AI 中文总结
该研究证明了直纹二次曲面上同一母线族直线的最近与最远点Voronoi图最坏情况复杂度为$\boldsymbol{\text{Θ}}(n^2)$，并给出了$\boldsymbol{\text{O}}(n^2)$时间的顶点枚举算法。
AI 中文摘要
我们研究欧氏度量下三维欧几里得空间（$\boldsymbol{\text{R}}^3$）中直线的最近点与最远点Voronoi图，其中所有$n$条直线均属于光滑双直纹实二次曲面的同一母线族。对于任意直线站点，已知最近点Voronoi图的组合复杂度介于$\boldsymbol{\text{Ω}}(n^2)$与$\boldsymbol{\text{O}}(n^{3+\boldsymbol{\text{ε}}})$之间。在一般位置假设下，我们证明母线族中的两种图均至多有$4n(n-3)$个顶点，总组合复杂度为$\boldsymbol{\text{O}}(n^2)$；反之，对每个$n\boldsymbol{\text{≥}}4$，固定非旋转单叶双曲面的一条母线族包含一组处于一般位置的$n$条直线，其具有至少$(n-2)(n-3)/2$个不同的正则最近顶点，其中正则指恰好四条直线支撑该顶点，且它们的三个定义平分线横截相交。因此，该类中最近点Voronoi图的最坏情况复杂度为$\boldsymbol{\text{Θ}}(n^2)$，而最远点图对每个一般位置输入均具有$\boldsymbol{\text{Θ}}(n^2)$复杂度，因其恰好有$n(n-1)$个三维单元。在普吕克嵌入下，母线为一条二次曲线，直线与欧氏球相切的条件退化为二元四次方程。在正则顶点处，四个支撑参数穷尽其根，符号交替迫使参数圆的两段弧无站点。这仅留下$n(n-3)/2$种可能的循环支撑类型，而贝祖定理将每种类型的中心数限制为8。同样的归约可得到一个精确的$\boldsymbol{\text{O}}(n^2)$时间算法，该算法在对站点参数进行循环排序后，将所有有限最近点与最远点顶点枚举为常次数实单变量表示。
英文摘要
We study nearest and farthest Voronoi diagrams of lines in $\mathbb{R}^3$ under the Euclidean metric when all $n$ lines belong to one ruling of a smooth doubly ruled real quadric. For arbitrary line sites, the combinatorial complexity of the nearest Voronoi diagram is known only to lie between $Ω(n^2)$ and $O(n^{3+\varepsilon})$. Under general-position assumptions, we prove that both diagrams in the ruling class have at most $4n(n-3)$ vertices and $O(n^2)$ total combinatorial complexity. Conversely, for every $n \ge 4$, one ruling of a fixed non-rotational one-sheeted hyperboloid contains a general-position set of $n$ lines with at least $(n-2)(n-3)/2$ distinct regular nearest vertices, where regular means that exactly four lines support the vertex and their three defining bisectors meet transversely. Thus the worst-case complexity of the nearest Voronoi diagram in this class is $Θ(n^2)$, while the farthest diagram has $Θ(n^2)$ complexity for every general-position input, since it has exactly $n(n-1)$ three-dimensional cells. Under the Plücker embedding, the ruling is a conic, and the condition for a line to be tangent to a Euclidean sphere restricts to a binary quartic. At a regular vertex, the four supporting parameters exhaust its roots, and sign alternation forces two arcs of the parameter circle to be site-free. This leaves only $n(n-3)/2$ possible cyclic support types, while Bézout's theorem bounds the number of centers for each type by eight. The same reduction yields an exact $O(n^2)$-time algorithm that, after cyclically sorting the site parameters, enumerates all finite nearest and farthest vertices as constant-degree real univariate representations.
