Regions surrounded by cylinders of circles of fixed radii and exposition of their shapes by natural graphs

TL;DR

This paper explores regions enclosed by cylinders of fixed-radius circles, analyzing their shape representations through natural graphs derived from level sets, and connects these findings to real algebraic geometry and graph realizations.

Contribution

It introduces a method to realize specific trees as graphs from level sets of projections of regions formed by cylinders of circles.

Findings

Certain trees are shown to be realizable as natural graphs of these regions.

Connections are made to real algebraic curves and their regions in the plane.

The study extends understanding of shape representation via algebraic maps.

Abstract

We investigate regions formed by cylinders of circles of fixed radii. We investigate graphs obtained by collapsing each level set of the functions represented by the natural projections of them to the $1$-dimensional line. Some specific trees obtained in simple ways from so-called balanced trees are shown to be realized as such graphs. Related studies on regions in the Euclidean plane surrounded by real algebraic curves are presented by several researchers. One of pioneering studies is presented by Bodin, Popescu-Pampu and Sorea in 2022--3 as an elementary and surprisingly new study. The author has been interested in related studies and also in constructing natural and explicit real algebraic maps onto such regions, generalizing the canonical projections of the unit spheres. Such studies in real algebraic geometry, different from theory of existence in the last century, mainly studied by Nash and Tognoli, are remarked.