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Curves and Surfaces in Geometric Modeling: Theory & Algorithms
 
 
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Curves and Surfaces in Geometric Modeling: Theory & Algorithms [Hardcover]

Jean Gallier (Author)

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Curves and Surfaces for Geometric Design offers both a theoretically unifying understanding of polynomial curves and surfaces and an effective approach to implementation that you can bring to bear on your own workwhether you're a graduate student, scientist, or practitioner.


Inside, the focus is on "blossoming"the process of converting a polynomial to its polar formas a natural, purely geometric explanation of the behavior of curves and surfaces. This insight is important for far more than its theoretical elegance, for the author proceeds to demonstrate the value of blossoming as a practical algorithmic tool for generating and manipulating curves and surfaces that meet many different criteria. You'll learn to use this and related techniques drawn from affine geometry for computing and adjusting control points, deriving the continuity conditions for splines, creating subdivision surfaces, and more.


The product of groundbreaking research by a noteworthy computer scientist and mathematician, this book is destined to emerge as a classic work on this complex subject. It will be an essential acquisition for readers in many different areas, including computer graphics and animation, robotics, virtual reality, geometric modeling and design, medical imaging, computer vision, and motion planning.



* Achieves a depth of coverage not found in any other book in this field.

* Offers a mathematically rigorous, unifying approach to the algorithmic generation and manipulation of curves and surfaces.

* Covers basic concepts of affine geometry, the ideal framework for dealing with curves and surfaces in terms of control points.

* Details (in Mathematica) many complete implementations, explaining how they produce highly continuous curves and surfaces.

* Presents the primary techniques for creating and analyzing the convergence of subdivision surfaces (Doo-Sabin, Catmull-Clark, Loop).

* Contains appendices on linear algebra, basic topology, and differential calculus.

Book Info

Offers a theoretically unifying understanding of polynomial curves and surfaces as well as an effective approach to implementation that you can apply to your own work as a graduate student, scientist, or practitioner. DLC: Curves, Algebraic--Data processing.
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Geometry, what a glorious subject! Read the first page
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Front Cover | Copyright | Table of Contents | Excerpt | Index | Back Cover
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Amazon.com: 5.0 out of 5 stars (3 customer reviews)

12 of 12 people found the following review helpful
5.0 out of 5 stars Best text on geometric design, Jan 16 2001
By A Customer - Published on Amazon.com
This review is from: Curves and Surfaces in Geometric Modeling: Theory & Algorithms (Hardcover)
This is a great book, definitely the best among the various books on geometric design and CAGD (other good ones include Farin, Mortsenson, Piegl and Tiller, Hoscheck and Lasser). It is not as encyclopedic as the sources listed above, but it a lot more coherent and a lot clearer, because it follows the unifying concept of blossoming. As a result, one gets multiple complementary views of polynomial curves and surfaces: algebraic, geometric, combinatorial, and algorithmic. For example, we can see where the Bernstein polynomials come from, instead of mysteriously being dropped from the sky. The systematic use of blossoms (polar forms) is particularly elegant in the presentation of surfaces, where it clarifies greatly the differences between rectangular and triangular patches. The discussion of subdivision versions of the de Casteljau algorithm is very thorough and unique. Gallier's book is also the only book to discuss subdivision surfaces in some detail (Doo-Sabin, Catmull-Clark, and Loop). In particular, an analysis of the convergence of Loop's scheme is given. For this, the author gives a remarkable crash course on the discrete Fourier transform. However, this chapter is too dense and should have been split. Also, much more pictures are needed. It seems that the author was in a rush. The appendix on vector spaces is gorgeous, and the one on differentials is also excellent. This book is highly recommended to mathematically inclined readers interested in geometric modeling and computer graphics. Too bad that applications to medicine such as organ modeling, or to computer animation, are not presented. Nevertheless, Mathematica code is provided for most of the algorithms. A web site would be helpful.

10 of 10 people found the following review helpful
5.0 out of 5 stars A brillian geometry book, Feb 9 2000
By Lisa Cohen - Published on Amazon.com
This review is from: Curves and Surfaces in Geometric Modeling: Theory & Algorithms (Hardcover)
I found this book an exellent introduction to advanced geometry concepts used in computer graphics, vision, robotics, geometric modeling and many other related areas. Gallier has struck a perfect balance between formal mathematical rigour and intuition and readability which the book lends easily with its many beautiful illustrations and examples. The concept of "blossoming" is a rarely-seen but extremely elegant way of presenting the curves and surfaces. This book is a must for anyone who loves the elegance of geometry.

12 of 13 people found the following review helpful
5.0 out of 5 stars A good mathematical review for practicing graphics engineers, July 19 2000
By Shankar N. Swamy - Published on Amazon.com
This review is from: Curves and Surfaces in Geometric Modeling: Theory & Algorithms (Hardcover)
This book is a good review of the concepts of geometry for Modeling. The presentation is original. The mathematical treatment is sound. This a "required reading" for those in Computer Graphics research and did not have a good course in geometry. Those who have had a good course in geometry will appreciate the original style of presentation. This book fills a long felt gap in the treatment of geometry from the perspective of Computer Graphics. The book assumes minimal background in mathematics, and is almost self-contained.

There are fewer graphics programmers who have an adequate understanding of the underlying mathematical concepts. This book can partially help the graphics programmers to cross over to that select group. Problems at the end of each chapter enhance the value of the book. The material is updated with latest developments in the field such as subdivision surfaces.

People interested in Computer Graphics, Geometric Modeling, Computer Vision, and Robotics will benefit from studying this book.

› Go to Amazon.com to see all 3 reviews  5.0 out of 5 stars 

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