Subdivision surfaces permit a designer to specify the approximate form of a surface defining an object and to refine and smooth the form to obtain a more useful or attractive version of the surface. A considerable amount of mathematical theory is required to understand the characteristics of the resulting surfaces, and this book provides a careful and rigorous presentation of the mathematics underlying subdivision surfaces as used in computer graphics and animation, explaining the concepts necessary to easily read the subdivision literature. It also organizes subdivision methods in a unique and unambiguous hierarchy in order to provide insight and understanding. The material is not restricted to questions related to regularity of subdivision surfaces at so-called extraordinary points but instead gives a broad discussion of the various methods. It is excellent preparation for reading more advanced texts that delve more deeply into special questions of regularity. The authors provide exercises and projects at the end of each chapter. Course material, including solutions to the exercises, is available on an associated Web page.