# Arc-Intersection Queries Amid Triangles in Three Dimensions and Related Problems

### Wednesday, January 5th, 4:10pm Tel Aviv time (3:10pm CET, 9:10am NY time)

### Esther Ezra, Bar Ilan University

### Abstract:

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 \emph{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 first result is an O^*(n^{4/3})-size data structure that answers a query in

O^*(n^{2/3}) time. Next, we devise an O^*(n)-size data structure that answers a

query in O^*(n^{4/5}) time. Incorporating this structure at the leaf nodes of

the main structure reduces its overall size to O^*(n^{11/9}), without affecting

its query time which remains O^*(n^{2/3}). We also present a data structure

that provides a trade-off between the query time and the size of the data

structure. For example, if \Gamma is a family of algebraic arcs defined by t > 1

real parameters, increasing the storage to O^*(n^{3/2}) decreases the query

time to O^*(n^{\rho}), where \rho=\max {1/2, (2t-3)/3(t-1)} < 2/3. We also show

that this query time can be further improved for circular query arcs.

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.