Shape theory and mathematical design of a general geometric kernel through regular stratified objects

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.This dissertation focuses on the mathematical design of a unified shape kernel for geometric computing, with possible applications to computer aided design (CAM) and manufacturing (CAM), solid geometric modelling, free-form modelling of curves and surfaces, feature-based modelling, finite element meshing, computer animation, etc. The generality of such a unified shape kernel grounds on a shape theory for objects in some Euclidean space. Shape does not mean herein only geometry as usual in geometric modelling, but has been extended to other contexts, e. g. topology, homotopy, convexity theory, etc. This shape theory has enabled to make a shape analysis of the current geometric kernels. Significant deficiencies have been then identified in how these geometric kernels represent shapes from different applications. This thesis concludes that it is possible to construct a general shape kernel capable of representing and manipulating general specifications of shape for objects even in higher-dimensional Euclidean spaces, regardless whether such objects are implicitly or parametrically defined, they have ‘incomplete boundaries’ or not, they are structured with more or less detail or subcomplexes, which design sequence has been followed in a modelling session, etc. For this end, the basic constituents of such a general geometric kernel, say a combinatorial data structure and respective Euler operators for n-dimensional regular stratified objects, have been introduced and discussed

A second-order in time, BGN-based parametric finite element method for geometric flows of curves

Over the last two decades, the field of geometric curve evolutions has attracted significant attention from scientific computing. One of the most popular numerical methods for solving geometric flows is the so-called BGN scheme, which was proposed by Barrett, Garcke, and Nurnberg (J. Comput. Phys., 222 (2007), pp. 441{467), due to its favorable properties (e.g., its computational efficiency and the good mesh property). However, the BGN scheme is limited to first-order accuracy in time, and how to develop a higher-order numerical scheme is challenging. In this paper, we propose a fully discrete, temporal second-order parametric finite element method, which incorporates a mesh regularization technique when necessary, for solving geometric flows of curves. The scheme is constructed based on the BGN formulation and a semi-implicit Crank-Nicolson leap-frog time stepping discretization as well as a linear finite element approximation in space. More importantly, we point out that the shape metrics, such as manifold distance and Hausdorff distance, instead of function norms, should be employed to measure numerical errors. Extensive numerical experiments demonstrate that the proposed BGN-based scheme is second-order accurate in time in terms of shape metrics. Moreover, by employing the classical BGN scheme as a mesh regularization technique when necessary, our proposed second-order scheme exhibits good properties with respect to the mesh distribution.Comment: 35 pages, 9 figure

Intrinsic shape analysis in archaeology: A case study on ancient sundials

This paper explores a novel mathematical approach to extract archaeological insights from ensembles of similar artifact shapes. We show that by considering all the shape information in a find collection, it is possible to identify shape patterns that would be difficult to discern by considering the artifacts individually or by classifying shapes into predefined archaeological types and analyzing the associated distinguishing characteristics. Recently, series of high-resolution digital representations of artifacts have become available, and we explore their potential on a set of 3D models of ancient Greek and Roman sundials, with the aim of providing alternatives to the traditional archaeological method of ``trend extraction by ordination'' (typology). In the proposed approach, each 3D shape is represented as a point in a shape space -- a high-dimensional, curved, non-Euclidean space. By performing regression in shape space, we find that for Roman sundials, the bend of the sundials' shadow-receiving surface changes with the location's latitude. This suggests that, apart from the inscribed hour lines, also a sundial's shape was adjusted to the place of installation. As an example of more advanced inference, we use the identified trend to infer the latitude at which a sundial, whose installation location is unknown, was placed. We also derive a novel method for differentiated morphological trend assertion, building upon and extending the theory of geometric statistics and shape analysis. Specifically, we present a regression-based method for statistical normalization of shapes that serves as a means of disentangling parameter-dependent effects (trends) and unexplained variability.Comment: accepted for publication from the ACM Journal on Computing and Cultural Heritag

Distributed boundary tracking using alpha and Delaunay-Cech shapes

For a given point set $S$ in a plane, we develop a distributed algorithm to compute the $\alpha-$shape of $S$. $\alpha-$shapes are well known geometric objects which generalize the idea of a convex hull, and provide a good definition for the shape of $S$. We assume that the distances between pairs of points which are closer than a certain distance $r>0$ are provided, and we show constructively that this information is sufficient to compute the alpha shapes for a range of parameters, where the range depends on $r$. Such distributed algorithms are very useful in domains such as sensor networks, where each point represents a sensing node, the location of which is not necessarily known. We also introduce a new geometric object called the Delaunay-\v{C}ech shape, which is geometrically more appropriate than an $\alpha-$shape for some cases, and show that it is topologically equivalent to $\alpha-$shapes