On inverse construction of isoptics and isochordal-viewed curves

Given a regular closed curve & alpha; in the plane, a & phi;-isoptic of & alpha; is a locus of points from which pairs of tangent lines to & alpha; span a fixed angle & phi;. If, in addition, the chord that connects the two points delimiting the visibility angle is of constant length l, then & alpha; is said to be (& phi;, l)-isochordal viewed. Some properties of these curves have been studied, yet their full classification is not known. We approach the problem in an inverse manner, namely that we consider a & phi;-isoptic curve c as an input and construct a curve whose & phi;-isoptic is c. We provide thus a sufficient condition that constitutes a partial solution to the inverse isoptic problem. In the process, we also study a relation of isoptics to multihedgehogs. Moreover, we formulate conditions on the behavior of the visibility lines so as their envelope is a (& phi;, l)-isochordal-viewed curve with a prescribed & phi;-isoptic c. Our results are constructive and offer a tool to easily generate this type of curves. In particular, we show examples of (& phi;, l)-isochordal-viewed curves whose & phi;-isoptic is not circular. Finally, we prove that these curves allow the motion of a regular polygon whose vertices lie along the (& phi;, l)-isochordal-viewed curve.& COPY; 2023 Elsevier B.V. All rights reserved.

Automated summary

This article investigates the problem of closed curves with specific attributes and their related isochordal-viewed curves. It provides a condition for constructing the curves and solves the inverse isochordal problem. Furthermore, it demonstrates that these curves can accommodate the motion of regular polygons.