Local shape of generalized offsets to algebraic curves
type: Item
instance of: scholarly article
title: Local shape of generalized offsets to algebraic curves
zbMATH Open document ID: 1236.14051
published in: Journal of Symbolic Computation
publication date: 2012-03-22T00:00:00Z
review text: The offset to a curve \(C\) is the locus traced by a point on the curve normal and at constant distance \(d\) to the curve point. Here, the author considers generalized (or ``non-classical'') offsets to algebraic curves, where the curve normal is additionally rotated by an angle \(\theta \neq 0,\pi\) about the curve point. His particular interest concerns the offset's local shape in the sense of \textit{J.~G.~Alcazar} and \textit{J.~R.~Sendra} [J. Symb. Comput. 42, No.~3, 338--351 (2007; Zbl 1123.14031)], who conducted a similar study for the classical offset case. The local shape of a curve in a regular point is either an ``elbow'' or a ``flex''. In singular points one distinguishes ``thorn'' and ``beak''. The main question is now whether these local shapes are preserved under non-classical offsetting or not? In contrast to the classical case, regular curve points never generate singular places or cusps, never preserve flexes and tangent directions and, in general, do not preserve turning points (points with vertical or horizontal tangent). The study of the local shape of non-classical offsets generated from singular points is more involved. As in the classical case, they are usually smoothed. A complete discussion requires numerous case distinctions. Again, many differences to the classical case can be observed.
reviewed by: Hans-Peter Schröcker
Mathematics Subject Classification ID: 14Q05
Mathematics Subject Classification ID: 14Q99
Mathematics Subject Classification ID: 65D17
zbMATH DE Number: 6017593
zbMATH Keywords: offsets
zbMATH Keywords: generalized offsets
zbMATH Keywords: local shape
zbMATH Keywords: offset topology
author: Juan Gerardo Alcazar
MaRDI profile type: Publication
full work available at URL: j.jsc.2011.12.001
OpenAlex ID: W2084147012
cites work: Good global behavior of offsets to plane algebraic curves
cites work: Good local behavior of offsets to rational regular algebraic surfaces
cites work: Good local behavior of offsets to implicit algebraic curves
cites work: Local shape of offsets to algebraic curves
cites work: Parametric generalized offsets to hypersurfaces
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cites work: A Laguerre geometric approach to rational offsets
cites work: Algebraic analysis of offsets to hypersurfaces
cites work: Rationality analysis and direct parametrization of generalized offsets to quadrics.
cites work: Q5799167
DOI: 10.1016/J.JSC.2011.12.001
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description: scientific article; zbMATH DE number 6017593