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

Naoki Kitazawa

TL;DR

The paper studies regions in Euclidean spaces bounded by fixed-radius circles (cylinders of circles) and analyzes their 1D collapsed representations via Poincaré-Reeb graphs. It proves two main theorems: first, that any augmented balanced rooted tree can be realized as the Poincaré-Reeb digraph of an RA-region bounded by cylinders of a circle with a large radius; second, a two-tree variant extends this realization to a joined structure with reversed orientation. The approach combines explicit geometric constructions with stratified Morse-theoretic ideas and real algebraic maps to connect combinatorial graph structures with RA-regions. These results advance explicit realizations of Reeb-type graphs in real algebraic geometry and offer practical tools for visualization and topological analysis of algebraic regions.

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.

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

TL;DR

The paper studies regions in Euclidean spaces bounded by fixed-radius circles (cylinders of circles) and analyzes their 1D collapsed representations via Poincaré-Reeb graphs. It proves two main theorems: first, that any augmented balanced rooted tree can be realized as the Poincaré-Reeb digraph of an RA-region bounded by cylinders of a circle with a large radius; second, a two-tree variant extends this realization to a joined structure with reversed orientation. The approach combines explicit geometric constructions with stratified Morse-theoretic ideas and real algebraic maps to connect combinatorial graph structures with RA-regions. These results advance explicit realizations of Reeb-type graphs in real algebraic geometry and offer practical tools for visualization and topological analysis of algebraic regions.

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 -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.

Paper Structure

This paper contains 11 sections, 1 theorem, 3 figures.

Table of Contents

  1. Introduction.
  2. Preliminaries.
  3. Fundamental terminologies and notation.
  4. Topological spaces, and smooth manifolds and maps.
  5. Real algebraic objects.
  6. Graphs, digraphs, V-digraphs and Reeb (V-di )graphs.
  7. An RA-region and its Poincaré-Reeb V-digraph.
  8. Main Theorems and their proof.
  9. Some remark on Main Theorems.
  10. Some real algebraic maps.
  11. Some cases related to our main result.

Key Result

Proposition 1

A tree $G$ has the structure of a digraph by the following.

Figures (3)

  • Figure 3: The cylinders $S_1$ and $S_2$ of $C_R$.
  • Figure 4: The remaining cylinders $S_j$ ($3 \leq j \leq 2+{\Sigma}_{i=1}^{d_1} (n_{c,1,i}+1)$) of the circle $C_R$ defined for the balanced tree of type $(\{n_{c,1,i}\}_{i=1}^{d_1},d_1)$ with the orientation reversed.
  • Figure 5: The remaining cylinders $S_j$ ($j \geq 3+{\Sigma}_{i=1}^{d_2} (n_{c,1,i}+1)$) of the circle $C_R$ defined for the balanced tree of type $(\{n_{c,2,i}\}_{i=1}^{d_2},d_2)$.

Theorems & Definitions (3)

  • Proposition 1
  • proof
  • proof