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The Exact Computation Paradigm
, 1994
"... We describe a paradigm for numerical computing, based on exact computation. This emerging paradigm has many advantages compared to the standard paradigm which is based on fixedprecision. We first survey the literature on multiprecision number packages, a prerequisite for exact computation. Next ..."
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Cited by 94 (10 self)
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We describe a paradigm for numerical computing, based on exact computation. This emerging paradigm has many advantages compared to the standard paradigm which is based on fixedprecision. We first survey the literature on multiprecision number packages, a prerequisite for exact computation. Next we survey some recent applications of this paradigm. Finally, we outline some basic theory and techniques in this paradigm. 1 This paper will appear as a chapter in the 2nd edition of Computing in Euclidean Geometry, edited by D.Z. Du and F.K. Hwang, published by World Scientific Press, 1994. 1 1 Two Numerical Computing Paradigms Computation has always been intimately associated with numbers: computability theory was early on formulated as a theory of computable numbers, the first computers have been number crunchers and the original massproduced computers were pocket calculators. Although one's first exposure to computers today is likely to be some nonnumerical application, numeri...
Towards Exact Geometric Computation
, 1994
"... Exact computation is assumed in most algorithms in computational geometry. In practice, implementors perform computation in some fixedprecision model, usually the machine floatingpoint arithmetic. Such implementations have many wellknown problems, here informally called "robustness issues". To rec ..."
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Cited by 88 (6 self)
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Exact computation is assumed in most algorithms in computational geometry. In practice, implementors perform computation in some fixedprecision model, usually the machine floatingpoint arithmetic. Such implementations have many wellknown problems, here informally called "robustness issues". To reconcile theory and practice, authors have suggested that theoretical algorithms ought to be redesigned to become robust under fixedprecision arithmetic. We suggest that in many cases, implementors should make robustness a nonissue by computing exactly. The advantages of exact computation are too many to ignore. Many of the presumed difficulties of exact computation are partly surmountable and partly inherent with the robustness goal. This paper formulates the theoretical framework for exact computation based on algebraic numbers. We then examine the practical support needed to make the exact approach a viable alternative. It turns out that the exact computation paradigm encomp...
Featurebased Similarity Assessment of Solid Models
, 1997
"... This paper presents our initial efforts to develop a systematic approach for assessing the similarity of solid models based on how they will be manufactured. The goal of this work is to develop methods that, given a solid model representing the design of a new product, query a product information da ..."
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Cited by 39 (13 self)
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This paper presents our initial efforts to develop a systematic approach for assessing the similarity of solid models based on how they will be manufactured. The goal of this work is to develop methods that, given a solid model representing the design of a new product, query a product information database (of solid models, associated manufacturing plans, and related attributes) and identify existing designs with manufacturing plans similar to some reasonable plan for the new designor useful as a starting point for creation of a new plan for the new design.
Swept Volumes: Foundations, Perspectives, and Applications
"... Several fundamental developments in the past decade have led to a better understanding of swept volumes. While the underlying formulation for characterizing the volume generated as a result of the motion of a geometric entity in space has appeared in various fields under different names, this review ..."
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Cited by 31 (1 self)
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Several fundamental developments in the past decade have led to a better understanding of swept volumes. While the underlying formulation for characterizing the volume generated as a result of the motion of a geometric entity in space has appeared in various fields under different names, this review seeks to unify the terminology and demonstrate the applicability to different fields. This paper reviews the various formulations that have appeared, outlines the basic research involved, and highlights the implications on research in engineering, mathematics, and computer science. The applicability of this seemingly simple formulation to the fields of solid modeling, robotics, manufacturing automation, and visualization is demonstrated through results reported by the authors, each in their own field. Keywords: Swept volumes, workspace analysis, NC verification, solid modeling. 1.0 Historical Remarks Attempts were made to obtain the exact swept volume of the motion of a rigid body in spa...
Extracting Alternative Machining Features: An Algorithmic Approach
 IEEE PARALLEL AND DISTRIBUTED TECHNOLOGY
, 1995
"... Automated recognition of features from CAD models has been attempted for a wide range ..."
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Cited by 27 (14 self)
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Automated recognition of features from CAD models has been attempted for a wide range
Finding Compact Coordinate Representations for Polygons and Polyhedra
 IBM Journal of Research and Development
, 1989
"... Practical solid modeling systems are plagued by numerical problems that arise from using floatingpoint arithmetic. For example, polyhedral solids are often represented by a combination of geometric and combinatorial information. The geometric information might consist of explicit plane equations, wi ..."
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Cited by 23 (4 self)
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Practical solid modeling systems are plagued by numerical problems that arise from using floatingpoint arithmetic. For example, polyhedral solids are often represented by a combination of geometric and combinatorial information. The geometric information might consist of explicit plane equations, with floatingpoint coefficients; the combinatorial information might consist of face, edge, and vertex adjacencies and orientations, with edges defined by faceface adjacencies and vertices by edgeedge adjacencies. Problems arise when numerical error in geometric operations causes the geometric information to become inconsistent with the combinatorial information. These problems could be avoided by using exact arithmetic instead of floatingpoint arithmetic. However, some operations, like rotation, increase the number of bits required to represent the plane equation coefficients. Since the execution time of exact arithmetic operators increases with the number of bits in the operands, the inc...
Building MRSEV Models for CAM Applications
, 1994
"... In integrating CAD and CAM applications, one major problem is how to interpret CAD information in a manner that makes sense for CAM. Our goal is to develop a general approach that can be used with a variety of CAD and CAM applications for the manufacture of machined parts. In particular, we present ..."
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Cited by 20 (13 self)
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In integrating CAD and CAM applications, one major problem is how to interpret CAD information in a manner that makes sense for CAM. Our goal is to develop a general approach that can be used with a variety of CAD and CAM applications for the manufacture of machined parts. In particular, we present a methodology for taking a CAD model, extracting alternative interpretations of the model as collections of MRSEVs (Material Removal Shape Element Volumes, a STEPbased library of machining features), and evaluating these interpretations to determine which one is optimal. The evaluation criteria may be defined by the user, in order to select the best interpretation for the particular application at hand. Keywords: Design Critiquing, CAD/CAM Integration, Feature Recognition. 1 Introduction Although various CAD and CAM applications may have compatible goals and functionality, the specific details are often different enough that it can be difficult to integrate them. One major problem is how...
An Approach for Solving Systems of Parametric Polynomial Equations
, 1993
"... An approach for solving nonlinear polynomial equations involving parameters is proposed. A distinction is made between parameters and variables. The objective is to generate from a system of parametric equations, solved forms from which solutions for specific values of parameters can be obtained wit ..."
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Cited by 11 (3 self)
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An approach for solving nonlinear polynomial equations involving parameters is proposed. A distinction is made between parameters and variables. The objective is to generate from a system of parametric equations, solved forms from which solutions for specific values of parameters can be obtained without much additional computations. It should be possible to analyze the parametrized solved forms so that it can be determined for different parameter values whether there are infinitely many solutions, finitely many solutions, or no solutions at all. The approach is illustrated for two different symbolic methods for solving parametric equations  Grobner basis computations and characteristic set computations. These methods are illustrated on a number of examples. 1.2 Introduction Many complex phenomena can be modeled using nonlinear polynomial equations. Examples include imaging transformations in computer vision, computing geometric invariants, geometric and solid modeling, constraintb...
Recognition of volumetric features from CAD models: Problem formalization and algorithms
, 1993
"... Automated recognition of features from CAD models has been attempted for a wide range of application domains in mechanical engineering. However, the absence of a clear mathematical formalism for the problem has made it di cult to develop a general approachand thus most of these methods are limited ..."
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Cited by 9 (7 self)
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Automated recognition of features from CAD models has been attempted for a wide range of application domains in mechanical engineering. However, the absence of a clear mathematical formalism for the problem has made it di cult to develop a general approachand thus most of these methods are limited in scope. In this paper, we develop a formalization of the problem of recognizing a class of machinable features expressed as MRSEVs (a PDES/STEP library of machining features) [19], and an algorithm for solving this problem. Some of the characteristics of this approach are: the algorithm handles a large variety ofholeandpocket features features along with elementary accessibility constraints and blends for those features� it is provably complete, even if the features intersect with each other in arbitrarily complex ways� it has O(n 4)worstcase time complexity. 1
Multiple sweeping using the DenavitHartenberg representation method
, 1999
"... The method of consecutive revolving or extrusion of a geometric entity in a CAD system is typically used by a designer to represent complex solids. While it is evident that consecutive sweeping is a very effective tool, it has been restricted by the associated difficult mathematics in representing c ..."
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Cited by 6 (4 self)
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The method of consecutive revolving or extrusion of a geometric entity in a CAD system is typically used by a designer to represent complex solids. While it is evident that consecutive sweeping is a very effective tool, it has been restricted by the associated difficult mathematics in representing consecutive sweeps and in analyzing the resulting equations. The DenavitHartenberg (DH) method, first introduced in 1955 in the field of kinematics to mathematically relate motion between two coordinate frames, provides an effective formulation for characterizing the resulting solid model. Design variations imposed on sweep properties are easily propagated through the formulation to update the solid. This paper illustrates the use of the DH formulation for multiple sweeps and demonstrates the method through examples. # 1999 Elsevier Science Ltd. All rights reserved.