B-Spline curves and surfaces are being used throughout the aircraft industry for geometric modeling. Geometric models having accurate surface representations in the non-uniform B-Spline surface format can contain very large quantities of data. The computing power required by a CAD system for visualization and analysis is directly influenced by these large amounts of data. Accordingly, a method for reducing the amount of data in a geometric model while maintaining accuracy is needed to reduce the computing power necessary to visualize and analyze a design. This thesis describes the refinement and implementation of a data reduction algorithm for non-uniform cubic B-Spline curves and non-uniform bi-cubic B-Spline surfaces. The topic of determining the significance of knots in non-uniform cubic B-Spline curves and non-uniform bi-cubic B-Spline surfaces is addressed. Also, a method for determining the order in which knots should be removed from non-uniform cubic B-Spline curves or non-uniform bi-cubic B-Spline surfaces during data reduction is presented. Finally, an algorithm for performing data reduction by removing knots from non-uniform cubic B-Spline curves and non-uniform bi-cubic B-Spline surfaces is presented. / Master of Science

This research studies the topic of geometric surface trimming. Most of the work that has been done on surface trimming refers to visual trimming, in which, unwanted portions of a surface patch are not removed but simply hidden. Geometric trimming actually removes the unwanted portion of a surface patch, and in the process, produces a new mathematical surface or set of surface patches. The new surface or surface patches produced should closely resemble the original surface which is being trimmed. However, since the entire procedure involves the creation of a new geometry, the resulting surface will not, in general, be identical with the remaining portion of the original one. For this reason, a means to measure the error in approximating the original surface is implemented and this error is minimized. Trimming criteria is provided by mapping the trimming curves from the two-dimensional parametric space on to the three-dimensional B-spline surface. Since visualization for this kind of procedure is vital a tool kit has been developed, using PHIGS, to model B-spline surfaces along with their approximate trimmed surfaces as well as the approximation error involved. / Master of Science

Thesis (M.S.)--Massachusetts Institute of Technology, Dept. of Mechanical Engineering, 1981. / Bibliography: leaves 105-107. / by Vincent C. Lin. / Thesis (M.S.)--Massachusetts Institute of Technology, Dept. of Mechanical Engineering, 1981.

Chan, Pui Lam. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leaves 72-75). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts also in Chinese.

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The concept of intelligent CAD systems to assist a designer in automating the design process has been discussed for years. It has been recognized that knowledge engineering techniques and the study of design theory can provide certain solutions to this approach. A major issue in developing intelligent CAD systems for geometric modeling is the integration of the design geometry with the representation of the design constraints. Current commercial computer aided design (CAD) systems are used primarily for recording the results of the design process. Using conventional CAD systems, a design engineer either must create the geometry of the design object with precise coordinates and dimensions, or start his design from an existing geometry of a previous design. It is difficult to propagate a dimensional change throughout an entire model -- especially solid models. This rigidity imposed by conventional CAD systems discourages a designer from exploring different approaches in creating a novel product. / Ph. D.

In an integrated design environment, the common thread between the different design stages is usually the geometric model of the part. However, the requirements for the geometric definition of the design is usually different for each stage. The transformation of data between these different stages is essential for the success of the integrated design environment. For example, conceptual design systems usually deal with geometric dimensional parameters (e.g. length, radius, etc.) whereas preliminary design systems frequently require the geometry definition to be in the form of surface models. This dissertation presents the necessity and scope of creating and implementing methodologies to obtain dimensional geometric parameters from the surface description of an object. Since the study of geometric modeling and parametric surfaces is a new field, few classical methods are applicable. Methods and algorithms for the extraction of various geometry parameters are created. A few methods to pre-process and manipulate these surfaces before the parameter extraction methods can be applied are outlined. One of the most important applications of parameter extraction is in the field of aircraft design. There are two important aspects of geometry data conversion in the design cycle. The first is the conversion from conceptual CAD models to CFD compatible models. The second is the conversion from surface representations of CFD models to obtain component parameters (e.g. wing span, fuselage fineness ratio, moments of inertia, etc.). The methods created in this dissertation are used to extract geometric parameters of importance in aircraft design. This enables the design cycle to be complete and promotes integrated design. These methods have been implemented in the aircraft design software, ACSYNT. Examples of the conversion of data from B-spline surface models to dimensional geometric parameters using these methods are included. The emphasis of this dissertation is on non-uniform B-spline surfaces. Methods for obtaining geometric parameters from aircraft models described by characteristic points are also considered briefly. / Ph. D.

In school algebra, using different methods including factorization to solve quadratic equations is one common teaching and learning topic at upper secondary school level. This study is about analyzing the algebra content related to solving quadratic equations and the method of factorization as presented in Swedish mathematics textbooks with subject matter content knowledge (CK) and pedagogical content knowledge (PCK) as analytical tools. Mathematics textbooks as educational resources and artefacts are widely used in classroom teaching and learning. What is presented in a textbook is often taught by teachers in the classroom. Similarly, what is missing from the textbook may not be presented by the teacher. The study is based on an assumption that pedagogical content knowledge is embedded in the subject content presented in textbooks. Textbooks contain both subject content knowledge and pedagogical content knowledge. The primary aim of the study is to explore what pedagogical content knowledge regarding solving quadratic equations that is embedded in mathematics textbooks. The secondary aim is to analyze the algebra content related to solving quadratic equations from the perspective of mathematics as a discipline in relation to algebra history. It is about what one can find in the textbook rather than how the textbook is used in the classroom. The study concerns a teaching perspective and is intended to contribute to the understanding of the conditions of teaching solving quadratic equations. The theoretical framework is based on Shulman’s concept pedagogical content knowledge and Mishra and Koehler’s concept content knowledge. The general theoretical perspective is based on Wartofsky’s artifact theory. The empirical material used in this study includes twelve mathematics textbooks in the mathematics B course at Swedish upper secondary schools. The study contains four rounds of analyses. The results of the first three rounds have set up a basis for a deep analysis of one selected textbook. The results show that the analyzed Swedish mathematics textbooks reflect the Swedish mathematics syllabus of algebra. It is found that the algebra content related to solving quadratic equations is similar in every investigated textbook. There is an accumulative relationship among all the algebra content with a final goal of presenting how to solve quadratic equations by quadratic formula, which implies that classroom teaching may focus on quadratic formula. Factorization method is presented for solving simple quadratic equations but not the general-formed quadratic equations. The study finds that the presentation of the algebra content related to quadratic equations in the selected textbook is organized by four geometrical models that can be traced back to the history of algebra. These four geometrical models are applied for illustrating algebra rules and construct an overall embedded teaching trajectory with five sub-trajectories. The historically related pedagogy and application of mathematics in both real world and pure mathematics contexts are the pedagogical content knowledge related to quadratic equations.

Indiana University-Purdue University Indianapolis (IUPUI) / A crucial task in computer vision and biomedical image applications is to represent images in a numerically compact form for understanding, evaluating and/or mining their content. The fundamental step of this task is the segmentation of images into regions, given some homogeneity criteria, prior appearance and/or shape information criteria. Specifically, segmentation of cells in microscopic images is the first step in analyzing many biomedical applications. This thesis is a part of the project entitled "Construction and profiling of biodegradable cardiac patches for the co-delivery of bFGF and G-CSF growth factors" funded by National Institutes of Health (NIH). We present a method that simultaneously segments the population of cells while partitioning the cell regions into cytoplasm and nucleus in order to evaluate the spatial coordination on the image plane, density and orientation of cells. Having static microscopic images, with no edge information of a cytoplasm boundary and no time sequence constraints, traditional cell segmentation methods would not perform well. The proposed method combines deformable models with a probabilistic framework in a simple graphical model such that it would capture the shape, structure and appearance of a cell. The process aims at the simultaneous cell partitioning into nucleus and cytoplasm. We considered the relative topology of the two distinct cell compartments to derive a better segmentation and compensate for the lack of edge information. The framework is applied to static fluorescent microscopy, where the cultured cells are stained with calcein AM.

Modeling the free surface flows is important in order to estimate the total drag of the sea Vessels. It is also necessary to study the effects of various maritime maneuvers. In this work, different ways of approximating an unstructured free surface grid with a B-spline surface are investigated. The Least squares and Galerkin approaches are studied in this regard. B-spline nite element method (BSPFEM) is studied for the solution of the steady-state kinematic free surface equation. The volume grid has to be moved in order to match the free boundary when the surface-tracking approach is adopted for the solution of free surface problem. Inherent smoothness of the B-spline representation of the free surface aids this process. B-spline representation of the free surface aids in building viscous volume grids hose boundaries closely match the steady state free surface. The B-spline approximation algorithm and BSPFEM solution of free surface equation have been tested with hypothetical algebraic testcases and real cases such as Gbody, Wigley hull and David Taylor Model Basin(DTMB) 5415 hull series.