Georgia Institute of Technology College of Sciences

Let T be a set of n triangles in 3-space, and let \Gamma be a family of
algebraic arcs of constant complexity in 3-space. We show how to preprocess T
into a data structure that supports various "intersection queries" for
query arcs \gamma \in \Gamma, such as detecting whether \gamma intersects any
triangle of T, reporting all such triangles, counting the number of
intersection points between \gamma and the triangles of T, or returning the
first triangle intersected by a directed arc \gamma, if any (i.e., answering
arc-shooting queries). Our technique is based on polynomial partitioning and
other tools from real algebraic geometry, among which is the cylindrical
algebraic decomposition.

Our approach can be extended to many other intersection-searching problems in
three and higher dimensions. We exemplify this versatility by giving an
efficient data structure for answering segment-intersection queries amid a set
of spherical caps in 3-space, and we lay a roadmap for extending our approach
to other intersection-searching problems.

Joint work with Pankaj Agarwal, Boris Aronov, Matya Katz, and Micha Sharir.