A Cookbook for Variational Subdivision

These course notes will attempt to answer the following questions: What is subdivision? How can the rules for subdivision schemes be derived in a systematic manner? How can these rules be extended to handle special topological features such as extraordinary points or creases? We will argue that most subdivision schemes correspond to a special type of multigrid method that generates shapes which solve (or nearly solve) a variational problem. These subdivision schemes are influenced by two factors: the variational functional and the local grid topology. Using a recipe based on this observation, we will build subdivision schemes for several interesting examples including B-splines, minimum energy curve networks, membrane splines and fluid flow.