Analysis and Design of Univariate Subdivision Schemes Prependices Functions and Curves Differences B Splines Eigenfactorisation Enclosures Hölder Continuity Matrix Norms Joint Spectral Radius Radix Notation z transforms Dramatis Personae An introduction to some regularly appearing characters Analyses Support Enclosure Continuity 1 at Support Ends Continuity 2 Eigenanalysis Continuity 3 Difference Schemes Continuity 4 Difference Eigenanalysis Continuity 5 the Joint Spectral Radius What Converges  Reproduction of Polynomials Artifacts Normalisation of Schemes Summary of Analysis Results Design The Design Space Linear Subspaces of the Design Space Non linear Conditions Non Stationary Schemes Geometry Sensitive Schemes Implementation Making Polygons Rendering Interrogation End Conditions Modifying the Original Polygon Appendices Proofs Historical Notes Solutions to Exercises Coda. This book covers the theory of subdivision curves in detail which is a prerequisite for that of subdivision surfaces. The book reports on the currently known ways of analysing a subdivision scheme (i.e. measuring criteria which might be important for the application of a scheme to a given context). It then goes on to consider how those analyses can be used in reverse to design a scheme best matching the particular criteria for a given application. The book is presented in an accessible fashion even for those whose mathematics is a tool to be used not a way of life. It should provide the reader with a full and deep understanding of the state of the art in subdivision analysis and separate sections on mathematical techniques provide revision for those needing it. The book will be of great interest to those starting to do research in CAD CAE. It will also appeal to those lecturing in this subject and industrial workers implementing these methods. The author has spent his professional life on the numerical representation of shape and his book fills a need for a book covering the fundamental ideas in the simplest possible context that of curves.