Introduction and Overview

Subdivision surfaces can be viewed from at least three different vantage points. A designer may focus on the increasingly smooth shape of refined polyhedra. The programmer sees local operators applied to a graph data structure. This book views subdivision surfaces as spline surfaces with singularities and it will focus on these singularities to reveal the analytic nature of subdivision surfaces. Leveraging the rich interplay of linear algebra, analysis and differential geometry that the spline approach affords, we will, in particular, be able to clarify the necessary and sufficient constraints on subdivision algorithms to generate smooth surfaces. Viewing subdivision surfaces as spline surfaces with singularities is, at present, an unconventional point of view. Visualizing a sequence of polyhedra or tracking a sequence of control nets appears to be more intuitive. Ultimately, however, both views fail to capture the properties of subdivision surfaces due to their discrete nature and lack of attention to the underlying function space. In Sects. 1.1/1 and 1.2/2, we now briefly discuss the two points of view not taken in this book while in Sect. 1.3/4 the analytic view of subdivision surfaces as splines with singularities is sketched out. Section 1.4/6 delineates the focus and scope and Sect. 1.5/7 gives an overview over the topics covered in the book. A useful section to read is Sect. 1.6/7 on notation.

The trailing two sections are special. We felt a need to recall the state of the art in subdivision in the regular, shift invariant setting, and to give an overview on the historical development of the topic discussed in this book. In view of our own, limited expertise in these fields, we decided to seek prominent help. Nira Dyn and Malcolm Sabin, two pioneers and leading researchers in the subdivision community agreed to contribute, and their insightful overviews form Sects. 1.7/8 and 1.8/11.