Partitioning Regular Polygons into Circular Pieces I: Convex Partitions

We explore an instance of the question of partitioning a polygon into pieces, each of which is as ``circular'' as possible, in the sense of having an aspect ratio close to 1. The aspect ratio of a polygon is the ratio of the diameters of the smallest circumscribing circle to the largest inscribed disk. The problem is rich even for partitioning regular polygons into convex pieces, the focus of this paper. We show that the optimal (most circular) partition for an equilateral triangle has an infinite number of pieces, with the lower bound approachable to any accuracy desired by a particular finite partition. For pentagons and all regular k-gons, k > 5, the unpartitioned polygon is already optimal. The square presents an interesting intermediate case. Here the one-piece partition is not optimal, but nor is the trivial lower bound approachable. We narrow the optimal ratio to an aspect-ratio gap of 0.01082 with several somewhat intricate partitions.Comment: 21 pages, 25 figure

Gap Processing for Adaptive Maximal Poisson-Disk Sampling

In this paper, we study the generation of maximal Poisson-disk sets with varying radii. First, we present a geometric analysis of gaps in such disk sets. This analysis is the basis for maximal and adaptive sampling in Euclidean space and on manifolds. Second, we propose efficient algorithms and data structures to detect gaps and update gaps when disks are inserted, deleted, moved, or have their radius changed. We build on the concepts of the regular triangulation and the power diagram. Third, we will show how our analysis can make a contribution to the state-of-the-art in surface remeshing.Comment: 16 pages. ACM Transactions on Graphics, 201

Multiresolution analysis as an approach for tool path planning in NC machining

Wavelets permit multiresolution analysis of curves and surfaces. A complex curve can be decomposed using wavelet theory into lower resolution curves. The low-resolution (coarse) curves are similar to rough-cuts and high-resolution (fine) curves to finish-cuts in numerical controlled (NC) machining.;In this project, we investigate the applicability of multiresolution analysis using B-spline wavelets to NC machining of contoured 2D objects. High-resolution curves are used close to the object boundary similar to conventional offsetting, while lower resolution curves, straight lines and circular arcs are used farther away from the object boundary.;Experimental results indicate that wavelet-based multiresolution tool path planning improves machining efficiency. Tool path length is reduced, sharp corners are smoothed out thereby reducing uncut areas and larger tools can be selected for rough-cuts

Polynomial Meshes: Computation and Approximation

We present the software package WAM, written in Matlab, that generates Weakly Admissible Meshes and Discrete Extremal Sets of Fekete and Leja type, for 2d and 3d polynomial least squares and interpolation on compact sets with various geometries. Possible applications range from data fitting to high-order methods for PDEs

2D hardware acceleration

The objective of the project is to develop an IP-core that provides hardware acceleration for common 2D rendering operations in an embedded system. The requirements for graphical user interfaces (GUI) on modern display and touchscreen based systems are increasing steadily. Rendering complex and attractive GUIs requires a lot of processing power. At the same time, energy consumption for most of these embedded systems should decrease. Being able to off-load processor intensive tasks such as rendering of 2D shapes to dedicated hardware vastly decreases rendering time and frees a lot of processor resources which leads to a faster GUI and a less power consuming system

Intersection of three-dimensional geometric surfaces

Calculating the line of intersection between two three-dimensional objects and using the information to generate a third object is a key element in a geometry development system. Techniques are presented for the generation of three-dimensional objects, the calculation of a line of intersection between two objects, and the construction of a resultant third object. The objects are closed surfaces consisting of adjacent bicubic parametric patches using Bezier basis functions. The intersection determination involves subdividing the patches that make up the objects until they are approximately planar and then calculating the intersection between planes. The resulting straight-line segments are connected to form the curve of intersection. The polygons in the neighborhood of the intersection are reconstructed and put back into the Bezier representation. A third object can be generated using various combinations of the original two. Several examples are presented. Special cases and problems were encountered, and the method for handling them is discussed. The special cases and problems included intersection of patch edges, gaps between adjacent patches because of unequal subdivision, holes, or islands within patches, and computer round-off error

Elliptic integrals and the Schwarz-Christoffel transformation

AbstractThe real elliptic integrals of the first and second kind in Jacobi's normal form are computed efficiently, using the convolution number in conjunction with the method of Frobenius. For this purpose certain treatments of the Laurent series are included. Different regions of convergence on the real axis are determined, and for each one a different series is developed. The real elliptic integral of the third kind is solved within a limited parameter plane by the same method.The integral of the Schwarz-Christoffel transformation is solved in the complex variable by complex convolution number algebra, using the unit disk as mapping region. Different regions of convergence of Frobenius, Laurent, and Taylor series are determined to cover the whole disk. The complex evaluation of the elliptic integral of the third kind is included. A Schwarz-Christoffel formula for an infinite periodic mapping is given. The solutions for exterior, interior, periodic, and cyclic polygons are separately treated. Examples of several polygon mappings are presented graphically, and compared with previous numerically integrated solutions.The parameter problem is solved by the Newton-Raphson method, using a quotient matrix as approximation for the Jacobian matrix. The coordinate relations are simplified by using an overdetermined system. An exact analytical Jacobian matrix is computed, solving Leibniz' derivative of the Schwarz-Christoffel integral, and results are compared with the approximate quotient matrix method

The development and implementation of a reverse engineering method for near net shape parts

This research presents a new method for the reverse engineering of Near Net Shape (NNS) parts that bridge the current 3D scanning and Rapid Prototyping technologies. Near Net Shape is a group of manufacturing technologies that includes forging, casting, hot isostatic pressing, and additive manufacturing. This research focuses on casting process and provides a software tool along with the new method for reverse engineering a legacy casting design to the “as was” casted state instead of the “as is” current state, and at the same time, reducing the cost and time for repairing a legacy casting part. The three main objective for this research is to 1.Create a new reverse engineering method 2.Develop a software tool that is designed for feature free model editing 3.Validate the process through metal casting. The Point Cloud Library is applied for assisting point cloud processing and feature free model editing. A series of algorithms is developed for draft adding and pattern generation for the process of casting. The Rapid Pattern Manufacturing system developed in Iowa State University, Rapid Manufacturing and Prototyping Lab is applied for pattern manufacturing. This method is validated to be correct and able to reverse engineer legacy casting parts rapidly and economically through a metal casting process. The layout of this thesis is as follows: Chapter 1: provides introduction, background, research problem statement and objective of this research. Chapter 2: a literature review for the current reverse engineering method and introduces the modules of point cloud library that are used in this research. Chapter 3: presents the overview of method and algorithms that developed for this method in detail. Chapter 4: presents the implementation of this method and gives the analysis of the demo metal casting process. Chapter 5: provides future work and conclusions