Mellish theorem for generalized constant width curves

Abstract

Abstract. In this paper we give a generalization of the theorem characterizing ovals of constant width proved by Mellish (Ann Math (2) 32: [181][182][183][184][185][186][187][188][189][190] 1931).Mathematics Subject Classification. 53A04.

On isoptics and isochordal-viewed curves

“…Note that this definition implies the following: an oval α is of constant φwidth if and only if its φ-isoptic is a circle. This holds true even for the case φ = π, which corresponds to the classical constant width curves (see [16] for more details). The characterization above motivates the next definition for any closed curve, convex or not.…”

“…In this setting, Mozgawa gave in [16] a generalized notion of constant width based on the definition of a curve of constant angle (the reader can see [9] and [13]). Given an oval α and a φ-isoptic of α, with 0 < φ < π, the function κ φ above is called the sine-curvature of the φ-isoptic of α.…”

On curves with circles as their isoptics

“…We fix a curve C ∈ M(α, r). From Theorem 3.1 of [37] we know that the Steiner centroid O of C and the center of the circle coincide. Thus we assume that the origin of the coordinate system is chosen at O, so the center of this circle is (0, 0).…”

“…Our goal is to describe all curves C ∈ M possessing a circle as one of its isoptics. Note that such curves were called curves of generalized constant width in the paper [37]. Now, we will consider a subfamily M(α, r) of M defined as follows M(α, r) = {C ∈ M : C α is a circle of radius r}.…”

“…Moreover, we find explicitly a support function of a curve C ∈ M, different from a circle, which has a circle as its isoptic. These curves were considered in a very interesting paper [32] and in a paper [37] by the second author.…”