On curves with circles as their isoptics

Abstract

In the present paper we describe the family of all closed convex plane curves of class $$C^1$$ C 1 which have circles as their isoptics. In the first part of the paper we give a certain characterization of all ellipses based on the notion of isoptic and we give a geometric characterization of curves whose orthoptics are circles. The second part of the paper contains considerations on curves which have circles as their isoptics and we show the form of support functions of all considered curves.

An explicit characterization of isochordal-viewed multihedgehogs with circular isoptics

“…for all t ∈ J ϕ such that h ′ (t) + h (3) (t) ̸ = 0. We know that the right-hand side of ( 12) is equal to b sin(ϕ) .…”

“…The same curve is obtained if c 3 = c 4 = 0. But notice that it does not correspond to a (ϕ, ℓ)-isochordal-viewed curve of constant ϕ-width; in fact, the assumption h ′ (t) + h (3) (t) ̸ = 0 to construct the system of equations above does not hold.…”

“…Indeed, from (11) and (13) we can get expressions for h ′ (t + ϕ) and h ′′ (t + ϕ). A substitution in (14) yields the equation h ′ (t) + h (3) (t) = 0. But the solution of this differential equation cannot fulfill (13) for b = 1.…”

“…In the recent years, the study of isoptic curves, support functions and related topics has attracted the interest of many researchers (see [3], [5], [15] or [6] among others). A curve α parameterized by a support function h can be written as α(t) = h(t) (cos t, sin t) + h ′ (t) (− sin t, cos t), t ∈ I, where I = [0, 2 m π], for some m ∈ Z, and the function h is 2 π m-periodic.…”

“…If the ϕ-isoptic of a closed curve α is circular, then α is called of constant ϕ-width [9]. A full characterization of this kind of curves in terms of their support function by a Fourier series has been given recently in [3].…”