Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2403.04494

Trigonometry of partially truncated triangles and tetrahedra

Jason DeBlois
View original source ↗Read source document ↗Export metadata ↓

Abstract

The first main results of this note establish forms of the hyperbolic laws of cosines and sines for certain classes of quadrilaterals and pentagons in the hyperbolic plane, having at least one ideal vertex and right angles at non-ideal vertices, in which the length of a horocyclic cross-section at an ideal vertex plays the role filled by the dihedral angle in the usual versions of these laws. The second set of main results concern transversal length, meaning the distance from a designated internal edge to its opposite, of partially truncated tetrahedra in three-dimensional hyperbolic space whose non-truncated vertices are ideal. Transversal lengths of such tetrahedra are proved to depend only on the entire collection of internal edge lengths (interpreted at ideal vertices in terms of horospherical cross-sections), and bounds on these lengths are established. The case of ideal tetrahedra (no truncated vertices) is also considered. All main results are established using the unifying perspective of the hyperboloid model and Lorentzian geometry. A thorough introduction to this perspective is provided, with references as appropriate.

Explore related subjects

math.GTmath.MG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Jason DeBlois. 2025-08-06. Trigonometry of partially truncated triangles and tetrahedra. https://arxiv.org/abs/2403.04494

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Word Length Formulae, Normal Forms, Conjugation and Root-finding Algorithms in Surface Groups

In this paper, we mainly study the following symmetric presentation of the surface group $$π_1(Σ_g)=\left\langle c_1,\dots, c_{2g}\mid c_1\cdots c_{2g}c_1^{-1}\cdots c_{2g}^{-1}\right\rangle.$$ For every nontrivial element $x\in π_1(Σ_g)$ and $k\geq 2$, we obtain a uniform representative of the normal forms $\mathfrak{nf}(x^k)$ of $x^k$ under the length-lexicographical order: $$\mathfrak{nf}(x^k) = \overline{LW^{k-2}R}.$$ Building on this result, we establish a new relation among these normal forms, and then derive the following three formulae related to the word length: $|x^2|>|x|$; $|x^k|=(k-1)(|x^2|-|x|)+|x|$; $\lim_{k\to\infty}\frac{|x^k|}{k}=|x^2|-|x|$. Furthermore, we extend these results to obtain a coarser analogue for every minimal geometric presentation. We then define normal forms of conjugacy classes in $π_1(Σ_g)$ and provide a criterion for determining the conjugacy of group elements. As a consequence, we provide efficient algorithms for solving the root-finding and conjugacy problems. Finally, we present applications to the computation of several growth rates.

math.GT↗
arXiv2026

Plane separating continua inscribe rectangles

We prove the following: If $X$ is a plane separating continuum, then every embedding of $X$ into $\mathbb{R}^2$ contains the vertices of a Euclidean rectangle. We arrive to this result by extending a known result by H. Vaughan for Jordan curves to a wider class of topological objects via shape theory and Steenrod homology.

math.GT↗
arXiv2026

Every Link Has Infinitely Many Explicit Generalised T-Link Presentations

Generalised $T$-links provide a simple description of all links in $S^3$ as closures of products of standard twisting blocks, parametrised by finite sequences of integers. We prove that every link admits infinitely many pairwise distinct generalised $T$-link presentations. Starting from any such presentation, we give explicit parameter transformations that preserve the represented link and generate families of pairwise distinct presentations depending on arbitrarily many independent integer parameters.

math.GT↗