Finding the convex hull of a simple polygon in linear time

Though linear algorithms for finding the convex hull of a simply-connected polygon have been reported, not all are short and correct. A compact version based on Sklansky's original idea⁽⁷⁾ and Bykat's counter-example⁽⁸⁾ is given. Its complexity and correctness are also shown.     

Nonlinear control of a simulated industrial evaporation process

Three differential geometry based nonlinear control strategies, including input-output linearization, generic model control and Su-Hunt-Meyer transformation were studied on a simulation of the evaporation stage of the liquor burning process associated with the Bayer process for the production of alumina. The simulation was implemented using a symbolic algebraic software, Maple V.3. The overall results show that all three nonlinear strategies, especially the input-output linearization, provide better control performance than the classical linear control theory. A simplified solution algorithm for the Su-Hunt-Meyer transformation is proposed. A brief discussion on the industrial implementation of the nonlinear technology is also given.     

VLSI systems for some problems of computational geometry

Fast solution of the computational geometry problems is important for computer graphics, image processing and pattern recognition. The capability of the network Mesh of Trees for application in VLSI systems solving fastly the computational geometry problems is shown on two examples: determination of the convex hull of a weakly externally visible polygon and determination of the visibility polygon of a polygon.     

Complexity,convexity, and unimodality

A class of polygons termedunimodal is introduced. LetP = P₁,p ₂,...,p _n be a simplen-vertex polygon. Given a fixed vertex or edge, several definitions of the distance between the fixed vertex or edge and any other vertex or edge are considered. For a fixed vertex (edge), a distance measure defines a distance function as the remaining vertices (edges) are traversed in order. If for every vertex (edge) ofP a specified distance function is unimodal thenP is a unimodal polygon in the corresponding sense. Relationships between unimodal polygons, in several senses, andconvex polygons are established. Several properties are derived for unimodal polygons when the distance measure is the euclidean distance between vertices of the polygons. These properties lead to very simple 0(n) algorithms for solving a variety of problems that occur in computational geometry and pattern recognition. Furthermore, these algorithms establish that convexity is not the key factor in obtaining linear-time-complexity for solving these problems. The paper closes with several open questions in this area.     

Book Reviews

Steinmetz, Ralf and Nahrstedt, Klara. Multimedia: Computing, Communications and Applications
Stollnitz, Eric J, Derose, Tony D and Salesin, David H. Wavelets for Computer Graphics
Prusinkiewicz, P and Lindenmayer, A. The Algorithmic Beauty of Plants
Geary, David M and McClellan, Alan L. Graphic Java: Mastering the AWT
Santos, Adelino. Multimedia and Groupware for Editing
Magnenat Thalmann, Nadia and Thalmann, Daniel. Interactive Computer Animation     

Proximity problems for points on a rectilinear plane with rectangular obstacles

We consider the following four problems for a setS ofk points on a plane, equipped with the rectilinear metric and containing a setR ofn disjoint rectangular obstacles (so that distance is measured by a shortest rectilinear path avoiding obstacles inR): (a) find aclosest pair of points inS, (b) find anearest neighbor for each point inS, (c) compute the rectilinearVoronoi diagram ofS, and (d) compute a rectilinearminimal spanning tree ofS. We describeO ((n+k) log(n+k))-time sequential algorithms for (a) and (b) based onplane-sweep, and the consideration of geometrically special types of shortest paths, so-calledz-first paths. For (c) we present anO ((n+k) log(n+k) logn)-time sequential algorithm that implements a sophisticateddivide-and-conquer scheme with an addedextension phase. In the extension phase of this scheme we introduce novel geometric structures, in particular so-calledz-diagrams, and techniques associated with the Voronoi diagram. Problem (d) can be reduced to (c) and solved inO ((n+k) log(n+k) logn) time as well. All our algorithms arenear-optimal, as well as easy to implement. An extended abstract appeared inProc. 13th Conf. on the Foundations of Software Technology and Theoretical Computer Science, Bombay, 1993, Springer-Verlag, pp. 218–227. Sumanta Guha was supported in part by a UW-Milwaukee Graduate School Research Committee Award. Ichiro Suzuki was supported in part by the National Science Foundation under Grants CCR-9004346 and IRI-9307506, the Office of Naval Research under Grant N00014-94-1-0284, and an endowed chair supported by Hitachi Ltd. at the Faculty of Engineering Science, Osaka University.     

Polycapillary Optics for Materials Science Studies: Instrumental Effects and Their Correction

The instrumental effects related to the use of a polycapillary x-ray lens as primary beam collimator are here studied and the features observed in the measurements modelled via Monte-Carlo ray-tracing. Comparison with existing procedures is presented and experimental evidence of the accuracy improvements due to the use of a correction algorithm is shown.     

Sound–Sights An Interdisciplinary Project

An interdisciplinary project between geometry, architecture and music resulted in a concert and exhibition with sound installations. Professors and students of architecture and mathematics of the Technical University of Kaiserslautern worked with a professor and students of music composition of the Music Academy of Cologne in Germany. The theoretical and historical analyses of the relationships between geometry and music formed the basis for original creative works in interdisciplinary groups. Music was composed according geometrical-architectural concepts and geometrical images, forms and processes were developed after musical ideas. Geometrical forms are combined with the music into a kinetic, visual and acoustic work of art. Through such interdisciplinary art projects it is possible to experience scientific coherence in a sensual way. The combination of geometry, architecture and music enables a visual and aural approach to formal thinking of sciences.     

Fatigue life enhancement of aluminium joints through mechanical and thermal prestressing

This paper presents some simple and flexible methods to enhance the fatigue life of welded aluminium components. Besides enhancing the fatigue life, the proposed methods can easily be implemented into manufacturing processes. The key element of the methods is to change residual stresses from tension to compression at locations vulnerable to fatigue. This is accomplished by mechanical prestressing using elastic pre-deformation or by thermal prestressing using induction heating. The specimens tested are welded aluminium rectangular hollow section T-joints. Prior to fatigue testing, welding FE-simulations were carried out to verify the magnitude and pattern of the residual stress fields (through process modeling). Fatigue testing was later carried out on four different batches. One batch was produced using elastically pre-deformed chords, two batches were treated by means of thermal prestressing (induction heating), and one batch was “as welded” representing a “reference case”. Based on statistical evaluation of S–N data, the introduction of superimposed compressive stress fields results in a significantly improved fatigue life. Among the different batches, induction heating turned out to be the most promising method with a fatigue strength improvement factor of 1.5 on stress, compared to “as welded” components.     

A geometrical model of the cut in five-axis milling accounting for the influence of tool orientation

This paper presents a model of the cut geometry in five-axis milling. This allows the establishment of a better model of cutting force to account for the influence of the tool orientation. The formulation of the width and the thickness of the cut were derived and implemented in a computer simulation. The results of simulations were verified experimentally and a good agreement was obtained. The result shows the importance of including the influence of the tool orientation in the cut cross-section calculation.