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Analysis and Design of Univariate Subdivision Schemes

  • Textbook
  • © 2010

Overview

Authors:
  1. Malcolm Sabin

    1. Ely, Cambridgeshire, United Kingdom

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  • 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

Part of the book series: Geometry and Computing (GC, volume 6)

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Table of contents (37 chapters)

  1. Front Matter

    Pages I-XIV
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  2. Prependices

    1. Front Matter

      Pages 3-3
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    2. Functions and Curves

      • Malcolm Sabin
      Pages 5-6

    3. Differences

      • Malcolm Sabin
      Pages 7-10

    4. B-Splines

      • Malcolm Sabin
      Pages 11-16

    5. Eigenfactorisation

      • Malcolm Sabin
      Pages 17-24

    6. Enclosures

      • Malcolm Sabin
      Pages 25-28

    7. Hölder Continuity

      • Malcolm Sabin
      Pages 29-32

    8. Matrix Norms

      • Malcolm Sabin
      Pages 33-36

    9. Joint Spectral Radius

      • Malcolm Sabin
      Pages 37-38

    10. Radix Notation

      • Malcolm Sabin
      Pages 39-40

    11. z-transforms

      • Malcolm Sabin
      Pages 41-45

  3. Dramatis Personae

    1. Front Matter

      Pages 47-47
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    2. An introduction to some regularly-appearing characters

      • Malcolm Sabin
      Pages 49-59

  4. Analyses

    1. Front Matter

      Pages 61-61
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    2. Support

      • Malcolm Sabin
      Pages 63-72

    3. Enclosure

      • Malcolm Sabin
      Pages 73-76

    4. Continuity 1 - at Support Ends

      • Malcolm Sabin
      Pages 77-80

    5. Continuity 2 - Eigenanalysis

      • Malcolm Sabin
      Pages 81-93

    6. Continuity 3 - Difference Schemes

      • Malcolm Sabin
      Pages 95-101
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Keywords

About this book

‘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.

Reviews

From the reviews:

“The analysis and design of subdivision schemes, by Malcolm Sabin, provides a practical approach to subdivision curve schemes based on the state of the subject mainly from the years 1972–2010 … . The bibliography includes a fairly comprehensive list of references sorted by year of publication. … It is appropriate for the computer science student wishing to learn about the subject, and can whet the appetite of the more mathematically inclined reader who is interested in further investigation of the subject.” (Scott N. Kersey, Mathematical Reviews, Issue 2011 j)

Authors and Affiliations

  • Ely, Cambridgeshire, United Kingdom

    Malcolm Sabin

About the author

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

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