We consider the Voronoi diagram of lines in $\mathbb{R}^3$ under the Euclidean metric, and give a full classification of its structure in the base case of four lines in general position. We first show that the number of vertices in the Voronoi diagram of four lines in general position is always even, between 0 and 8, and all such numbers can be realized. We identify a key structure for the diagram formation, called a \emph{twist}, which is a pair of consecutive intersections among trisector branches; only two types of twists are possible, so-called \emph{full} and \emph{partial} twists. A full twist is a purely local structure, which can be inserted or removed without affecting the rest of the diagram. Assuming no full twists, the nearest and the farthest Voronoi diagrams of four lines, each have 15 distinct topologies, which are in one-to-one correspondence; the two-dimensional faces are all unbounded, and the total number of vertices is at most six. The unbounded features of the farthest diagram, encoded in a two-dimensional spherical map, are also in one-to-one correspondence. The identified topologies are all realizable. Any Voronoi diagram of four lines in general position in $\mathbb{R}^3$ can be obtained from one of these topologies by inserting full twists; each twist induces a bounded face of exactly two vertices in both the nearest and farthest diagrams. We obtain the classification by an exhaustive search algorithm using some new structural and combinatorial observations of line Voronoi diagrams.

翻译：我们考虑欧几里得度量下$\mathbb{R}^3$中直线的维诺图，并对一般位置下四条直线的基例情况给出了完整的结构分类。首先证明，一般位置下四条直线的维诺图顶点数目始终为偶数，取值范围介于0到8之间，且所有取值均可实现。我们识别出图形成的一个关键结构，称为\emph{扭转}，即三叉分支间连续成对出现的交点；仅存在两种可能的扭转类型，即所谓的\emph{完整}扭转与\emph{部分}扭转。完整扭转是纯局部结构，可在不影响图其余部分的情况下插入或移除。假设不存在完整扭转时，四直线的最邻近维诺图与最远维诺图各有15种不同的拓扑结构，且两者一一对应；所有二维面均为无界，顶点总数至多为六个。最远图的无界特征编码于二维球面映射中，同样具有一一对应关系。所识别的拓扑结构均可实现。一般位置下$\mathbb{R}^3$中任意四条直线的维诺图均可通过插入完整扭转得到上述拓扑之一；每个扭转均在最邻近图与最远图中诱导出一个恰好包含两个顶点的有界面。我们利用直线维诺图的新结构组合观测结果，通过穷举搜索算法完成了该分类。