The idea of an oriented version of projective geometry is not new. At least part of it was known to earlier generations of mathematicians. The emphasis in this book is on a formalized treatment, but it contains examples and many hints that aid in the derivation of pertinent algorithms. The book directs itself mainly to computer graphics researchers, although a wider audience could be anyone interested in computational geometry.

The book starts with a chapter on projective geometry, which presents the three models of projective space most frequently used in this book--the straight model, spherical geometry, and the homogeneous coordinates model. It mentions advantages and disadvantages of classical projective geometry; disadvantages include the non-orientedness of the projective plane and the ambiguity of line segments and directions. Then “oriented” projective geometry is introduced. Here points, lines, and planes do have a direction or orientation, resulting in, among other things, signed homogeneous coordinates. In chapter 2, oriented projective geometry is illustrated by means of the three models mentioned above. The following chapters generalize to spaces of arbitrary dimensions, introduce useful operations, and study properties. Chapter 8 defines projective maps, used in the next chapter to introduce abstract oriented projective spaces. Chapters 10 through 13 deal with duality and additional concepts having to do with projective maps. The book closes with a discussion of computer processing of lines and planes.

The book does a good job of explaining and applying the concepts developed. Perhaps it hints too heavily about the substantial improvement of algorithms based on oriented projective geometry; in many cases, the enhancements seem marginal. It puts all the issues having to do with the application of projective geometry in computer graphics into one consistent framework, however. This framework is badly needed because computer graphics texts often treat this subject in a haphazard manner, if at all.

This book is a slightly revised version of the author’s Ph.D. thesis, which appeared in 1988. Various projective geometry problems surfacing in computer graphics have recently been discussed in another Ph.D. thesis [1].

The index is sufficient and the references are adequate. The typography is reasonable, yet the print quality is weak. Figures could benefit from professional drafting. In addition, the book has too many typos (the first one appears halfway down the first page), many of which could have been tracked down with a spelling checker.

Although exercises are missing, the book could be used in an upper-level course or seminar in computer science or computational geometry. The best use, however, is as a text and reference for computer scientists (especially computer graphics researchers) to learn about the revived art of oriented projective geometry.