Preface

Akin to B-splines, the appeal of subdivision surfaces reaches across disciplines from mathematics to computer science and engineering. In particular, subdivision surfaces have had a dramatic impact on computer graphics and animation over the last 10 years: the results of a development that started three decades ago can be viewed today at movie theaters, where feature length movies cast synthetic characters ‘skinned’ with subdivision surfaces. Correspondingly, there is a rich, evergrowing literature on its fundamentals and applications. Yet, as with every vibrant new field, the lack of a uniform notation and standard analysis tools has added unnecessary, at times inconsistent, repetition that obscures the simplicity and beauty of the underlying structures. One goal in writing this book is to help shorten introductory sections and simplify proofs by proposing a standard set of concepts and notation. When we started writing this book in 2001, we felt that the field had sufficiently settled for standardization. After all, Cavaretta, Dahmen, and Micchelli’s monograph [CDM91] had appeared 10 years earlier and we could build on the habilitation of the second author as well as a number of joint papers. But it was only in the process of writing and seeing the issues in conjunction, that structures and notation became clearer. In fact, the length of the book repeatedly increased and decreased, as key concepts and structures emerged. Chapter 2/15, for example, was a late addition, as it became clear that the differential geometry for singular parameterizations, of continuity, smoothness, curvature, and injectivity, must be established upfront and in generality to simplify the exposition and later proofs. By contrast, the key definition of subdivision surfaces as splines with singularities, in Chap. 3/39, was a part of the foundations from the outset. This point of view implies a radical departure from any focus on control nets and instead places the main emphasis on nested surface rings, as explained in Chap. 4/57. Careful examination of existing proofs led to the explicit formulation of a number of assumptions, in Chap. 5/83, that must hold when discussing subdivision surfaces in generality. Conversely, placing these key assumptions upfront, shortened the presentation considerably. Therefore, the standard examples of subdivision algorithms