An explicit characterization of isochordal-viewed multihedgehogs with circular isoptics

On inverse construction of isoptics and isochordal-viewed curves

“…The parametric construction of a ϕ-isoptic of α implicitly assumes the existence of a homeomorphism f such that α(t) and α f (t) are the contact points of α where two tangent lines to α meet at an angle ϕ. This homeomorphism is called the Holditch function for the parameterization α and the angle ϕ and its consideration assumes that no retrograde movements are done by the moving chord that joins both contact points, see [17] for more details.…”

“…Notice that from the definition of a ϕ-isoptic, we are only imposing that the pair of tangent lines to α make a constant angle ϕ, not that the tangent vectors to α that define these lines span an angle ϕ. In fact, once a homeomorphism f has been chosen, the angle between tangents could change and be either ϕ or π − ϕ for different parameter values for curves with cusps (see [17]).…”

“…Prop. 2.1 of [17]. This typically leads to an underconstrained problem with two variables (parameters t 1 and t 2 ) and one constraint (the two tangent lines at α(t 1 ) and α(t 2 ) spanning the given angle ϕ).…”

“…If, in addition, the chord that connects the two contact points delimiting the visibility angle is of constant length ℓ, then α is said to be (ϕ, ℓ)-isochordal viewed. We refer the reader for more details and further properties of these curves to [6,17,18,19]. Requiring the distance between the contact points to be of constant length, i.e., α(t 1 ) − α(t 2 ) = ℓ, however, makes the problem difficult.…”