Intersection Queries for Flat Semi-Algebraic Objects in Three Dimensions and Related Problems

Abstract: Let $\mathcal{T}$ be a set of $n$ flat (planar) semi-algebraic regions in $\mathbb{R}<sup>3$ of constant complexity (e.g., triangles, disks), which we call plates. We wish to preprocess $\mathcal{T}$ into a data structure so that for a query object $\gamma$, which is also a plate, we can quickly answer various intersection queries, such as detecting whether $\gamma$ intersects any plate of $\mathcal{T}$, reporting all the plates intersected by $\gamma$, or counting them. We also consider two simpler cases of this general setting: (i) the input objects are plates and the query objects are constant-degree parametrized algebraic arcs in $\mathbb{R}<sup>3$ (arcs, for short), or (ii) the input objects are arcs and the query objects are plates in $\mathbb{R}<sup>3$. Besides being interesting in their own right, the data structures for these two special cases form the building blocks for handling the general case. By combining the polynomial-partitioning technique with additional tools from real algebraic geometry, we present many different data structures for intersection queries, which also provide trade-offs between their size and query time. For example, if $\mathcal{T}$ is a set of plates and the query objects are algebraic arcs, we obtain a data structure that uses $O<sup>*(n<sup>{4/3})$ storage (where the $O<sup>*(\cdot)$ notation hides factors of the form $n<sup>\epsilon$, for an arbitrarily small $\epsilon&gt;0$) and answers an arc-intersection query in $O<sup>*(n<sup>{2/3})$ time. This result is significant since the exponents do not depend on the specific shape of the input and query objects. We generalize and slightly improve this result: for a parameter $s\in [n^{4/3}, n<sup>{t_q}]$, where ${t_q}\ge 3$ is the number of real parameters needed to specify a query arc, the query time can be decreased to $$O<sup>*((n/s<sup>{1/{t_q}})<sup>{\tfrac{2/3}{1-1/{t_q}}})$$ by increasing the storage to $O<sup>*(s)$.