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On face angles of tetrahedra with a given base

E. V. Nikitenko, Yu. G. Nikonorov

TL;DR

This work analyzes the set of triples $ (\cos \overline{\alpha},\cos \overline{\beta},\cos \overline{\gamma})$ formed by the face angles $\overline{\alpha}=\angle BDC$, $\overline{\beta}=\angle ADC$, $\overline{\gamma}=\angle ADB$ of tetrahedra with a fixed base $ABC$. By embedding the problem into distance coordinates and a Cartesian map $F$, the authors show the image lies in the pillow $\mathbb{P}$ and describe the closure $\operatorname{CFT}$ of $F(\mathrm{GT})$, its boundary $\operatorname{BFT}$, and the limit ellipses $\operatorname{Lim}(A),\operatorname{Lim}(B),\operatorname{Lim}(C)$. They analyze the role of the right cylinder over the circumscribed circle, toroids, and the special regions on the cylinder in determining the boundary geometry, proving that $\operatorname{CFT}$ is homeomorphic to a 3-ball with a spherical boundary, and giving an explicit decomposition of $\Sigma(\triangle ABC)=F(\Pi^+)$ into the interior plus the images of special cylinder regions. The results yield a complete description of the boundary for any fixed base and connect the geometry of face angles to the ambient round-cylinder and toroidal structures, with potential implications for geometric modeling and perspective problems.

Abstract

Let us consider the set $Ω(\triangle ABC)$ of all tetrahedra $ABCD$ with a given non-degenerate base $ABC$ in $\mathbb{E}^3$ and $D$ lying outside the plane $ABC$. Let us denote by $Σ(\triangle ABC)$ the set $\left\{\Bigl(\cos \overlineα,\cos \overlineβ,\cos \overlineγ \Bigr)\in \mathbb{R}^3\,|\, ABCD \in Ω(\triangle ABC)\right\}$, where $\overlineα=\angle BDC$, $\overlineβ=\angle ADC$, and $\overlineγ=\angle ADB$. The paper is devoted to the problem of determining of the closure of $Σ(\triangle ABC)$ in $\mathbb{R}^3$ and its boundary.

On face angles of tetrahedra with a given base

TL;DR

This work analyzes the set of triples formed by the face angles , , of tetrahedra with a fixed base . By embedding the problem into distance coordinates and a Cartesian map , the authors show the image lies in the pillow and describe the closure of , its boundary , and the limit ellipses . They analyze the role of the right cylinder over the circumscribed circle, toroids, and the special regions on the cylinder in determining the boundary geometry, proving that is homeomorphic to a 3-ball with a spherical boundary, and giving an explicit decomposition of into the interior plus the images of special cylinder regions. The results yield a complete description of the boundary for any fixed base and connect the geometry of face angles to the ambient round-cylinder and toroidal structures, with potential implications for geometric modeling and perspective problems.

Abstract

Let us consider the set of all tetrahedra with a given non-degenerate base in and lying outside the plane . Let us denote by the set , where , , and . The paper is devoted to the problem of determining of the closure of in and its boundary.

Paper Structure

This paper contains 12 sections, 35 theorems, 125 equations, 16 figures, 1 table.

Table of Contents

  1. Introduction
  2. Using distance coordinates
  3. Cartesian coordinates and some general observations
  4. Degenerate points of $F$
  5. Properties of the map $F$ on the plane $ABC$
  6. On the limit points of the image $F(\mathop{\mathrm{GT}}\nolimits)$
  7. On the images of the points of $\mathop{\mathrm{Cyl}}\nolimits$
  8. Mappings of special parts of the cylinder $\mathop{\mathrm{Cyl}}\nolimits$
  9. On special regions on the cylinder $\mathop{\mathrm{Cyl}}\nolimits$
  10. On regions bounded by special toroids
  11. On the region inside $\mathbb{BP}$
  12. The main construction

Key Result

Proposition 1

For any non-degenerate tetrahedron $ABCD$ with an acute-angled base $ABC$, we have the following statements (in the above notations): as well as all statements obtained from these three ones by simultaneous cyclic permutations $\overline{\alpha} \to \overline{\beta} \to \overline{\gamma} \to \overline{\alpha}$ and $\angle BAC \to \angle ABC \to \angle ACB \to \angle BAC$.

Figures (16)

  • Figure 1: Tetrahedron $ABCD$.
  • Figure 2: The closure of the set $\Sigma(\triangle ABC)$ (in purple) and the closure of its complement $CS$ (in turquoise) in $\mathbb{P}$ (in orange) for the case, when the base $ABC$ is a regular triangle, for different points of view (angles): a) the closure of $\Sigma(\triangle ABC)$, b) $\mathbb{P}$ and the closure of $\Sigma(\triangle ABC)$, c) and d) $\mathbb{P}$ and the closure of $CS$. Red, green, and magenta curves show non-smoothness points of the boundary of $\Sigma(\triangle ABC)$ as well as of the boundary of the closure of $CS$.
  • Figure 3: A tetrahedron and Menelaus's theorem.
  • Figure 4: The description of suitable tetrahedra using distance coordinates.
  • Figure 5: Important subsets of the plane $ABC$.
  • ...and 11 more figures

Theorems & Definitions (55)

  • Proposition 1: Rieck
  • Proposition 2
  • Theorem 1
  • Lemma 1
  • Remark 1
  • Remark 2
  • Lemma 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • ...and 45 more