1507134
Comparison of interval methods for plotting algebraic curves
This paper compares the performance and efficiency of different function range interval methods for plotting f(x, y)=0 on a rectangular region based on a subdivision scheme, where f(x, y) is a polynomial. The solution of this problem has many applications in CAGD. The methods considered are interval arithmetic methods (using the power basis, Bernstein basis, Homer form and centred form), an affine arithmetic method, a Bernstein coefficient method, Taubin's method, Rivlin's method, Gopalsamy's method, and related methods which also take into account derivative information. Our experimental results show that the affine arithmetic method, interval arithmetic using the centred form, the Bernstein coefficient method, Taubin's method, Rivlin's method, and their related derivative methods have similar performance, and generally they are more accurate and efficient than Gopalsamy's method and interval arithmetic using the power basis, the Bernstein basis, and Horner form methods.
750888
Detecting approximate incomplete symmetries in discrete point sets
Motivated by the need to detect design intent in approximate boundary representation models, we give an algorithm to detect incomplete symmetries of discrete points, giving the models' potential local symmetries at various automatically detected tolerances. Here, incomplete symmetry is defined as a set of incomplete cycles which are constructed by, e.g., a set of consecutive vertices of an approximately regular polygon, induced by a single isometry. All seven 3D elementary isometries are considered for symmetry detection. Incomplete cycles are first found using a tolerance-controlled point expansion approach. Subsequently, these cycles are clustered for incomplete symmetry detection. The resulting clusters have welldefined, unambiguous approximate symmetries suitable for design intent detection, as demonstrated experimentally.
27145850
A non-Euclidean implementation of LOGO
Abstract This paper describes an implementation of the LOGO Turtle Graphics programming environment in non-Euclidean geometry. Turtle graphics and non-Euclidean geometry are described briefly, and consideration is given to some of the details and problems of implementation. We describe the ways in which the LOGO language must be modified for a non-Euclidean environment, and the geometrical problems encountered during implementation.
5659182
Finding approximate shape regularities in reverse engineered solid models bounded by simple surfaces
Current reverse engineering systems are able to generate simple valid boundary representation (B-rep) models from 3D range data. Such models suffer from various inaccuracies caused by noise in the input data and algorithms. The quality of reverse engineered geometric models can potentially be improved by finding candidate shape regularities in such an initial model, and imposing a suitable subset of them on the model by using constraints, in a postprocessing step called beautification. Finding such candidate regularities is a necessary first step, and is discussed in this paper. Algorithms for analysis are presented which use feature objects to describe properties of faces, edges and vertices, and small groups of these elements in a B-rep model with only planar, spherical, cylindrical, conical and toroidal faces. The methods seek similarities between feature objects, e.g. axes which are parallel, for each property type. For each group of similar feature objects they also try to find a special feature object which might represent the group, e.g. an integer value which approximates the radius of similar cylinders. The feature objects used represent shape parameters, directions, axes and positions present in the model. Experiments show that the regularities found by these algorithms include the desired regularities. Although other spurious regularities which must be discarded in subsequent beautification steps are also produced, their number can be reduced by appropriate choice of tolerance values.
2214669
Fast degree elevation and knot insertion for B-spline curves
We give a new, simple algorithm for simultaneous degree elevation and knot insertion for B-spline curves. The method is based on the simple approach of computing derivatives using the control points, resampling the knot vector, and then computing the new control points from the derivatives. We compare our approach with previous algorithms and illustrate it with examples.
17590358
Constructing Regularity Feature Trees for Solid Models
Approximate geometric models, e.g. as created by reverse engineering, describe the approximate shape of an object, but do not record the underlying design intent. Automatically inferring geometric aspects of the design intent, represented by feature trees and geometric constraints, enhances the utility of such models for downstream tasks. One approach to design intent detection in such models is to decompose them into regularity features. Geometric regularities such as symmetries may then be sought in each regularity feature, and subsequently be combined into a global, consistent description of the model's geometric design intent. This paper describes a systematic approach for finding such regularity features based on recovering broken symmetries in the model. The output is a tree of regularity features for subsequent use in regularity detection and selection. Experimental results are given to demonstrate the operation and efficiency of the algorithm.
16901231
A Noise Suppression System for the AMR Speech Codec
In this paper we describe a noise reduction preprocessing algorithm for the adaptive multirate (AMR) speech codec of the GSM system. The algorithm is based on spectral weighting and explicitly takes into account the properties of the human auditory system. The weighting rule results in the smallest possible speech distortion under the constraint that the background noise should exhibit no audible distortions. The algorithm was implemented in 16 Bit fixed-point arithmetic and submitted to the ETSI AMR noise reduction standardization contest. Compared to other algorithms, our noise reduction method gave very good results in CCR tests and good results in ACR tests.
6378032
Adding random edges to dense graphs
This paper investigates the addition of random edges to arbitrary dense graphs; in particular, we determine the number of random edges required to ensure various monotone properties including the appearance of a fixed size clique, small diameter and k-connectivity.
17312370
A note on short cycles in a hypercube
How many edges can a quadrilateral-free subgraph of a hypercube have? This question was raised by Paul Erdos about 27 years ago. His conjecture that such a subgraph asymptotically has at most half the edges of a hypercube is still unresolved. Let f(n,C"l) be the largest number of edges in a subgraph of a hypercube Q"n containing no cycle of length l. It is known that f(n,C"l)=o(|E(Q"n)|), when l=4k, k>=2 and that f(n,C"6)>=13|E(Q"n)|. It is an open question to determine f(n,C"l) for l=4k+2, k>=2. Here, we give a general upper bound for f(n,C"l) when l=4k+2 and provide a coloring of E(Q"n) by four colors containing no induced monochromatic C"1"0.
18074193
Dynamic Diffusion Load Balancing
We consider the problem of dynamic load balancing in arbitrary (connected) networks on n nodes. Our load generation model is such that during each round, n tasks are generated on arbitrary nodes, and then (possibly after some balancing) one task is deleted from every non-empty node. Notice that this model fully saturates the resources of the network in the sense that we generate just as many new tasks per round as the network is able to delete. We show that even in this situation the system is stable, in that the total load remains bounded (as a function of n alone) over time. Our proof only requires that the underlying “communication” graph be connected. (It of course also works if we generate less than n new tasks per round, but the major contribution of this paper is the fully saturated case.) We further show that the upper bound we obtain is asymptotically tight (up to a moderate multiplicative constant) by demonstrating a corresponding lower bound on the system load for the particular example of a linear array (or path). We also show some simple negative results (i.e., instability) for work-stealing based diffusion-type algorithms in this setting.
94046
Rapidly mixing Markov chains for sampling contingency tables with a constant number of rows
We consider the problem of sampling almost uniformly from the set of contingency tables with given row and column sums, when the number of rows is a constant. (2002) have recently given a fully polynomial randomized approximation scheme (fpras) for the related counting problem, which only employs Markov chain methods indirectly. But they leave open the question as to whether a natural Markov chain on such tables mixes rapidly. Here we answer this question in the affirmative, and hence provide a very different proof of the main result of Cryan and Dyer. We show that the "2 /spl times/ 2 heat-bath" Markov chain is rapidly mixing. We prove this by considering first a heat-bath chain operating on a larger window. Using techniques developed by Morris and Sinclair (2002) (see also Morris (2002)) for the multidimensional knapsack problem, we show that this chain mixes rapidly. We then apply the comparison method of Diaconis and Saloff-Coste (1993) to show that the 2 /spl times/ 2 chain is rapidly mixing. As part of our analysis, we give the first proof that the 2 /spl times/ 2 chain mixes in time polynomial in the input size when both the number of rows and the number of columns is constant.