部分截断三角形与四面体的三角学
Trigonometry of partially truncated triangles and tetrahedra
AI总结：
本文利用双曲面模型和洛伦兹几何，建立了具有理想顶点的四边形和五边形的双曲余弦与正弦定理，并证明了部分截断四面体的横向长度仅取决于内部边长集合，且给出了界限。
摘要：
本文的第一个主要结果建立了双曲平面中某些四边形和五边形的双曲余弦定理和正弦定理的形式，这些四边形和五边形至少有一个理想顶点，且在非理想顶点处为直角，其中理想顶点处的水平圆截面长度所起的作用，相当于通常版本定理中二面角所起的作用。第二组主要结果涉及三维双曲空间中部分截断四面体的横向长度，即从指定的内部边到其对边的距离，这些四面体的非截断顶点为理想顶点。本文证明了此类四面体的横向长度仅取决于整个内部边长的集合（在理想顶点处通过水平球截面来解释），并建立了这些长度的界限。还考虑了理想四面体（无截断顶点）的情形。所有主要结果均通过双曲面模型和洛伦兹几何的统一视角建立。本文对该视角进行了详尽的介绍，并酌情提供了参考文献。
英文摘要：
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.
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