Analysis and Design of Univariate Subdivision Schemes

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Springer Science & Business Media, Aug 16, 2010 - Mathematics - 218 pages
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‘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.
 

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Contents

Notation
3
Part I Prependices
4
Part II Dramatis Personae
46
Part III Analyses
60
Part IV Design
138
Part V Implementation
165
Part VI Appendices
183
Bibliography
207
Index
213
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About the author (2010)

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