Proposal to incorporate volumetric (three dimensional) subdivisions in Victoria

This project researched the development of three dimensional subdivisions in Victoria and analysed the current limitations by comparing it with legislation from other states. It then determined how current best practice from other states could be incorporated into the current Victorian system. Case studies of a similar nature were identified from both Victoria and Queensland to make comparisons on how the subdivisions were performed and their respective plans drawn. Research has shown that currently, there is very little written with respect to three dimensional subdivisions outside of legislation in Victoria. There are only two sections within the legislation that refer to three dimensional subdivisions. The first refers to how buildings defined by boundaries can be defined and how they are to be shown, and the second specifies that an elevation, section or diagram must be used when lots lay in stratum. Lots can take any shape as legislation does not define any limitations, provided they can be mathematically defined, if not defined by structure. Queensland and Western Australia were the only states that had legislation specifically written for three dimensional subdivisions outside of a standard building subdivision. Western Australia’s legislation was found to have more flexibility than Queensland. It has been found that Victoria lacks examples of plan presentation types for three dimensional subdivisions in the Survey Practice Handbook 1997. Because no examples exist, most surveyors are not aware of the options available to them, and continue to draw lots in plan and section format only, rather than an isometric view or another method that may be suitable

Orthogonal Range Reporting and Rectangle Stabbing for Fat Rectangles

In this paper we study two geometric data structure problems in the special case when input objects or queries are fat rectangles. We show that in this case a significant improvement compared to the general case can be achieved. We describe data structures that answer two- and three-dimensional orthogonal range reporting queries in the case when the query range is a \emph{fat} rectangle. Our two-dimensional data structure uses $O(n)$ words and supports queries in $O(\log\log U +k)$ time, where $n$ is the number of points in the data structure, $U$ is the size of the universe and $k$ is the number of points in the query range. Our three-dimensional data structure needs $O(n\log^{\varepsilon}U)$ words of space and answers queries in $O(\log \log U + k)$ time. We also consider the rectangle stabbing problem on a set of three-dimensional fat rectangles. Our data structure uses $O(n)$ space and answers stabbing queries in $O(\log U\log\log U +k)$ time.Comment: extended version of a WADS'19 pape

3D Delaunay triangulation of non-uniform point distributions

In view of the simplicity and the linearity of regular grid insertion, a multi-grid insertion scheme is proposed for the three-dimensional Delaunay triangulation of non-uniform point distributions by recursive application of the regular grid insertion to an arbitrary subset of the original point set. The fundamentals and difficulties of three-dimensional Delaunay triangulation of highly non-uniformly distributed points by the insertion method are reviewed. Current strategies and methods of point insertions for non-uniformly distributed spatial points are discussed. An enhanced kd-tree insertion algorithm with a specified number of points in a cell and its natural sequence derived from a sandwich insertion scheme is also presented. The regular grid insertion, the enhanced kd-tree insertion and the multi-grid insertion have been rigorously studied with benchmark non-uniform distributions of 0.4–20 million points. It is found that the kd-tree insertion is more efficient in locating the base tetrahedron, but it is also more sensitive to the triangulation of non-uniform point distributions with a large amount of conflicting elongated tetrahedra. Including the grid construction time, multi-grid insertion is the most stable and efficient for all the uniform and non-uniform point distributions tested.postprin

Triangulation of Simple 3D Shapes with Well-Centered Tetrahedra

A completely well-centered tetrahedral mesh is a triangulation of a three dimensional domain in which every tetrahedron and every triangle contains its circumcenter in its interior. Such meshes have applications in scientific computing and other fields. We show how to triangulate simple domains using completely well-centered tetrahedra. The domains we consider here are space, infinite slab, infinite rectangular prism, cube and regular tetrahedron. We also demonstrate single tetrahedra with various combinations of the properties of dihedral acuteness, 2-well-centeredness and 3-well-centeredness.Comment: Accepted at the conference "17th International Meshing Roundtable", Pittsburgh, Pennsylvania, October 12-15, 2008. Will appear in proceedings of the conference, published by Springer. For this version, we fixed some typo

Optimal collapse simulator for three-dimensional structures

In this project limit analysis for 3D structures is studied. The goal is to obtain for a certain structure the load factor that applied to the external loads induces collapse to the structure. The static theorem of limit analysis is the theoretical basis for the Structural Collapse Simulator (SCS), that is nding a stress distribution in equilibrium that does not violate yield criteria anywhere. This theorem is employed combined with linear programming techniques. Thereby a tutorial on LP problems is presented rst. Then a brief summary of the progresses in study of limit analysis for structures is o ered, being a useful introduction for understanding the very nature of SCS functioning. Moreover, limit analysis is developed and written as a LP problem, which consists of the maximization of the collapse load factor subject to equilibrium and yield criteria. Two major contributions are presented for nding the collapse load. Firstly, the yield curve of standard 2D beam cross sections is adaptively approximated with inscribed and circumscribed polygons that yield to lower and upper bounds of respectively. Secondly, an interesting approach for accounting with uniform distributed loads is shown, producing bounding of the load factor. Combining these two techniques the bound gap can be reduced arbitrarily, observing convergence of the upper and the lower bounds to the exact load factor. A tutorial for using SCS and computing structures is provided, and numerical examples are thoroughly studied in order to illustrate the functioning of the program and the limits of the method. Finally, recent developments and future branches of research are detailed in order to widen the applicability range of SCS, the most important being the adaptive approximation of the yield surface for 3D beams

The three-dimensional art gallery problem and its solutions

This thesis addressed the three-dimensional Art Gallery Problem (3D-AGP), a version of the art gallery problem, which aims to determine the number of guards required to cover the interior of a pseudo-polyhedron as well as the placement of these guards. This study exclusively focused on the version of the 3D-AGP in which the art gallery is modelled by an orthogonal pseudo-polyhedron, instead of a pseudo-polyhedron. An orthogonal pseudopolyhedron provides a simple yet effective model for an art gallery because of the fact that most real-life buildings and art galleries are largely orthogonal in shape. Thus far, the existing solutions to the 3D-AGP employ mobile guards, in which each mobile guard is allowed to roam over an entire interior face or edge of a simple orthogonal polyhedron. In many realword applications including the monitoring an art gallery, mobile guards are not always adequate. For instance, surveillance cameras are usually installed at fixed locations. The guard placement method proposed in this thesis addresses such limitations. It uses fixedpoint guards inside an orthogonal pseudo-polyhedron. This formulation of the art gallery problem is closer to that of the classical art gallery problem. The use of fixed-point guards also makes our method applicable to wider application areas. Furthermore, unlike the existing solutions which are only applicable to simple orthogonal polyhedra, our solution applies to orthogonal pseudo-polyhedra, which is a super-class of simple orthogonal polyhedron. In this thesis, a general solution to the guard placement problem for 3D-AGP on any orthogonal pseudo-polyhedron has been presented. This method is the first solution known so far to fixed-point guard placement for orthogonal pseudo-polyhedron. Furthermore, it has been shown that the upper bound for the number of fixed-point guards required for covering any orthogonal polyhedron having n vertices is (n3/2), which is the lowest upper bound known so far for the number of fixed-point guards for any orthogonal polyhedron. This thesis also provides a new way to characterise the type of a vertex in any orthogonal pseudo-polyhedron and has conjectured a quantitative relationship between the numbers of vertices with different vertex configurations in any orthogonal pseudo-polyhedron. This conjecture, if proved to be true, will be useful for gaining insight into the structure of any orthogonal pseudo-polyhedron involved in many 3-dimensional computational geometrical problems. Finally the thesis has also described a new method for splitting orthogonal polygon iv using a polyline and a new method for splitting an orthogonal polyhedron using a polyplane. These algorithms are useful in applications such as metal fabrication

Converting Boundary Representation Solid Models to Half-Space Representation Models for Monte Carlo Analysis

Solid modeling computer software systems provide for the design of three-dimensional solid models used in the design and analysis of physical components. The current state-of-the-art in solid modeling representation uses a boundary representation format in which geometry and topology are used to form three-dimensional boundaries of the solid. The geometry representation used in these systems is cubic B-spline curves and surfaces--a network of cubic B-spline functions in three-dimensional Cartesian coordinate space. Many Monte Carlo codes, however, use a geometry representation in which geometry units are specified by intersections and unions of half-spaces. This paper describes an algorithm for converting from a boundary representation to a half-space representation

Development and Application of Three-dimensional Impedance Maps Related to Tissue Pathology

Three-dimensional acoustic tissue models are a unique means to study ultrasonic scattering by tissue microstructure. In this work, the previous methods used to create and analyze these models were evaluated and refined. These techniques were then applied to a set of 10 human fibroadenomas, a benign tumor of the breast. These models, called three-dimensional impedance maps (3DZMs), are created from serial sets of histological images which must be properly transformed to recreate the original tissue volume. A properly reconstructed 3DZM can then be used to estimate properties, such as the effective scatterer size, of the ultrasonic scattering sites in the underlying tissue. These estimates can, in turn, be related to histological features of the tissue. For the fibroadenoma datasets, the average effective scatterer diameter was estimated to be 84 ?? 40 ??m when the entire volume was used for analysis. This result compared roughly to the size of the acini in the tissue, although a wide variation was observed in the histological layout of the tissue