Analysis and Design of Univariate Subdivision Schemes

This book covers the theory of subdivision curves in detail, which is a prerequisite for that of subdivision surfaces. It then considers how those analyses can be used in reverse to design a scheme best matching the particular criteria for a given application.

'Subdivision' is a way of representing smooth shapes in a computer. A curve or surface (both of which contain an in?nite number of points) is described in terms of two objects. One object is a sequence of vertices, which we visualise as a polygon, for curves, or a network of vertices, which we visualise by drawing the edges or faces of the network, for surfaces. The other object is a set of rules for making denser sequences or networks. When applied repeatedly, the denser and denser sequences are claimed to converge to a limit, which is the curve or surface that we want to represent. This book focusses on curves, because the theory for that is complete enough that a book claiming that our understanding is complete is exactly what is needed to stimulate research proving that claim wrong. Also because there are already a number of good books on subdivision surfaces. The way in which the limit curve relates to the polygon, and a lot of interesting properties of the limit curve, depend on the set of rules, and this book is about how one can deduce those properties from the set of rules, and how one can then use that understanding to construct rules which give the properties that one wants.

All the fundamental ideas Presented in an accessible fashion even for those whose mathematics is a tool to be used, not a way of life Separate sections on mathematical techniques providing revision for those needing it

Auteur
The author has spent his professional life on the numerical representation of shape.

Contenu
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.