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 $\mathbb{R}^3$ 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 $\mathbb{R}^3$ (arcs, for short), or (ii) the input objects are arcs and the query objects are plates in $\mathbb{R}^3$. 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 $O^*(n^{4/3})$ storage (where the $O^*(\cdot)$ notation hides factors of the form $n^\epsilon$, for an arbitrarily small $\epsilon>0$) and answers an arc-intersection query in $O^*(n^{2/3})$ 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 $s\in [n^{4/3}, n^{t_q}]$, where ${t_q}\ge 3$ is the number of real parameters needed to specify a query arc, the query time can be decreased to $O^*((n/s^{1/{t_q}})^{\tfrac{2/3}{1-1/{t_q}}})$ by increasing the storage to $O^*(s)$.