The EdgeConflict Predicate in the 3D Apollonius Diagram

Abstract

In this paper 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 three sub-predicates, which are then evaluated with the aid of four 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. Among the tools we use is the 3D inversion transformation. In the scope of this paper and due to space limitations, only non-degenerate configurations are considered, i.e. different Voronoi vertices are distinct and the predicates never return a degenerate answer. Most of our analysis is carried out in the inverted space, which is where our geometric observations and analysis is captured in algebraic terms.