发表机构
University of Las Palmas de Gran Canaria (ULPGC); IUMA. University of Las Palmas de Gran Canaria (ULPGC); Facultad de Informatica y Matematica, Universidad de Holguin（拉斯帕尔马斯大学; 加那利群岛数学研究所; 奥尔金大学信息数学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种基于六元组空间的四面体多重最长边二分法，通过二分模式编码多值细化行为，并证明R1+族与Liu-Joe族的细化树坍缩为同一八状态有向图，为深度迭代分析提供统一组合框架。
AI 中文摘要
我们提出了一种全新的四面体最长边二分法（LEB）表述，该方法完全在六元组空间R6中进行，其中四面体由其边长的平方表示。这种表示使得LEB细化方程完全线性化，并消除了对基于坐标的数据结构的需求。三维LEB中的一个核心难点出现在四面体具有多条最长边时，这使得细化规则本质上具有多值性。我们通过多重最长边二分法（MLEB）的概念将这一现象形式化，该方法系统地探索所有允许的最长边选择。为了编码这种多值行为，我们引入了二分模式的概念，将其定义为控制细化过程的六元组置换序列。我们证明了共享同一LEB模式的六元组集合在R6中构成一个凸区域。对于结构上重要的四面体族，包括R1+族和Liu-Joe族，我们证明了无限细化树会坍缩为一个具有八个状态的有限有向图。值得注意的是，这两个族由同一个图控制，仅在初始状态上有所不同。这种有向图表述为细化过程提供了统一的组合描述，并为深度迭代LEB分析提供了高效的计算框架。
英文摘要
We introduce a new formulation of the Longest Edge Bisection (LEB) of tetrahedra entirely in sextuple space R6, where tetrahedra are represented by the squares of their edge lengths. This representation renders the LEB refinement equations fully linear and eliminates the need for coordinate-based data structures. A central difficulty in three-dimensional LEB arises when a tetrahedron possesses multiple longest edges, making the refinement rule intrinsically multivalued. We formalize this phenomenon through the notion of Multiform Longest Edge Bisection (MLEB), which systematically explores all admissible longest-edge choices. To encode this multivalued behavior, we introduce the concept of bisection patterns, defined as sequences of sextuple permutations governing the refinement process. We prove that the set of sextuples sharing a common LEB pattern forms a convex region in R6. For structurally significant families of tetrahedra, including the R1+ family and the Liu-Joe family, we show that the infinite refinement tree collapses into a finite directed graph with eight states. Remarkably, both families are governed by the same graph, differing only in their initial state. This directed-graph formulation provides a unified combinatorial description of the refinement process and offers an efficient computational framework for deep iterative LEB analysis.
