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Analysis and Design of Univariate Subdivision Schemes / by Malcolm Sabin
Resource type: Ressourcentyp: Buch (Online)Book (Online)Language: English Series: Geometry and Computing ; 6 | SpringerLink BücherPublisher: Berlin, Heidelberg : Springer-Verlag Berlin Heidelberg, 2010Description: Online-Ressource (XIV, 218 p, digital)ISBN:- 9783642136481
- 1280391790
- 9781280391798
- 004
- 511.1 23
- 516.152 22/ger
- 510
- QA76.9.I52
- QA76.9.C65
Contents:
Summary: 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.Summary: 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.PPN: PPN: 1650103026Package identifier: Produktsigel: ZDB-2-SEB | ZDB-2-SXMS | ZDB-2-SMA
""Analysis and Design of Univariate Subdivision Schemes""; ""Preface""; ""Table of Contents""; ""Notation""; ""Part I. Prependices""; ""1. Functions and Curves""; ""1.1 Summary""; ""2. Differences""; ""2.1 First Differences""; ""2.2 Higher Differences""; ""2.3 Differences of Polynomials""; ""2.4 Divided Differences""; ""2.5 Summary""; ""3. B-Splines""; ""3.1 Definition""; ""3.2 Derivative Properties""; ""3.3 Construction""; ""3.4 Refinement""; ""3.5 Summary""; ""4. Eigenfactorisation""; ""4.1 Definition""; ""4.2 Uniqueness""; ""4.3 Properties of Eigenvectors""
""4.3.1 What happens when a general vector is multiplied by a matrix?""""4.3.2 What happens when a general vector is multiplied by a matrix repeatedly?""; ""4.4 Calculating Eigencomponents""; ""4.5 The Effect of Non-zero Off-diagonal Elements""; ""4.5.1 Jordan blocks""; ""4.5.2 Effect of a Jordan block on the multiplication of a (generalised) eigenvector by the matrix.""; ""4.5.3 Complex eigenvalues""; ""4.6 Summary""; ""5. Enclosures""; ""5.1 Definition""; ""5.2 Examples of Enclosures""; ""5.3 Representations""; ""5.4 Summary""; ""6. Hölder Continuity""; ""6.1 Continuity""
""6.2 Derivatives""""6.3 Hölder Continuity""; ""6.4 Summary""; ""7. Matrix Norms""; ""7.1 Vector Norms""; ""7.2 Matrix Norms""; ""7.2.1 Proofs of matrix norm properties""; ""7.2.2 Evaluating matrix norms""; ""7.3 Summary""; ""8. Joint Spectral Radius""; ""8.1 Summary""; ""9. Radix Notation""; ""9.1 Summary""; ""10. z-transforms""; ""10.1 The z-transform""; ""10.2 Why z?""; ""10.3 What Sort of Object is z?""; ""10.4 Some Special Sequences""; ""10.5 Normalisation""; ""10.6 Summary""; ""Part II. Dramatis Personae""; ""11. An introduction to some regularly appearingcharacters""
""11.1 Polygons""""11.2 Labelling and Parametrisation""; ""11.3 Primal and Dual Schemes""; ""11.4 Ternary Schemes and Higher Arities""; ""11.5 Interpolatory Schemes""; ""11.6 Range""; ""11.7 Representations of Subdivision Schemes""; ""11.8 Exercises""; ""11.9 Summary""; ""Part III. Analyses""; ""12. Support""; ""12.1 The Basis Function""; ""12.2 Support Width""; ""12.2.1 Primal binary schemes""; ""12.2.2 Dual binary schemes""; ""12.2.3 Ternary schemes""; ""12.2.4 �Neither� ternary schemes""; ""12.2.5 Higher arities""; ""12.3 Facts which will be Relevant to Other Analyses""
""12.4 The Matrices of Powers of a Scheme""""12.5 Practical Support""; ""12.6 Exercises""; ""12.7 Summary""; ""13. Enclosure""; ""13.1 Positivity""; ""13.2 If the Basis Functions are Somewhere Negative""; ""13.3 Exercises""; ""13.4 Summary""; ""14. Continuity 1 - at Support Ends""; ""14.1 Derivative Continuity of the Basis Function at its Ends""; ""14.1.1 Why does the limit curve converge just because the control points do?""; ""14.1.2 How do you know something nasty doesn�t happen at places in between the extreme control points examined?""; ""14.2 Exercises""; ""14.3 Summary""
""15. Continuity 2 - Eigenanalysis""
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