  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.
