On inverse construction of isoptics and isochordal-viewed curves

Abstract

Given a regular closed curve α 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 $\ell$, then $\alpha$ is said to be $(\phi,\ell)$-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 $(\phi,\ell)$-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 $(\phi,\ell)$-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 $(\phi,\ell)$-isochordal-viewed curve.