On inverse construction of isoptics and isochordal-viewed curves

Abstract

Given a regular closed curve α in the plane, a ϕ-isoptic of α is a locus of points from which pairs of tangent lines to α span a fixed angle ϕ. If, in addition, the chord that connects the two points delimiting the visibility angle is of constant length ℓ, then α is said to be (ϕ,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 ϕ-isoptic curve c as an input and construct a curve whose ϕ-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 (ϕ,ℓ)-isochordal-viewed curve with a prescribed ϕ-isoptic c. Our results are constructive and offer a tool to easily generate this type of curves. In particular, we show examples of (ϕ,ℓ)-isochordal-viewed curves whose ϕ-isoptic is not circular. Finally, we prove that these curves allow the motion of a regular polygon whose vertices lie along the (ϕ,ℓ)-isochordal-viewed curve.