Discrete And Computational Geometry Research Articles

Digital sieving of irregular 3D particles — A study using XRCT and statistically similar synthetic data

The properties of particulate materials are influenced by purely geometric properties like size, size distribution, and shape. However, size is an ambiguous quantity for non-spherical particles. Therefore, the impact of particle shape on the size as measured by several common and original estimators is investigated. Particularly, a high-fidelity digital sieving method based on the first principal axis of a particle is proposed. An XRCT scan of asphalt concrete is analysed for the shape distribution and its compliance with typical standards. Then, a larger synthetic particle set with a statistically similar, but slightly more extreme shape distribution is generated for testing the size estimators. It is found that the equivalent volume sphere should be abandoned and alternatives are proposed. The analysis is based on simple discrete computational geometry and can easily be implemented.

On the positive fraction Erdös-Szekeres theorem for convex sets

Let F = {F1, ..., Fn} be a collection of disjoint compact convex sets in the plane. We say that F is in general position if no Fi is in the convex hull of two other Fi's. We say that F is in convex position if no Fi is in the convex hull of the other n - 1Fi's. For k ≥ 4, F is called a k-cluster if it is a disjoint union of k subfamilies F1, F2, ..., Fk ⊂ F of equal size such that each transversal {F1, F2,...,Fk}, Fi ∈ Fi, is in convex position. In this paper we show that for any F in general position there is a k-cluster F' ⊂ F of size at least 2-37.8k-o(1)|F|. This improves the result of J. Pach and J. Solymosi [Canonical theorems for convex sets, Discrete and Computational Geometry 19 (1998) 427-435].

Adaptive-randomised self-calibration of electro-mechanical shutters for space imaging

This work describes the self-calibration of a high-precision open-loop mechanism. The self-calibration method is applied to a mechanical shutter for space applications, which was launched onboard the ESA-ROSETTA mission (launch: 2 March 2004). It is based on an adaptive ‘model reference’ and a ‘randomised’ search method which may be generalised to applications in which high performance and functionality are strongly interconnected. The method makes use of an adaptive ‘model-reference’ control approach [K.J. Astrom, B. Wittenmark, On self-tuning regulators Automatica 9 (1973) 185–199 [16]; K.J. Astrom, Theory and application of adaptive control, in: Proceedings of the Eighth IFAC World Conference, Kyoto, Japan, 1981 [17]; D.E. Seborg, S.L. Shah, T.F. Edgar, Adaptive control strategies for process control, AIChE Journal 6(32) (1986) 881–895 [18]] to guarantee mechanism performance. The proposed control system comprises both a deterministic adaptive part and a random-search one [K.L. Clarkson, Applications of random sampling in computational geometry, II, in: Proceedings of the Fourth Annual ACM Symposium on Computational Geometry, 1998, pp. 1–11 [19]; P.K. Agarwal, M. Sharir, Efficient randomised algorithms for some geometric optimisation problems, Discrete Computational Geometry 16 (1996) 317–337 [15]] to guarantee shutter mechanism functionality and performance over testing lifetime (5×10 4 cycles).

Coarsening unstructured meshes by edge contraction

AbstractA new unstructured mesh coarsening algorithm has been developed for use in conjunction with multilevel methods. The algorithm preserves geometrical and topological features of the domain, and retains a maximal independent set of interior vertices to produce good coarse mesh quality. In anisotropic meshes, vertex selection is designed to retain the structure of the anisotropic mesh while reducing cell aspect ratio. Vertices are removed incrementally by contracting edges to zero length. Each vertex is removed by contracting the edge that maximizes the minimum sine of the dihedral angles of cells affected by the edge contraction. Rarely, a vertex slated for removal from the mesh cannot be removed; the success rate for vertex removal is typically 99.9% or more.For two‐dimensional meshes, both isotropic and anisotropic, the new approach is an unqualified success, removing all rejected vertices and producing output meshes of high quality; mesh quality degrades only when most vertices lie on the boundary. Three‐dimensional isotropic meshes are also coarsened successfully, provided that there is no difficulty distinguishing corners in the geometry from coarsely‐resolved curved surfaces; sophisticated discrete computational geometry techniques appear necessary to make that distinction. Three‐dimensional anisotropic cases are still problematic because of tight constraints on legal mesh connectivity. More work is required to either improve edge contraction choices or to develop an alternative strategy for mesh coarsening for three‐dimensional anisotropic meshes. Copyright © 2003 John Wiley & Sons, Ltd.

Geometric Construction of Be´zier Motions

This paper deals with discrete computational geometry of motion. It combines concepts from the fields of kinematics and computer aided geometric design and develops a computational geometric framework for geometric construction of motions useful in mechanical systems animation, robot trajectory planning and key framing in computer graphics. In particular, screw motion interpolants are used in conjunction with deCasteljau-type methods to construct Be´zier motions. The properties of the resulting Be´zier motions are studied and it is shown that the Be´zier motions obtained by application of the deCasteljau construction are not, in general, of polynomial type and do not possess the useful subdivision property of Bernstein-Be´zier curves. An alternative from of deCasteljau algorithm is presented that results in Be´zier motions with subdivision property and Bernstein basis function. The results are illustrated by examples.