Binary Space Partitions for Fat Rectangles

This is the published version. Copyright © 2000 Society for Industrial and Applied Mathematic

Finite Boolean Algebras for Solid Geometry using Julia's Sparse Arrays

The goal of this paper is to introduce a new method in computer-aided geometry of solid modeling. We put forth a novel algebraic technique to evaluate any variadic expression between polyhedral d-solids (d = 2, 3) with regularized operators of union, intersection, and difference, i.e., any CSG tree. The result is obtained in three steps: first, by computing an independent set of generators for the d-space partition induced by the input; then, by reducing the solid expression to an equivalent logical formula between Boolean terms made by zeros and ones; and, finally, by evaluating this expression using bitwise operators. This method is implemented in Julia using sparse arrays. The computational evaluation of every possible solid expression, usually denoted as CSG (Constructive Solid Geometry), is reduced to an equivalent logical expression of a finite set algebra over the cells of a space partition, and solved by native bitwise operators.Comment: revised version submitted to Computer-Aided Geometric Desig

Computational Geometry Column 42

A compendium of thirty previously published open problems in computational geometry is presented.Comment: 7 pages; 72 reference

On Communication Protocols that Compute Almost Privately

A traditionally desired goal when designing auction mechanisms is incentive compatibility, i.e., ensuring that bidders fare best by truthfully reporting their preferences. A complementary goal, which has, thus far, received significantly less attention, is to preserve privacy, i.e., to ensure that bidders reveal no more information than necessary. We further investigate and generalize the approximate privacy model for two-party communication recently introduced by Feigenbaum et al.[8]. We explore the privacy properties of a natural class of communication protocols that we refer to as "dissection protocols". Dissection protocols include, among others, the bisection auction in [9,10] and the bisection protocol for the millionaires problem in [8]. Informally, in a dissection protocol the communicating parties are restricted to answering simple questions of the form "Is your input between the values \alpha and \beta (under a predefined order over the possible inputs)?". We prove that for a large class of functions, called tiling functions, which include the 2nd-price Vickrey auction, there always exists a dissection protocol that provides a constant average-case privacy approximation ratio for uniform or "almost uniform" probability distributions over inputs. To establish this result we present an interesting connection between the approximate privacy framework and basic concepts in computational geometry. We show that such a good privacy approximation ratio for tiling functions does not, in general, exist in the worst case. We also discuss extensions of the basic setup to more than two parties and to non-tiling functions, and provide calculations of privacy approximation ratios for two functions of interest.Comment: to appear in Theoretical Computer Science (series A

Constructing Binary Space Partitions for Orthogonal Rectangles in Practice

The original publication is available at www.springerlink.comIn this paper, we develop a simple technique for constructing a I3inary Space Partition (nSP) for a set of orthogonal rectangles in IR3. OUf algorithm has the novel feature that it tunes its performance to the geometric properties of the rectangles, e.g., their aspect ratios. "Fe have implemented our algorithm and tested its performance on real data scti). V\.Tc have also systematically compared the performance of our algorithm with that of other techniques presented in the literature. Our studies show that our algorithm constructs nsps of near-linear size and small height in practice, has fast running times, and answers queries efficiently. It is a method of choice for constructing BSPs for orthogonal rectangles

Parallel surface reconstruction through virtual milling

Surface definition deals with representing a surface analytically using a finite number of parameters and with acceptable levels of error. In the past few years it has become a key discipline in Computational Fluid Dynamics (CFD). Recent advances in computers and numerical algorithms have made it possible for CFD practitioners to attempt flow solutions about complex three-dimensional geometries. The first step in this process is having a numerical representation of the shape. In many cases of interest such a representation already exists; i.e., aircraft designed on a computer. Such Computer-Aided Design (CAD) descriptions do not exist, though, for objects found in nature or predating CAD. In such situations a technique for measuring the object and then constructing a surface conforming to these measurements is needed;Existing techniques for 3-D surface definition often require considerable human intervention, both in the measuring and the reconstruction process. This is a time consuming proposition. It is desirable to develop a fully automated alternative;Three-dimensional objects can be measured accurately and quickly from multiple viewpoints using a Cyberware laser digitizer. The digitizer returns the coordinates of a set of surface points. The problem is then to construct a faithful representation of the original object from these points. The algorithm proposed here has two distinct stages. In the first stage, surface fragments, using information from a single view, are produced by employing a visibility constraint and a 2-D Delaunay triangulation technique. In the next stage, surfaces from multiple views are combined through an approach that emulates the machining operation of milling. The final result is a non-convex, triangular faceted, polyhedron that approximates the object shape;A sequential version of the virtual milling algorithm exists on a Silicon Graphics workstation. The algorithm is of O(NlogN) complexity, where N is the number of data points. Experimental results have been obtained for a scaled F117-A model scanned from multiple viewpoints. Several topological issues have been addressed;A parallel version of the algorithm has been implemented on the Intel Gamma Prototype, a 128 node, distributed-memory, MIMD computer. Run times are compared to those obtained on an Iris 310/VGX workstation

Reduced-Order Equivalent-Circuit Models Of Thermal Systems Including Thermal Radiation

We established a general, automatic, and versatile procedure to derive an equivalent circuit for a thermal system using temperature data obtained from FE simulations. The EC topology was deduced from the FE mesh using a robust and general graph-partitioning algorithm. The method was shown to yield models that are independent of the boundary conditions for complicated 3D thermal systems such as an electronic chip. The results are strongly correlated with the geometry, and the EC can be extended to yield variable medium-order models. Moreover, a variety of heat sources and boundary conditions can be accommodated, and the EC models are inherently modular. A reliable method to compute thermal resistors connecting different regions was developed. It appropriately averages several estimates of a thermal resistance where each estimate is obtained using data obtained under different boundary or heating conditions. The concept of fictitious heat sources was used to increase the number of simulation datasets. The method was shown to yield models that are independent of the BCs for complicated 2-D thermal systems such as a 2D cavity. A reliable method to compute thermal resistors connecting different regions was developed. In general, the number of regions required for getting an accurate reduced-order model depends on the complexity of the system to be modeled. We have extended the reduced-order modeling procedure to include a view-factor based thermal radiation heat transfer model by including voltage controlled current sources in the equivalent circuit