Parallelograms inscribed in a curve having a circle as π/2-isoptic

Abstract

Abstract. Jean-Marc Richard observed in [7] that maximal perimeter of a parallelogram inscribed in a given ellipse can be realized by a parallelogram with one vertex at any prescribed point of ellipse. Alain Connes and Don Zagier gave in [4] probably the most elementary proof of this property of ellipse. Another proof can be found in [1]. In this note we prove that closed, convex curves having circles as π 2 -isoptics have the similar property.1. Introduction. Let C be a closed and strictly convex curve. We fix an interior point of C as an origin of a coordinate system. Denote e it = (cos t, sin t), ie it = (− sin t, cos t). The function p :is called the support function of C. For a strictly convex curve p is differentiable. We assume that the function p is of class C 2 and the curvature of C is positive. We have the following equation of C in terms of its support function (1.1) z(t) = p(t)e it +ṗ(t)ie it .Then z = p 2 (t) +ṗ 2 (t) and R(t) = p(t) +p(t) is a radius of curvature of C at t.2000 Mathematics Subject Classification. 53C12.

On curves with circles as their isoptics

“…In the paper [32] there was given an example of a support function of a curve C ∈ M whose orthoptic is a circle, namely p(t) = a sin 2 3t + b cos 2 9t + c.…”

“…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.…”

Mellish theorem for generalized constant width curves

“…(2) Is it possible to give a counterpart of a theorem given by Miernowski ( [2,19]) showing that for closed, convex curves having circles as π 2 -isoptics, the maximal perimeter of a parallelogram inscribed in this curve can be realized by a parallelogram with one vertex at any prescribed point of the curve? (3) Is it possible to extend the above results to ovaloids, obtaining a suitable generalization of the second part of Mellish' paper [17]?…”