Local shape of generalized offsets to algebraic curves
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.
scientific article; zbMATH DE number 6017593
scientific article