The idea of lines and angles in the two dimensional space is a fairly simple and intuitive idea. However, as we move to N-dimensions, this idea may not be as intuitive. In this project, we are going to take the intuitive ideas in a plane and move to a higher dimensional space. We will introduce some machinery, such as the wedge product and the Hodge star operator, which will assist us to investigate these concepts in N-dimensions. We will then look at what we mean by oriented volume of an object in N-dimensions. This will lead us to interesting generalizations of two well-known ideas: the Pythagorean theorem and the law of cosines

The images which comprise the majority of these pages have been given the name Light Sculptures. They are not computer generated or hand drawn, they are photographs of the interaction between light and a reflective cylinder. Originally color transparencies, the Sculptures have been interpreted by the computer as inverted, greyscale images. What was once a black background is now white paper, and what were once lines of light are now lines of ink. The sculptures were synthesized through experimentation with the fundamental elements of architecture: Light, Material, and Geometry. Following the series of Light Sculptures are images of Objects through which materials, tectonics, and proportion were explored. The design and fabrication of each piece was guided by the intent to make something which functioned structurally, visually, and tactily as an integrated whole. / Master of Architecture

A synthetic approach to the construction of projective geometry, its methods and selected results are given in the proposed thesis. The main historical drawbacks of the original proof of Chasles's theorem for non-developable ruled surfaces and von Staudt's formalization of projective geometry are commented. The corre- sponding theoretical background is elaborated on visual demonstrations with the accent to interrelations of classical synthetic, axiomatic and analytic points of view. Synthetic methods of projective geometry and their mixture with analytic methods are described on examples including numerous alternative proofs and generalizations of some theorems. A method of four-dimensional visualization is introduced in details. Elementary constructions of images of points, lines, planes and 3-spaces are followed by models of polychora, their sections and shadows. Chasles's theorem is proven for non-developable ruled quadrics on synthetic vi- sualizations, then generalized and proven within the pure projective framework for algebraic surfaces. The synthetic classification of regular quadrics is derived from descriptive geometry constructions of sections of four-dimensional cones and analytically verified in the projective extension of the real space. An integral part of the thesis is a...

The aim of this study was to investigate the relationship between teachers' instructional practices in terms of specific areas of focus pertaining to the teaching and learning of geometry described in literature and, their learners' levels of geometry thinking as elaborated in the Van Hiele theory. A review of literature on the development of geometry understanding was conducted to frame what is meant by 'teachers' instructional practices' as they pertain to the teaching and learning of geometry in this study. These instructional practices are understood to include the appropriate allocation of time for the facilitation of geometry concept development, the use of concrete apparatus, the use of relevant and level appropriate language as well as the use of level appropriate geometry activities. The structure of the curriculum in terms of its content and opportunity for conceptual progression was also considered. Literature reveals continuing discourse regarding the levels of thinking described in the Van Hiele theory, and even though there is no consensus regarding the nature of the levels and that assessing learners' levels of thinking remains difficult and inconclusive, it is generally accepted that the Van Hiele test is a reliable measure in assessing learners' levels of geometry thinking. An exploratory case study design was chosen for this study. The phenomenon being explored is the teaching and learning of geometry in the Foundation and Intermediate Phases of a particular private school. In order to do this, the teachers' timetables and Work Schedules were analysed to determine how much time was allocated to the instruction of Mathematics in general and for the instruction of geometry in particular. These documents also yielded data regarding the type of geometry experiences included in the implemented curriculum. The learners' level of geometry understanding according to the Van Hiele theory was assessed using an instrument designed by Usiskin (1982). This assessment was facilitated by the researcher in the learners' home class and happened in June after six months of instruction in a particular grade level. Data regarding the teachers' perception of geometry and the best method to facilitate the learning of geometry was gathered through a teacher's questionnaire. The teachers were requested to facilitate geometry lessons, which were digitally recorded by the researcher. Each grade level (0-5) was regarded as a sub-unit and analysed as the case for that grade level. The data was then assimilated to present the case of geometry teaching and learning in the Foundation and Intermediate Phases in the school. The findings report that when juxtaposed alongside research, geometry instructional practices in this school, compare favourably with regards to the teachers' professed and observed practice of using concrete aids and tasks that engage the learners actively in developing geometry insight. There is also evidence that these instructional practices support progression through the levels however the shortfall of time allocated to facilitating this progression and the lack of conclusive data regarding the language used and the types of experiences may justify further research into whether this progression is satisfactory. Copyright / Dissertation (MEd)--University of Pretoria, 2011. / Science, Mathematics and Technology Education / unrestricted

published_or_final_version / Education / Doctoral / Doctor of Philosophy

In the setting of tropical mathematics, geometric objects are rich with inherent combinatorial structure. For example, each polynomial $p(x,y)$ in the tropical setting corresponds to a tropical curve; these tropical curves correspond to unbounded graphs embedded in $\R^2$. Each of these graphs is dual to a particular subdivision of its Newton polytope; we classify tropical curves by combinatorial type based on these corresponding subdivisions. In this thesis, we aim to gain an understanding of the likeliness of the combinatorial type of a randomly chosen tropical curve by using methods from polytope geometry. We focus on tropical curves corresponding to quadratics, but we hope to expand our exploration to higher degree polynomials.

This thesis presents a series of results obtained in [13, 18, 19, 23{25, 87]. In [19], we prove a Darboux theorem for derived schemes with symplectic forms of degree k < 0, in the sense of [142]. We use this to show that the classical scheme X = t₀(X) has the structure of an algebraic d-critical locus, in the sense of Joyce [87]. Then, if (X, s) is an oriented d-critical locus, we prove in [18] that there is a natural perverse sheaf P·_X,s on X, and in [25], we construct a natural motive MF_X,s, in a certain quotient ring M^μ_X of the μ-equivariant motivic Grothendieck ring M^μ_X, and used in Kontsevich and Soibelman's theory of motivic Donaldson-Thomas invariants [102]. In [13], we obtain similar results for k-shifted symplectic derived Artin stacks. We apply this theory to categorifying Donaldson-Thomas invariants of Calabi-Yau 3-folds, and to categorifying Lagrangian intersections in a complex symplectic manifold using perverse sheaves, and to prove the existence of natural motives on moduli schemes of coherent sheaves on a Calabi-Yau 3-fold equipped with 'orientation data', as required in Kontsevich and Soibelman's motivic Donaldson-Thomas theory [102], and on intersections L??M of oriented Lagrangians L,M in an algebraic symplectic manifold (S,ω). In [23] we show that if (S,ω) is a complex symplectic manifold, and L,M are complex Lagrangians in S, then the intersection X= L??M, as a complex analytic subspace of S, extends naturally to a complex analytic d-critical locus (X, s) in the sense of Joyce [87]. If the canonical bundles K_L,K_M have square roots K^1/2_L, K^1/2_M then (X, s) is oriented, and we provide a direct construction of a perverse sheaf P·_L,M on X, which coincides with the one constructed in [18]. In [24] we have a more in depth investigation in generalized Donaldson-Thomas invariants DT^α(τ) defined by Joyce and Song [85]. We propose a new algebraic method to extend the theory to algebraically closed fields <b>K</b> of characteristic zero, rather than <b>K = C</b>, and we conjecture the extension of generalized Donaldson-Thomas theory to compactly supported coherent sheaves on noncompact quasi-projective Calabi-Yau 3-folds, and to complexes of coherent sheaves on Calabi-Yau 3-folds.

We study log varieties, over a field of characteristic zero, which are generically logarithmically smooth and fs in the Kummer normally log étale topology. As an application, we prove an analog of Abramovich-Temkin-Wlodarczyk’s log resolution of singularities of fs log schemes in the Kummer fs setting.<br>

於參數化和特徵模型的變形中保持幾何特徵是CAD建模中一項新的挑戰。這篇論文提出了一個以限制為基礎去進行變形的系統。此系統結合了自由曲面和特徵模型建模的好處，而且容許更自由的工程設計。 / 本方法可分為三個主要步驟。以常用的變形技術去改變一個模型的形狀，包括自由變形及軸向變形，然後參數特徵會根據用戶的要求去分拆為一系列基本的限制，最後目標特徵將會以逐步增量的優化技術去重建。 / 這篇論文提出了一個逐步增量的方法為優化中提供導引。這個優化是於維持所有提供的限制下盡量減少變形後模型的改變。另外，於一組的限制中以一個基准為參考，能使本系統更有效的運行。最後，我們也會展示一些使用本系統以限制為基礎去進行變形的結果。 / To retain geometric features in the deformation of a parametric and feature-based model is a new challenge for CAD modeling. This thesis presents a constraints based deformation framework. This framework combines the advantage of free-form modeling with feature based modeling, and allows engineering design to be performed in a free-form manner. / The proposed method can be divided into three major steps. An object is deformed by common deformation techniques such as FFD and axial deformation. Parametric features are divided into systems of primitive constraints based on user specification. The targeting features are reconstructed by the use of incremental optimization technique. / An incremental constrained deformation is introduced. It is used to provide hints for the optimization. The optimization is to minimize the changes in the deformed model subjected to all the provided constraints. For a structural constraint specified with a group of constraints, it would be better to use a reference datum for all its component constraints. We show numerous results of constraints retained models using our framework. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Tang, Wing Shing. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leaves 84-86). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts also in Chinese. / Chapter 1 --- INTRODUCTION --- p.3 / Chapter 1.1 --- Aims and Objectives --- p.4 / Chapter 1.2 --- Report Organization --- p.5 / Chapter 2 --- BACKGROUND AND LITERATURE REVIEW --- p.7 / Chapter 2.1 --- Mesh Editing Techniques --- p.7 / Chapter 2.1.1 --- Mesh Deformation Techniques --- p.7 / Chapter 2.1.2 --- Detail Preserving Techniques --- p.9 / Chapter 2.2 --- Optimization Techniques --- p.11 / Chapter 2.2.1 --- Optimization Techniques --- p.11 / Chapter 2.2.2 --- Linear Programming --- p.12 / Chapter 2.2.2.1 --- Simplex Method --- p.12 / Chapter 2.2.2.2 --- Interior Point Method --- p.12 / Chapter 2.2.2.2.1 --- Primal-Dual Interior Point Method --- p.13 / Chapter 2.2.3 --- Nonlinear Programming --- p.14 / Chapter 2.2.3.1 --- Sequential Quadratic Programming --- p.14 / Chapter 2.2.3.2 --- Reduced Gradient Methods --- p.14 / Chapter 2.2.3.3 --- Interior Point Methods --- p.15 / Chapter 2.2.4 --- Optimization Solver --- p.15 / Chapter 2.2.4.1 --- KNITRO --- p.16 / Chapter 3 --- SPECIFICATION OF CONSTRAINTS --- p.18 / Chapter 3.1 --- Constraints --- p.18 / Chapter 3.1.1 --- Constraints with Reference Points --- p.22 / Chapter 3.1.2 --- Constraints with Reference Variables --- p.24 / Chapter 3.1.3 --- Reference Vector Constraints --- p.26 / Chapter 3.1.4 --- Constraints with Reference Datum --- p.27 / Chapter 3.1.4.1 --- Planer Constraint with References --- p.28 / Chapter 3.1.4.2 --- Collinear Constraint with References --- p.29 / Chapter 3.1.4.3 --- Circular Constraint with References --- p.30 / Chapter 3.2 --- Redundant Constraints --- p.31 / Chapter 4 --- CONSTRAINED OPTIMZATION --- p.32 / Chapter 4.1 --- Objective Function --- p.32 / Chapter 4.2 --- Incremental Constrained Deformation --- p.39 / Chapter 4.3 --- The Scaling Problem --- p.43 / Chapter 5 --- CASE STUDIES --- p.44 / Chapter 5.1 --- Maintain Individual Engineering Features --- p.44 / Chapter 5.2 --- Maintain Pattern between Engineering Features --- p.49 / Chapter 5.3 --- Maintain Relationship between Engineering Features --- p.51 / Chapter 5.4 --- Implementation Issue --- p.66 / Chapter 6 --- TESTS AND RESULTS --- p.68 / Chapter 6.1 --- Constraints with References --- p.68 / Chapter 6.2 --- Level Of Detail --- p.71 / Chapter 6.3 --- Incremental Method --- p.73 / Chapter 6.4 --- Comparison --- p.76 / Chapter 7 --- FURTHER WORK AND CONCLUSIONS --- p.81 / Chapter 7.1 --- Recommendation for Further Work --- p.81 / Chapter 7.2 --- Conclusions --- p.82 / REFERENCES --- p.84

Space curves have a variety of uses within mathematics, and much attention has been paid to calculating quantities related to such objects. The quantities of curvature and energy are of particular interest to us. While the notion of curvature is well-known, the Mobius energy is a much newer concept, having been first defined by Jun O'Hara in the early 1990s. Foundational work on this energy was completed by Freedman, He, and Wang in 1994, with their most important result being the proof of the energy's conformal invariance. While a variety of results have built those of Freedman, He, and Wang, two topics remain largely unexplored: the interaction of curvature and Mobius energy and the generalization of the Mobius energy to curves with a varying thickness. In this thesis, we investigate both of these subjects. We show two fundamental results related to curvature and energy. First, we show that any simple, closed, twice-differentiable curve can be transformed in an energy-preserving and length-preserving way that allows us to make the pointwise curvature arbitrarily large at a point. Next, we prove that the total absolute curvature of a twice-differentiable curve is uniformly bounded with respect to conformal transformations. This is accomplished mainly via an analytic investigation of the effect of inversions on total absolute curvature. In the second half of the thesis, we define a generalization of the Mobius energy for simple curves of varying thickness that we call the "nonuniform energy." We call such curves "weighted knots," and they are defined as the pairing of a curve parametrization and positive, continuous weight function on the same domain. We then calculate the first variation formulas for several different variations of the nonuniform energy. Variations preserving the curve shape and total weight are shown to have no minimizers. Variations that "slide" the weight along the curve are shown to preserve energy is special cases.