The Voronoi Diagram of Three Lines

Tuesday, May 29, 2007 - 3:10pm - 3:40pm
EE/CS 3-180
Sylvain Lazard (Institut National de Recherche en Informatique Automatique (INRIA)-Lorraine)
We give a complete description of the Voronoi diagram of three lines in
three-dimensional real space. In particular, we show that the topology of the Voronoi diagram is
invariant for three lines in general position, that is, that are pairwise skew
and not all parallel to a common plane. The trisector consists of four
unbounded branches of either a non-singular quartic or of a cubic and line
that do not intersect in real space. Each cell of dimension two consists of
two connected components on a hyperbolic paraboloid that are bounded,
respectively, by three and one of the branches of the trisector. The proof
technique, which relies heavily upon modern tools of computer algebra, is of
interest in its own right.

This characterization yields some fundamental properties of the Voronoi
diagram of three lines. In particular, we present linear semi-algebraic tests
for separating the two connected components of each two-dimensional Voronoi
cell and for separating the four connected components of the trisector. This
enables us to answer queries of the form, given a point, determine in which
connected component of which cell it lies. We also show that the arcs of the
trisector are monotonic in some direction. These properties imply that points
on the trisector of three lines can be sorted along each branch using only
linear semi-algebraic tests.