Semi-analytic geometry with R-functions

Abstract

V. L. Rvachev called R-functions 'logically charged functions' because they encode complete logical information within the standard setting of real analysis. He invented them in the 1960s as a means for unifying logic, geometry, and analysis within a common computational framework -in an effort to develop a new computationally effective language for modelling and solving boundary value problems. Over the last forty years, R-functions have been accepted as a valuable tool in computer graphics, geometric modelling, computational physics, and in many areas of engineering design, analysis, and optimization. Yet, many elements of the theory of R-functions continue to be rediscovered in different application areas and special situations. The purpose of this survey is to expose the key ideas and concepts behind the theory of Rfunctions, explain the utility of R-functions in a broad range of applications, and to discuss selected algorithmic issues arising in connection with their use. CONTENTS1 From Descartes to Rvachev 240 2 Functions for shapes with corners 241 3 R-functions 246 4 From inequalities to normalized functions 258 5 The inverse problem of analytic geometry 263 6 Geometric modelling 275 7 Boundary value problems 284 8 Conclusions 294 References 296 240 V. ShapiroHowever, he could not explain his own constructions using the classical Semi-analytic geometry with R-functions 241 methods of analytic and algebraic geometry that focus on direct problems of investigating given equations and inequalities. In contrast, Rvachev wanted to devise a methodology for solving what he termed the inverse problem of analytic geometry: constructing equations and inequalities for given geometric objects. This quest resulted in his seminal publication, Rvachev (1963), followed by the comprehensive theory of R-functions that has been developed over the last forty-plus years. 1 In a nutshell, R-functions operate on real-valued inequalities as differentiable logic operations; the resulting theory solves the inverse problem of analytic geometry and has a wide range of applications, with particular emphasis on solutions of boundary value problems. As of 2001, the bibliography on the theory of R-functions included more than fifteen monographs and over five hundred technical articles coauthored by Rvachev and his followers (Matsevity 2001). R-functions were introduced into the Western literature by the author (Shapiro 1988), and are now widely used in geometric modelling, computer graphics, robotics, engineering analysis, and other computational applications. The goal of this paper is to expose the key concepts in the theory of R-functions, without trying to be comprehensive. This subject will take us up to Section 4. Additional references in English are now accessible, notably the review by Rvachev and Sheiko (1995). Among the references in Russian, the monograph by Rvachev (1982) continues to serve as an encyclopedic source of many key ideas and results. The utility of R-functions cannot be fully appreciated without some discussion of...

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