In this thesis we study one of the fundamental predicates required for the construction of the 3D Apollonius diagram (also known as the 3D Additively Weighted Voronoi diagram), namely the EDGECONFLICT predicate: given five sites $S_i, S_j,S_k,S_l,S_m$ that define an edge $e_{ijklm}$ in the 3D Apollonius diagram, and a sixth query site $S_q$, the predicate determines the portion of $e_{ijklm}$ that will disappear in the Apollonius diagram of the six sites due to the insertion of $S_q$. Our focus is on the algorithmic analysis of the predicate with the aim to minimize its algebraic degree. We decompose the main predicate into sub-predicates, which are then evaluated with the aid of additional primitive operations. We show that the maximum algebraic degree required to answer any of the sub-predicates and primitives, and, thus, our main predicate is 10 in non-degenerate configurations when the trisector is of Hausdorff dimension 1. We also prove that all subpredicates developed can be evaluated using 10 or 8-degree demanding operations for degenerate input for these trisector types, depending on whether they require the evaluation of an intermediate INSPHERE predicate or not. Among the tools we use is the 3D inversion transformation and the so-called qualitative symbolic perturbation scheme. Most of our analysis is carried out in the inverted space, which is where our geometric observations and analysis is captured in algebraic terms.

中文翻译：

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3D Apollonius 图的谓词

在本论文中，我们研究了构建 3D Apollonius 图（也称为 3D Additively Weighted Voronoi 图）所需的基本谓词之一，即 EDGECONFLICT 谓词：给定五个站点 $S_i, S_j,S_k,S_l,S_m$在 3D Apollonius 图中定义一条边 $e_{ijklm}$ 和第六个查询站点 $S_q$，谓词确定 $e_{ijklm}$ 将在六个站点的 Apollonius 图中消失的部分，因为$S_q$ 的插入。我们的重点是谓词的算法分析，旨在最小化其代数度。我们将主谓词分解为子谓词，然后在额外的原始操作的帮助下对其进行评估。我们证明了回答任何子谓词和原语所需的最大代数次数，因此，我们的主谓词在非退化配置中是 10，当三分线是 Hausdorff 维 1 时。我们还证明了所有开发的子谓词都可以使用 10 或 8 度要求操作对这些三分线类型的退化输入进行评估，这取决于它们是否需要中间 INSPHERE 谓词的评估与否。我们使用的工具包括 3D 反演和所谓的定性符号扰动方案。我们的大部分分析都是在倒置空间中进行的，这是我们的几何观察和分析以代数术语捕获的地方。取决于它们是否需要评估中间 INSPHERE 谓词。我们使用的工具包括 3D 反演和所谓的定性符号扰动方案。我们的大部分分析都是在倒置空间中进行的，这是我们的几何观察和分析以代数术语捕获的地方。取决于它们是否需要评估中间 INSPHERE 谓词。我们使用的工具包括 3D 反演变换和所谓的定性符号扰动方案。我们的大部分分析都是在倒置空间中进行的，这是我们的几何观察和分析以代数术语捕获的地方。