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Pythagorean-Hodograph Curves: Algebra and Geometry Inseparable

Rida T. Farouki
Publication Date: 
Number of Pages: 
Geometry and Computing 1
[Reviewed by
Luiz Henrique de Figueiredo
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We learn in elementary differential geometry that the nicest parametrization of curves is the unit-speed or arc-length parametrization. However, even for simple curves like cubic curves, the arc-length parametrization cannot be expressed in elementary form. Farouki proved in 1991 that no rational plane curve has an rational arc-length parametrization, with the trivial exception of straight lines. This result marked the beginning of Farouki's involvement with Pythagorean-hodograph curves.

Pythagorean-hodograph curves are parametric polynomial curves whose arc-length is a polynomial function as well; this happens when x′(t)2+y′(t)2 is the square of a polynomial, hence the connection with Pythagoras theorem and Pythagorean triples. ("Hodograph is another name for "derivative.") This special property has many important consequences: for instance, the offsets of Pythagorean-hodograph curves are rational curves. On the other hand, Pythagorean-hodograph curves naturally have less degrees of freedom than general polynomial curves. For instance, the role of the all-purpose cubic Bézier is played by Pythagorean-hodograph quintics.

This book is a learned treatise on Pythagorean-Hodograph curves, and explains the theory developed by the author and collaborators since 1990, which was, until now, only available scattered in many papers. The book is very well written and contains a wealth of interesting information, not only about its main subject matter but also about the background material on algebra, geometry, and computer-aided geometric design, including historical references and quotes.

The prime audience for this book is students, researchers, and professionals working in computer-aided design and manufacturing, but anyone interested in mathematics, geometric modeling, or computer graphics and animation will learn much interesting mathematics from it.

Luiz Henrique de Figueiredo is a researcher at IMPA in Rio de Janeiro, Brazil. His main interests are numerical methods in computer graphics, but he remains an algebraist at heart. He is also one of the designers of the Lua language.