Fundamentally, geometric modeling studies the methods used to construct and control numerical geometric models of real and imaginary objects.

Geometric modeling is entirely dependent of several areas of mathematics, but primarily differential geometry (that employs differential and integral calculus and linear algebra for developing planes, curves, and surfaces in 3D Euclidean space) and numerical methods (to find numerical approximations – the more precise the approximation, the better the model).

How basic math becomes numerical modeling that becomes geometric modeling is the subject of a new book, *Geometric Modeling*, by Dr. Nikolay Golovanov. The book is based on the author’s experience gained throughout his career, and especially during the development of the ASCON Group’s C3D geometric kernel as a principal architect.

In a nutshell, the book outlines the methods of geometric modeling, including methods for constructing curves, surfaces, and solids. It describes the algorithms and data structures behind geometric objects. It also presents the principles of the interconnections between the elements of geometric models. Finally, the book examines some of the applications of geometric models, such as determining model physical characteristics, rendering, simulation, etc.

The book follows a logical sequence of topics that build on each other as the book progresses. For example, curves, surfaces, constructions, and geometric models (building, constraints, and applications).

At the beginning of each chapter concepts are defined in relatively simple terms with words and then more detailed with mathematical equations and graphical representations.

Each chapter ends with a few exercises that gently test your understanding of the chapter’s basic concepts that you should come away with.

Some of the highlights of the book for me included:

- The theoretical and mathematical differences between Bezier and B-spline curves and surfaces
- What constitutes analytic surfaces
- Geometric models as boundary representations, notably shells and solids
- Different methods of surface and solid geometric modeling, including Boolean operations, chamfering, filleting, and direct modeling
- Constraints for controlling geometric models
- Geometric model content that includes geometric description, geometric constraints, build history, and attributes
- How optical properties are modeled
- Calculating physical properties in geometric models, such as moments of inertia
- Fundamentals of different coordinate systems and their implications in geometric modeling.

OK, I understand that this book may not be for everybody, but Dr. Golovanov presents what could be extremely dry and boring material in a way that is methodical and in some ways even entertaining. I think the book is suitable as both a text for students, as well as a reference source for practitioners.

The author, Dr. Nikolay Golovanov, is Russian, and I’m assuming that the book was translated from that language. Knowing this, I was expecting some bad consequences from the translation, but was surprised at how clean the text was with regard to usage, syntax, and presentation. And, on one hand, the book is concise, but on the other, comprehensive.

Regardless of your current station in your professional life, if you are a student of the history of CAx and are curious how all this stuff works, I highly recommend *Geometric Modeling*. You might have to re-familiarize yourself with some math concepts (as I certainly did), but you will be rewarded coming away with a much better understanding of what goes on behind the scenes with virtually all CAx applications. In other words, it’s definitely worth the effort.

The content can get complicated at times, but overall is very well organized, approachable, and understandable.

The book is available as a paperback at Amazon.com and as an e-book at www.scribd.com so you can have it with you as a ready reference on a digital device.

*Geometric Modeling: The Mathematics of Shapes *is available at the following Amazon link.

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