Equal Aperture Angles Curve for some Convex Sets in the Plane

TL;DR

This paper introduces a method to construct curves in the plane where the visibility angle of a convex set remains constant, providing explicit formulas, computational techniques, and practical examples, with open-source software support.

Contribution

It presents an explicit formula for equal visibility angle curves for convex sets and discusses computational methods and examples, advancing geometric analysis tools.

Findings

Explicit formulas for equal visibility angle curves

Effective computational methods demonstrated

Open-source software used for practical implementation

Abstract

For given convex set $K$ in the plane (not necessarily bounded), we can construct the curve $C$ for which the visibility (aperture) angle of this set has the same, prescribed value. We give the implicit formula for $C$, discuss some issues concerning practical computations of $C$ and bring several simple examples, when the equal visibility angle curves can be effectively computed explicitly. We conclude with some remarks about possible directions for further research in this area. Extensive use of Open Source software (Sage, Pylab, IPython,...) is a key feature of this article.