Doctor of Philosophy

dissertationWe identify the number of compatibility conditions in 2-D and 3-D discrete structures as a measurement device in determining the rigidity of structures and their resilience to damage. Different shortcuts in counting compatibility conditions are developed and proved. Computing discrete compatibility conditions in hexagonal structures that undergo small enough deformation to be safely linearized, and using these results to say something about the continuum compatibility condition, is the main part of this document. Namely, it is shown that the linearized discrete compatibility condition of hexagonal lattices imposes the 2-D infinitesimal compatibility condition. The computation of discrete 3-D compatibility conditions is done by studying a cuboctahedron. The results show that, because of its infinitesimal flexibility, the extra degree of freedom in a linearized system produces an extra compatibility condition. However, the breaking of nongeneric symmetry of a cuboctahedron by any nonzero perturbation of its nodes makes it infinitesimally rigid, and the extra compatibility condition is lost. The role that compatibility conditions play in damaged structures is demonstrated by experiments on hexagonal lattices. The correlation between the loss of compatibility conditions and the spread of damage is made, as well as the importance of strategic placing of stronger and weaker links in an attempt to strengthen a structure with given boundary conditions and loading

AN ALTERNATIVE DECOMPOSITION OF CATALAN NUMBER

A particular integer sequence derived by the convex polygon triangulation is introduced and investigated. After some underlying results are presented, the forbidden (or improper) integer values relative to the triangulation are concerned. It is understood that the forbidden sequences do not correspond to any triangulation. Some of their properties are presented. These properties are used to count the forbidden values, which is, finally, exploited in stating another decomposition of the Catalan number

Triangulation of Convex Polygon with Storage Support

Unlike the algorithms for convex polygon triangulation which make the triangulation of an n-gon from the scratch, we propose the algorithm making the triangulation of an (n+1)-gon on the base of already found triangulations of an n-gon. For such a purpose we must maintain suitable file storage to store previously derived triangulations and later use them to generate the triangulations of polygon with one more vertex. The file storage is partially exploited for elimination of duplicates our algorithm produces. Yet, triangulation and elimination of duplicates do not critically decrease our algorithm performances for smaller values of n

COMPUTING TRIANGULATIONS OF THE CONVEX POLYGON IN PHP/MYSQL ENVIRONMENT

In this paper we implement Block method for convex polygon triangulation in web environment (PHP/MySQL). Our main aim is to show the advantages of usage of web technologies in performing complex algorithm from computer graphics. The basic assumption is that one obtained results we store in database and use it for other calculation. Databases are convenient and structured methods of sharing and retrieving data. We have performed a comparative analysis of developed program with respect to two criteria: CPU time in generating triangulation and CPU time in reading results from database

Primene metode inverzne poljske notacije i interpolacije u simboličkim izračunavanjima

Ova doktorska disertacija sadrži opis originalne metode za simbolička izračunavanja, zasnovane na inverznoj poljskoj notaciji. Opisana metoda je primenjena na niz problema iz različitih oblasti. Metod inverzne poljske notacije opisan u ovoj disertaciji omogućuje obavljanje simboličkih manipulacija nad različitm tipovima izraza. Pri tome su izbegnute dinamičke strukture podataka, kao što su povezane liste ili stabla, već se manipulacija obavlja direktno nad statičkim nizovima koji predstavljaju izraze u postfiksnoj notaciji. Svakako da dinamičke strukture podataka omogućiju efikasno korišćenje memorije, ali smo “premošćavanjem” ovih faza dobili na jednostavnosti programiranja i brzini izvršenja. Problem utroška memorije, karakterističan za simbolička izračnavanja, uglavnom je uspešno prebrodjen

A non-recursive algorithm for polygon triangulation

In this paper an algorithm for the convex polygon triangulation based on the reverse Polish notation is proposed. The formal grammar method is used as the starting point in the investigation. This idea is "translated" to the arithmetic expression field enabling application of the reverse Polish notation method.

Application of Delaunay Triangulation and Catalan Objects in Steganography

This paper presents a new method of steganography based on a combination of Catalan objects and Voronoi–Delaunay triangulation. Two segments are described within the proposed method. The first segment describes the process of embedding data and generating a complex stego key. The second segment explains the extraction of a hidden message. The main goal of this paper is to transfer a message via the Internet (or some other medium) using an image so that the image remains absolutely unchanged. In this way, we prevented the potential attacker from noticing some secret message hidden in that picture. Additionally, the complex stego key consists of three completely different parts (the image, the encrypted Delaunay triangulation, and the array Rk in Base64 code), which are very difficult to relate with each other. Finally, a few security analyses of the proposed method are conducted, as well as the corresponding steganalysis