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Abstract

In computer graphics, subdivision algorithms are common tools for smoothing down irregularly shaped meshes. Of special interest, due to their simple formulations, are algorithms that generalize B-spline subdivision. Their conceptual simplicity is in stark contrast to the complexity of analysing their results. A complete formal examination of smoothness properties for subdivision schemes was only recently performed by Jörg Peters and Ulrich Reif. This thesis presents a precise and detailed introduction to the analysis of subdivision algorithms. For this purpose, first of all, the necessary background in B-spline theory is established. Building on this, two of the most common subdivision algorithms, the Doo-Sabin and the Catmull-Clark scheme, are motivated. Their treatment is followed by an in-depth description of methods for analysing smoothness properties of subdivision schemes, as developed by Peters and Reif. Afterwards, these methods are applied to the two aforementioned algorithms, thereby establishing smoothness for both algorithms in their original form. Last, in order to demonstrate the effects of choosing unsuitable weights, a number of degenerate weights, which produce irregular shapes in almost all cases, are derived for both schemes—these have hitherto not been published.