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 fixed-precision. 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 mass-produced computers were pocket calculators. Although one's first exposure to computers today is likely to be some non-numerical application, numeri...

We propose interactive beautification, a technique for rapid geometric design, and introduce the technique and its algorithm with a prototype system Pegasus. The motivation is to solve the problems with current drawing systems: too many complex commands and unintuitive procedures to satisfy geometric constraints. Interactive beautification system receives the user's freestroke and beautifies it by considering geometric constraints among segments. A single stroke is beautified one after another, preventing accumulation of recognition errors or catastrophic deformation. Supported geometric constraints includes perpendicularity, congruence, symmetry, etc., which were not seen in existing freestroke recognition systems. In addition, the system generates multiple candidates as a result of beautification to solve the problem of ambiguity. Using the technique, the user can draw precise diagrams rapidly satisfying geometric relations without using any editing commands. Interactive beautificat...

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We present a correctness proof of a graph-directed variational geometric constraint solver. First, we prove that the graph reduction that establishes the sequence in which to apply the construction steps defines a terminating confluent reduction system, in the case of well-constrained graphs. For overconstrained problems there may not be a unique normal form. Underconstrained problems, on the other hand, do have a unique normal form. Second, we prove that all geometric solutions found using simple root-selection rules must place certain triples of elements in the same topological order, no matter which graph reduction sequence they are based on. Moreover, we prove that this implies that the geometric solutions derived by different reduction sequences must be congruent. Again, this result does not apply to overconstrained problems. Keywords: geometric constraint solving, computer aided design 1. Introduction Geometric constraint solving has broad applications in a wide range of subje...

A central issue in dealing with geometric constraint systems for CAD/CAM/CAE is the generation of an optimal decomposition plan that not only aids efficient solution, but also captures design intent and supports conceptual design. Though complex, this issue has evolved and crystallized over the past few years, permitting us to take the next important step: in this paper, we formalize, motivate and explain the decomposition–recombination (DR)-planning problem as well as several performance measures by which DR-planning algorithms can be analyzed and compared. These measures include: generality, validity, completeness, Church–Rosser property, complexity, best- and worst-choice approximation factors, (strict) solvability preservation, ability to deal with underconstrained systems, and ability to incorporate conceptual design decompositions specified by the designer. The problem and several of the performance measures are formally defined here for the first time—they closely reflect specific requirements of CAD/CAM applications. The clear formulation of the problem and performance measures allow us to precisely analyze and compare existing DR-planners that use two well-known types of decomposition methods: SR (constraint shape recognition) and MM (generalized maximum matching) on constraint graphs. This analysis additionally serves to illustrate and provide intuitive substance to the newly formalized measures. In Part II of this article, we use the new performance measures to guide the development of a new DR-planning algorithm which excels with respect to these performance measures. c ○ 2001 Academic Press 1.

One of the major unsolved problems in parametric solid modeling is a robust update (regeneration) of the solid's boundary representation, given a specified change in the solid's parameter values. The fundamental difficulty lies in determining the mapping between boundary representations for solids in the same parametric family. Several heuristic approaches have been proposed for dealing with this problem, but the formal properties of such mappings are not well understood. We propose a formal definition for Boundary Representation (BR-)deformation for solids in the same parametric family, based on the assumption of continuity: small changes in solid parameter values should result in small changes in the solid's boundary representation, which may include local collapses of cells in the boundary representation. The necessary conditions that must be satisfied by any BR-deforming mappings between boundary representations are powerful enough to identify invalid updates in many (but...

This paper deals with the constrained reconstruction of 3D geometric models of objects from range data. It describes a new technique of global shape improvement based upon feature positions and geometric constraints. It suggests a general incremental framework whereby constraints can be added and integrated in the model reconstruction process, resulting in an optimal trade-off between minimization of the shape fitting error and the constraint tolerances. After defining sets of constraints for planar and special case quadric surface classes based on feature coincidence, position and shape, the paper shows through application on synthetic model that our scheme is well behaved. The approach is then validated through experiments on different real parts. This work is the first to give such a large framework for the integration of geometric relationships in object modelling. The technique is expected to have a great impact in reverse engineering applications and manufactured object modelling where the majority of parts are designed with intended feature relationships. Keywords Reverse engineering, Geometric constraints, constrained shape reconstruction, shape optimization. 2

In design and manufacturing applications, users of computer aided design systems want to define relationships between dimension variables, since such relationships express design intent very flexibly. This work reports on a technique developed to enhance a class of constructive geometric constraint solvers with the capability of managing functional relationships between dimension variables. The method is shown to be correct.

Feature-based design is becoming one of the fundamental design paradigms of CAD systems. In this paradigm the basic unit is a feature and parts are constructed by a sequence of feature attachment operations. The type and number of possible features involved depend upon product type, the application reasoning process and the level of abstraction. Therefore to provide CAD systems with a basic mechanism to define features that fit the end-user needs seems more appropriate than trying to provide a large repertoire of features covering every possible application. A procedural mechanism is proposed for generating and deploying userdefined features in a feature-based design paradigm. The usefulness of the mechanism relies on two functional capabilities. First the shape and size of the user defined features are instantiated according to parameter values given by the end-user. Second the end-user positions and orients the feature in the part being designed by means of geometric ges...

. Three-dimensional geometric constraint solving is a rapidly developing field, with applications in areas such as kinematics, molecular modeling, surveying, and geometric theorem proving. While two-dimensional constraint solving has been studied extensively, there remain many open questions in the arena of three-dimensional problems. In this paper, we continue the development of our previous work on configuring a set of points and planes in three-space so that the configuration satisfies a given system of constraints. The constraint system considered consists of six geometric elements and pairwise constraints between triples of the elements. We first review the basic techniques developed in our earlier work germaine to the current problem and explain how the problem we consider in this paper occurs. We then demonstrate how to solve the case of a geometric constraint system with four points and two planes. 1. Introduction The spatial geometric constraint problem consists of a set of g...