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Algebraic analysis of offsets to hypersurfaces. (English) Zbl 0996.14027

From the paper: Let \(K\) be an algebraically closed field, and \({\mathcal V}\) an irreducible hypersurface in \(K^n\). The offset variety, or more formerly called parallel variety, to \({\mathcal V}\) at distance \(d\) is “essentially” the envelope of the system of spheres centered at the points of \({\mathcal V}\) with fixed radius \(d\). In the 1980s, CAGD (Computer Aided Geometric Design) community started to be interested on the topic, and began to address problems related to “offsets” to curves and surfaces, but “only” over the reals. Reason of this interest is that offsets play an important role in CAGD, since they arise in practical applications as tolerance analysis, geometric control, robot path-planning and numerical-control machining problems. In this paper, we present a complete algebraic analysis of degeneration and existence of simple and special components of generalized offsets to irreducible hypersurfaces over algebraically closed fields of characteristic zero. More precisely, we analyze the degeneration situations when offseting, and we state that there exist, at most, a finite set of distances for which the offset of a hypersurface may degenerate. As a consequence of this analysis, an algorithmic method to determine such distances is derived. Furthermore, as an application of these results, a complete degeneration analysis of the generalized offset to the sphere is developed. In addition, we study the existence of simple and special components of the offset. In this context we prove that, in the case of classical offsets, there always exists at least one simple component and, in the case of generalized offsets, we prove that for almost every distance and for almost every isometry, all components of the offset are simple.

MSC:

14Q10 Computational aspects of algebraic surfaces
68U07 Computer science aspects of computer-aided design
14J70 Hypersurfaces and algebraic geometry
70E60 Robot dynamics and control of rigid bodies
93B27 Geometric methods
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