to other russians work. As far as I can evaluate by glimpse and rudimentary mathematical knowledge there is proof and definition driven reasoning which is a must in mathematics in order to be of great use for the reader at hand.

Also for a novice like myself the extensive and clear, instant writing about necessary mathematical methods, functions and so on to solve the problems developed, functions in use give back great insight value; The art of thinking of how go about solving the issues at hand.

I think one would feel confident in reading this as one go by reading mathematical/Physics courses at University (if you do/did). This mean that for me the reading would be limited to pieces of reading as one grow with required knowledge.

10-20 points in mathematical analysis and additional introduction course in functions of complex variables would suffice (of course this is an estimation).

A Classic.......2000-10-09

This is a Classic introduction to Asymptotics. Its main strength, is the illustration of techniques through the detailed working out of examples. In addition, there is FULL rigour, so that the proofs are complete.

To study this book you should be comfortable with:

a) Undergraduate Level Real Analysis. b) Elementary Notions from Complex Variable Theory including Complex Integration, Calculus of Residues, Power Series Expansions and related ideas.

Knuth calls this a beautiful book........1998-11-28

Asymptotic methods in analysis is great book. I came to know about this book while reading TAOCP vol 1. As Knuth says this is a truly beautiful book on the subject.

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Intersting Studies Hampered by LEDA and LP Dependencies

Algorithms on Trees and Graphs
Gabriel Valiente
Manufacturer: Springer
ProductGroup: Book
Binding: Hardcover

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ASIN: 3540435506

Book Description

Graph algorithms is a well-established subject in mathematics and computer science. Beyond classical application fields, like approximation, combinatorial optimization, graphics, and operations research, graph algorithms have recently attracted increased attention from computational molecular biology and computational chemistry. Centered around the fundamental issue of graph isomorphism, this text goes beyond classical graph problems of shortest paths, spanning trees, flows in networks, and matchings in bipartite graphs. Advanced algorithmic results and techniques of practical relevance are presented in a coherent and consolidated way. This book introduces graph algorithms on an intuitive basis followed by a detailed exposition in a literate programming style, with correctness proofs as well as worst-case analyses. Furthermore, full C++ implementations of all algorithms presented are given using the LEDA library of efficient data structures and algorithms. Numerous illustrations, examples, and exercises, and a comprehensive bibliography support students and professionals in using the book as a text and source of reference

Customer Reviews:

Intersting Studies Hampered by LEDA and LP Dependencies.......2003-12-24

I'm pleased with the text in this book; the descriptions are almost all clear, and reading through the book gives me insight into some more interesting problems in trees and graphs. The high points of the book are its treaments of tree and graph isomorphism, but I also found the discussions of non-traditional traversal algorithms on trees and graphs very interesting. The author discussions leaf-first, breadth-first, and depth-first traversals and provides algorithms for their implementation. Many of the algorithms include correctness proofs.

These topics alone made the book worth its to me. A deep academic book that costs less than $50 is nearly unheared-of.

Unfortunately, there are two flaws that make the book hard to use. In a nutshell, the author expects the reader to buy into a couple of pretty invasive and expensive propositions.

First, the author decided to use literate programming for all of his presented algorithms and code fragments. This isn't so bad, since literate programming is about documenting code. If you suppose that the author wrote the code, then documented it, then calld it a book, using a tool like literate programming seems like a natural choice.

But if you're not familiar with literate programming, it's a bit of a chore. The author's introduction to literate programming doesn't help with some of the questions even an experienced programmer might have wend reading the text. More practically, literate programming enforces operators that are different than most C/C++ developers are familiar with, and can cause confusion when reading the text. ^ is used, for example, to indicate a logical and, where C/C++ developers expect it to indicate a bitwise-exclusive or.

While it's esay to eventually overcome such tricks of memory, I've been finding it hard to scan literate programs to find definitions and declarations. The author doesn't include a CD (and at this cover price, that is hard to fault) but also doesn't make his code available for download. His website includes a LEDA-based program that interactively demonstrates some algorithms, but doesn't include code for the algorithm his own books develops and discusses.

The other decision made by the author, to the overwhelming inconvenience of this reader, is the reliance on the LEDA library for his samples and programs. The algorithms are understandable without the library, but a reader without access to LEDA doesn't benefit from any of the visualizations the author provides. , and several include

In fact, the author spends about 40 pages discussing LEDA and the characteristics of its implementation. Maybe researchers working on tree and graph algorithms all use LEDA and have ready access to it, but the literate programming code provided to for some of the algorithms isn't useful to readers who aren't familiar with LEDA, as researching the definitions and declarations themsleves becomes arduous.

The bibliography is very diverse, with more than 380 entries and is well-cited throughout the book. Unfortunately, the author sometimes relies on the bibliography too much. On page 392, the author brings up "the so-called graph isopomorphism disease" without defining it himself; he instead relies on his bibliography entries to give the user any definition or background on the "disease".

Unfortunately, the index received not nearly as much attention as the bibliography; neglecting whitespace, it's scarcely more than a single page long!

The book appears to be something more than a research paper, but is written much like a research apper would be. The book probably also serves well for a class that teaches this subject, and assumes LEDA and literate programming as prerequisites. But as a commercial developer who's interested in applying advanced graph and tree algorithms to the work I'm doing, I found the book has limited its value by relying on LEDA and applying literate programming.

Book Description

Mixing up various disciplines frequently produces something that are profound and far-reaching. Cybernetics is such an often-quoted example. Mix of information theory, statistics and computing technology proves to be very useful, which leads to the recent development of information-theory based methods for estimating complicated probability distributions.

Estimating probability distribution of a random variable is the fundamental task for quite some fields besides statistics, such as reliability, probabilistic risk analysis (PSA), machine learning, pattern recognization, image processing, neural networks and quality control. Simple distribution forms such as Gaussian, exponential or Weibull distributions are often employed to represent the distributions of the random variables under consideration, as we are taught in universities. In engineering, physical and social science applications, however, the distributions of many random variables or random vectors are so complicated that they do not fit the simple distribution forms at al.

Exact estimation of the probability distribution of a random variable is very important. Take stock market prediction for example. Gaussian distribution is often used to model the fluctuations of stock prices. If such fluctuations are not normally distributed, and we use the normal distribution to represent them, how could we expect our prediction of stock market is correct? Another case well exemplifying the necessity of exact estimation of probability distributions is reliability engineering. Failure of exact estimation of the probability distributions under consideration may lead to disastrous designs.
There have been constant efforts to find appropriate methods to determine complicated distributions based on random samples, but this topic has never been systematically discussed in detail in a book or monograph. The present book is intended to fill the gap and documents the latest research in this subject.

Determining a complicated distribution is not simply a multiple of the workload we use to determine a simple distribution, but it turns out to be a much harder task. Two important mathematical tools, function approximation and information theory, that are beyond traditional mathematical statistics, are often used. Several methods constructed based on the two mathematical tools for distribution estimation are detailed in this book. These methods have been applied by the author for several years to many cases. They are superior in the following senses:

(1) No prior information of the distribution form to be determined is necessary. It can be determined automatically from the sample;
(2) The sample size may be large or small;
(3) They are particularly suitable for computers.

It is the rapid development of computing technology that makes it possible for fast estimation of complicated distributions.

The methods provided herein well demonstrate the significant cross influences between information theory and statistics, and showcase the fallacies of traditional statistics that, however, can be overcome by information theory.

Key Features:

- Density functions automatically determined from samples

- Free of assuming density forms

- Computation-effective methods suitable for PC

Book Description

Most functions that occur in mathematics cannot be used directly in computer calculations. Instead they are approximated by manageable functions such as polynomials and piecewise polynomials. The general theory of the subject and its application to polynomial approximation are classical, but piecewise polynomials have become far more useful during the last twenty years. Thus many important theoretical properties have been found recently and many new techniques for the automatic calculation of approximations to prescribed accuracy have been developed. This book gives a thorough and coherent introduction to the theory that is the basis of current approximation methods. Professor Powell describes and analyses the main techniques of calculation supplying sufficient motivation throughout the book to make it accessible to scientists and engineers who require approximation methods for practical needs. Because the book is based on a course of lectures to third-year undergraduates in mathematics at Cambridge University, sufficient attention is given to theory to make it highly suitable as a mathematical textbook at undergraduate or postgraduate level.

Approximating Integrals Via Monte Carlo and Deterministic Methods

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Everything in one place

Approximating Integrals Via Monte Carlo and Deterministic Methods
Michael Evans , and Tim Swartz
Manufacturer: Oxford University Press, USA
ProductGroup: Book
Binding: Hardcover

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ASIN: 0198502788

Book Description

This book is designed to introduce graduate students and researchers to the primary methods useful for approximating integrals. The emphasis is on those methods that have been found to be of practical use, and although the focus is on approximating higher- dimensional integrals the lower-dimensional case is also covered. Included in the book are asymptotic techniques, multiple quadrature and quasi-random techniques as well as a complete development of Monte Carlo algorithms. For the Monte Carlo section importance sampling methods, variance reduction techniques and the primary Markov Chain Monte Carlo algorithms are covered. This book brings these various techniques together for the first time, and hence provides an accessible textbook and reference for researchers in a wide variety of disciplines.

Customer Reviews:

Everything in one place.......2000-11-16

This book has lots of information about estimating integrals all in one place. The text is very readable and has clearly explained examples throughout. I would recommend this book to anyone who does work using computers to calculate integrals.

An Introduction to Stein's Method (Lecture Notes Series, Institute for Mathematical Sciences, Vol. 4) (Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore)

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An Introduction to Stein's Method (Lecture Notes Series, Institute for Mathematical Sciences, Vol. 4) (Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore)
A. D. Barbour , and Louis H. Y. Chen
Manufacturer: World Scientific Publishing Company
ProductGroup: Book
Binding: Paperback

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ASIN: 981256330X

Book Description

A common theme in probability theory is the approximation of complicated probability distributions by simpler ones, the central limit theorem being a classical example. Stein's method is a tool which makes this possible in a wide variety of situations. Traditional approaches, for example using Fourier analysis, become awkward to carry through in situations in which dependence plays an important part, whereas Stein's method can often still be applied to great effect. In addition, the method delivers estimates for the error in the approximation, and not just a proof of convergence. Nor is there in principle any restriction on the distribution to be approximated; it can equally well be normal, or Poisson, or that of the whole path of a random process, though the techniques have so far been worked out in much more detail for the classical approximation theorems.

This volume of lecture notes provides a detailed introduction to the theory and application of Stein's method, in a form suitable for graduate students who want to acquaint themselves with the method. It includes chapters treating normal, Poisson and compound Poisson approximation, approximation by Poisson processes, and approximation by an arbitrary distribution, written by experts in the different fields. The lectures take the reader from the very basics of Stein's method to the limits of current knowledge.

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