Abstract
In the present paper we describe the family of all closed convex plane curves of class 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.
Highlights
In the present paper we consider the family M of all closed convex plane curves of class C1
We denote by p a support function of the curve C ∈ M with respect to the origin O
In the second part of the paper using some considerations from the previous sections we find curves with special isoptics called orthoptics
Summary
In the present paper we consider the family M of all closed convex plane curves of class C1. Let Cα be the locus of vertices of a fixed angle π − α formed by two tangent lines of the curve C. In the second part of the paper using some considerations from the previous sections we find curves with special isoptics called orthoptics. We present there a certain characterization of a class of curves with circles as their orthoptics. 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. We would like to emphasize that all the papers in the bibliography, that is [1,2,3,4,5,6,7,8,9,10,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,47,49,50,51,52,53,54,55,56], with the exception of Santalo’s and Su’s books, [46,48], and the paper by Cyr, [11], present a wide spectrum of results in isoptics theory and are included here for the interested reader to have a complete overview of isoptics theory
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