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Reviewer: Adina Raclariu

Differential computational geometry is a powerful tool in the modeling and representation of curves, surfaces, and solid shapes. The methods and concepts of differential geometry have many practical applications in robotics and machine vision. Any program that successfully models even a small part of intelligence will be massive and complex; this complexity will require the use of methods and knowledge from many domains. The differential geometric methods explained in this book will play a key role in the development of object and scene representation. The author helps the readers develop their ability to use intuitive and heuristic tools to shape the problem. He explains the physical significance of every mathematical model used; the concrete pictures that illustrate all the examples help the reader get a firm grip on the subject matter. The first chapter establishes the basic entities that make up the physical space and introduces methods to describe or reveal different aspects of the shape. The second chapter reveals the author's approach to studying objects in three-dimensional space; objects are defined physically. Wherever possible, Koenderink uses representations that lend themselves to computer implementation. The third chapter deals with smooth entities such as curves and local patches in relation to the application domains. It provides a short but fairly complete review of space curves. The author introduces several remarkable curves on surfaces—geodesics, lines of curvature, asymptotic and parabolic curves, and ridge curves. The fourth chapter uses the concepts of compactness and completeness to delimit the surfaces of practical interest. It treats curves on a surface as the boundaries of special areas or as the geometric loci of extraneous conditions such as shadow boundaries. A complex curve is described and analyzed as the framework or skeleton that captures the global shape of the surface. Finally, the author discusses the problems of ranging data and illumination as they depend on the shape. The fifth chapter, “Dynamic Shape,” studies blobs in morphogenetic sequences. Koenderink stresses that any real density is already a one-parameter family or morphogenetic sequence with the resolution as the parameter. He then describes some typical evolutionary sequences that do not conserve the topology of the surfaces. The next section considers only transformations that do conserve the topology. The last chapter reviews some common models for shapes and indicates their typical use. The book has great scientific and practical value. The author suggests a lot of useful methods and models from differential geometry. He has found the most explicit and intuitive way to present difficult notions for readers with little experience in mathematics. The book also contains sections that teach the reader to write simple differential geometry programs using a dialect of “pidgin Pascal or Modula-2.” The technical material is sufficiently complete to point you in the right direction, but you will need to consult standard texts in order to start any serious work. This book is a true guide for those who are concerned with computer modeling of geometrical entities. It compiles the knowledge in this area that had previously been treated only in papers. The book also contains a useful glossary and an adequate index. The style is modern and pleasant, and the author's explanations are excellent.