Geometric Modeling

Geometric Modeling is a fun subject, with many applications (for instance, shape design for cars and airplanes), and plenty of interesting mathematics: polynomial interpolation and approximation, differential geometry of parametric curves and surfaces, topology of solids, planar subdivisions, graph representations, numerical methods for the solution of equations, etc. One of the main themes in Geometric Modeling is the expressive power of combinations of low-degree polynomial curves and surfaces, which are known in the trade as splines.

Mortenson's book takes the concrete approach to all these subjects and succeeds well in treating them. When the first edition came out in 1985, there were very few books on Geometric Modeling and Mortenson's book was one of the best. Alas, the book now feels a little dated, perhaps because there are now lots of books on splines (probably too many). Moreover, the book fails to discuss subdivision surfaces, which is the hot topic in Geometric Modeling these days, because they generalize splines (but are easier to explain) and support control meshes of general topology. This makes subdivision surfaces easy to use by animators. In fact, subdivision surfaces seem to have taken the computer animation industry by storm. (See for instance Pixar's "Jerry's Game", "A Bugs Life", and "Toy Story II".)

Nevertheless, the book remains a good, concrete introduction to Geometric Modeling and can be used in a course without worries, especially if supplemented with material on subdivision surfaces, should the instructor lean that way. People teaching numerical analysis or differential geometry may also find good examples in Geometric Modeling.

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.