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Subdivision Surfaces: C2 schemes and generalized control nets.

Hartmann, René :
Subdivision Surfaces: C2 schemes and generalized control nets.
Technische Universität, Darmstadt
[Ph.D. Thesis], (2011)

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Item Type: Ph.D. Thesis
Title: Subdivision Surfaces: C2 schemes and generalized control nets.
Language: English

After a brief introduction in Chapter 1, in Chapter 2 we introduce and discuss a new basis for C2 splines of orders seven–eight. Cardinal cubic B-splines are among the generating elements of this basis, which allows to decompose the space of polynomials of high degree into the direct sum of the subspace of cubic splines, and some “details”, whose purpose is to allow for curvature continuity at extraordinary points in the bivariate setting. Masks for binary subdivision are provided. We also prove convergence rates of the cubic part of the spline under repeated refinement. We show how it is possible to change from B-spline representations to this basis. Besides this main topic of the chapter, we point out new insights into polynomial subdivision in the regular setting. The analysis leads to techniques of a general nature that allow to deduce convergence rates for generalized control structures toward the limit curve, or surface. The third chapter centers on the characteristic map of a subdivision scheme. We present a method by which characteristic maps to arbitrary eigenvalues 0 < lambda < 1 can be constructed, which is, for instance, needed for the PTER-scheme. Further, a solution to verifying injectivity of a characteristic map for infinitely many valencies is presented and executed at hand of a sample characteristic map. In Chapter 4 we construct and test C2-subdivision schemes based on the PTER-principle by minimizing quadratic functionals. We discuss some selected differential operators that can be used, and example surfaces, as well as generating splines derived by them. Convergence rates of control nets have been studied extensively only in recent years. Chapter 5 further develops the concept of extraordinary proxies from the book Subdivision Surfaces. Proxies abstract the relevant properties that make control nets converge to the limit surface. Parametric and Hausdorff distances are estimated, with sharpness established for each. We continue by analyzing convergence speed of unit normals in the vicinity of extraordinary points. Finally, we conclude by pointing out how slow convergence—of distance or of normals—can be circumvented in situations where the Catmull- Clark algorithm is still used. This also provides a new perspective on using control-nets as approximations to the limit surface.

Alternative Abstract:
Alternative AbstractLanguage
Wir führen neue Basen für C2-Splines der Ordnungen sieben und acht ein, inkl. Masken für binäre Unterteilung. Diese Basen zerlegen den Funktionenraum in die direkte Summe des Raums der kubischen Splines, sowie von “Details”, die vorrangig für Krümmungsstetigkeit an irregulären Punkten im bivariaten Fall notwendig sind. Wichtige Eigenschaften wie Konvergenz des kubischen Teils werden gezeigt. Weiterhin werden neue Aspekte polynomialer Unterteilung im regulären Fall aufgezeigt. Das dritte Kapitel beschäftigt sich mit der Konstruktion von Charakteristischen Abbildungen; des weiteren wird ein Weg präsentiert, wie deren Injektivität sich für undendlich viele Wertigkeiten nachweisen lässt. Konstruktion und Diskussion von uns konstruierter C2-Verfahren sind Gegenstand von Kapitel 4. Kapitel 5 beschäftigt sich mit Konvergenzgeschwindigkeiten von parametrischer und Hausdorff- Distanz zwischen Kontrollnetzen und Subdivisionsfläche in der Umgebung von irregulären Punkten, wobei die dafür verantwortlichen Eigenschaften der Netze im Begriff der (irregulären) Proxies abstrahiert werden. Weiterhin analysiert werden Konvergenz-Geschwindigkeit von Einheitsnormalen. Ein Konzept, langsame Konvergenz zu vermeiden, sowie seine Diskussion, schließen diese Betrachtungen ab.German
Place of Publication: Darmstadt
Uncontrolled Keywords: Subdivisionsalgorithmen, Unterteilungsalgorithmen, Distanz zu Fläche, Splines, Einheitsnormalen, Proxies.
Alternative keywords:
Alternative keywordsLanguage
Subdivision algorithms, distance to surface, splines, unit normals, proxies.English
Classification DDC: 500 Naturwissenschaften und Mathematik > 510 Mathematik
Divisions: 20 Department of Computer Science
04 Department of Mathematics
04 Department of Mathematics > Applied Geometry
Date Deposited: 15 Jun 2011 08:08
Last Modified: 07 Dec 2012 12:00
URN: urn:nbn:de:tuda-tuprints-26170
Referees: Reif, Prof. Dr. Ulrich and Prautzsch, Prof. Dr. Hartmut
Refereed: 17 March 2011
URI: http://tuprints.ulb.tu-darmstadt.de/id/eprint/2617
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