Regular polygons on isochordal-viewed hedgehogs

Abstract

A curve $\alpha$ is called isochordal viewed if there is a smooth motion of a constant length chord with its endpoints along $\alpha$ such that their tangents to the curve at these points form a constant angle. In this paper some properties of isochordal-viewed hedgehogs and Holditch curves are studied. It is proved that, under some conditions, the construction of some closed regular polygons whose vertices move smoothly along the curve $\alpha$ is possible. The property is illustrated with some examples. Moreover, Holditch curves of isochordal-viewed hedgehogs are considered and it is seen that they feature similar regular polygon properties although they are, in general, not parameterized by a support function. Finally, a recursive iteration of some Holditch curves for isochordal-viewed hedgehogs is shown to converge to the curve of polygon centers.