Three dimensional extension of Bresenham’s algorithm with Voronoi diagram

Bresenham’s algorithm for plotting a two-dimensional line segment is elegant and efficient in its deployment of mid-point comparison and integer arithmetic. It is natural to investigate its three-dimensional extensions. In so doing, this paper uncovers the reason for little prior work. The concept of the mid-point in a unit interval generalizes to that of nearest neighbours involving a Voronoi diagram. Algorithmically, there are challenges. While a unit interval in two-dimension becomes a unit square in three-dimension, “squaring” the number of choices in Bresenham’s algorithm is shown to have difficulties. In this paper, the three-dimensional extension is based on the main idea of Bresenham’s algorithm of minimum distance between the line and the grid points. The structure of the Voronoi diagram is presented for grid points to which the line may be approximated. The deployment of integer arithmetic and symmetry for the three-dimensional extension of the algorithm to raise the computation efficiency are also investigated

Special issue: Uncertainties in geometric design

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/30611/1/0000250.pd

An algorithm for generating solid elements in objects with holes

An algorithm for dividing an object with holes into solid elements for finite element preprocessing is presented. Since a tetrahedron can always be subdivided into prisms and cuboids, the approach of first dividing the given object into disjoint tetrahedra is taken.Objects without holes are dealt with first. Two mesh operators, each generating a single tetrahedron, are presented. In addition to the construction procedure, it is shown that they handle all objects without holes. The algorithm for objects with holes requires a third operator. In addition to showing the necessary and sufficient condition for applying such an operator, it is shown that it effectively reduces the number of holes in an object by one while yielding three tetrahedra. The algorithm which sequences the three operators thus reduces a given polyhedron to a single tetrahedron iteratively. Data structure requirements and update procedures are also given in this paper.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/25002/1/0000429.pd

Algorithmic aspects of alternating sum of volumes. Part 2: Nonconvergence and its remedy

The paper is the second part of a 2-part paper. The first part focused on the issues of data structure and fast difference operation. The second studies the non-convergence of the alternating sum of volumes (ASV) process. An ASV is a series of convex components joined by alternating union and difference operations. It is desirable that an ASV series be finite. However, such is not always the case - the ASV algorithm can be nonconvergent. The paper investigates the causes of this nonconvergence, and finds and proves the conditions responsible for it. Linear time algorithms are then developed for detection.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/29257/1/0000314.pd

Algorithmic aspects of alternating sum of volumes. Part 1: Data structure and difference operation

In terms of basic theory, a unique conversion from a boundary representation to a CSG representation is of importance. In terms of application, the extraction of features by convex decomposition is of interest. The alternating sum of volumes (ASV) technique offers both. However, some algorithmic issues are still unresolved. The paper is the first section of a 2-part paper that addresses specialized set operations and the convergence of the ASV process. In the first part, a fast difference operation for the ASV process and a data structure for pseudopolyhedra are introduced.A fast difference operation between an object and its convex hull is made possible by the crucial observation it takes only linear time to distinguish them. However, it takes O(NlogN) time to construct a data structure with the proper tags. The data structure supporting the operation is a pseudopolyhedron, capturing the special relationship between an object and its convex hull. That the data structure is linear in space is also shown.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/29308/1/0000371.pd

Dimensional measurement of surfaces and their sampling

The number of the discrete samples for the dimensional measurement of machined surfaces and thier coordinates is investigated. Counter to intuition, there need not be quadratically more samples than in the case for sampling lines or curves. To justify this novel scheme, accuracy is defined as the discrepancy of a finite point set. Then, from number theory, a particular sequence of numbers is used to compute the sampling coordinates, resulting in a number that is linear in ID, at the same level of accuracy that is achieved by a 2D uniform distribution. Finally, experimental results of the measurement of machined surfaces modeled as random processes are compiled.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/30873/1/0000537.pd

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.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/26420/1/0000507.pd

Применение методов ядерной физики и ускорителей ННЦ ХФТИ для изучения состава, структуры и свойств твердых тел

Излагается применение методов ядерной физики и ускорителей ННЦ ХФТИ для решения научных и технологических задач в области физики металлов, сплавов, полупроводников, металлоксидных и магнитных материалов. Представляются результаты экспериментов по определению концентрации и распределения элементов, выполненных с помощью пучков ионов водорода и гелия, ускоренных в электростатическом ускорителе до энергии 0,5…4,0 МэВ. Для идентификации элементов и изотопов использовались резонансные ядерные реакции, кулоновское и резонансное ядерное рассеяние и возбуждаемое ускоренными частицами характеристическое рентгеновское излучение. Излагаются результаты и возможности применения каналированных частиц и ориентационных эффектов для изучения локализации, структуры, ориентации, образования, распада и аннигиляции простейших дефектов, определения концентрации и распределения дефектов в радиационно нарушенных кристаллах.Викладається застосування методів ядерної фізики і прискорювачів ННЦ ХФТИ для рішення наукових і технологічних задач в області фізики металів, сплавів, напівпровідників, металооксидних та магнітних матеріалів. Приведені результати експерементів з визначення концентрації та розподілу елементів, виконаних за допомогою пучків іонів водню та гелію, що прискорюються в електростатичному прискорювачю до енергії 0,5 - 4,0 МеВ. Для ідентифікації елементів і ізотопів використовувалися резонансні ядерні реакції, кулонівське і резонансне ядерне розсіювання та збуджуване прискореними частками характеристичне рентгенівське випромінюваня. Викладаються результати і можливості застосування часток що каналюють та орієнтаційных ефектів для вивчення локалізації, структури, орієнтації, утворення, розпаду й анігіляції найпростіших дефектів, визначення концентрації і розподілу дефектів у радіаційно порушених кристалах.Nuclear physics methods and use of the NSC KIPT accelerators as applied to scientific and technological problems in the field of physics of metals, alloys, semiconductors, metal oxide and magnetic materials are surveyed. The results of the experiments relevant to determination of element concentration and distribution using 0,5 - 4,0 MeV hydrogen and helium ions of electrostatic accelerator are presented. Resonance nuclear reactions, coulomb and nuclear resonance scattering as well as X-ray radiation were used for element and isotope identification. The possibilities of application of channeling particles and oriented effects for study of localization, structure, orientation, formation, disintegration and annihilation of elementary defects, determination of defect concentration and distribution in radiation disturbed crystals

Review of dimensioning and tolerancing: representation and processing

The paper surveys the current state of knowledge of techniques for representing, manipulating and analysing dimensioning and tolerancing data in computer-aided design and manufacturing. The use of solid models and variational geometry, and its implications for the successful integration of CAD and CAM, are discussed. The topics explored so far can be grouped into four categories: (a) the representation of dimensioning and tolerancing (D & T), (b) the synthesis and analysis of D & T, (c) tolerance control, and (d) the implications of D & T in CAM. The paper describes in detail the recent work in each group, and concludes with speculation on a general framework for future research.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/29159/1/0000204.pd

Parting directions for mould and die design

On the basis of the condition for demouldability, two levels of visibility, complete and partial visibility, are defined. The viewing directions from which a surface is completely visible can be represented as a convex region on the unit sphere called the visibility map of the surface. Algorithms are given for dividing a given object into pockets, for which visibility and demouldability can be determined independently, for constructing visibility maps, and for selecting an optimal pair of parting directions for a mould that minimizes the number of cores. An example illustrates the algorithms.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/30430/1/0000051.pd