On isoptics and isochordal-viewed curves

Abstract

In this paper, some results involving isoptic curves and constant φ-width curves are given for any closed curve. The non-convex case, as well as non-simple shapes with or without cusps are considered. Relating the construction of isoptics to the construction given in Holditch's theorem, a kind of curves is defined: the isochordal-viewed curves. The explicit expression of these curves is given together with some examples. Integral formulae on the area of their isoptics are obtained and a Barbier-type theorem is derived. Finally, a characterization for isochordal-viewed hedgehogs and curves of constant φ-width is given in terms of an angle function.Key words and phrases. Isoptic curves, Isochordal-viewed curves, Holditch's theorem, Curves of constant φ-width, Hedgehogs.The author has been partially funded by the BCAM Severo Ochoa accreditation of excellence, Spain (SEV-2017-0718).

Generalized plane offsets and rational parameterizations

“…□ Remark 1. If the curves of Theorem 1 are not simple (they present self-intersections), then the areas must be counted by sign and multiplicity as it is pointed out in (Rochera, 2022c), where notice that the previous result can be seen as a particular case of Lemma 4.1 stated in the same paper. In order to have pairs of simple curves, the self-intersection avoidance of generalized offsets will be studied in Section 2.4.…”