Planar Visibility: Testing and Counting

In this paper we consider query versions of visibility testing and visibility counting. Let $S$ be a set of $n$ disjoint line segments in $\R^2$ and let $s$ be an element of $S$. Visibility testing is to preprocess $S$ so that we can quickly determine if $s$ is visible from a query point $q$. Visibility counting involves preprocessing $S$ so that one can quickly estimate the number of segments in $S$ visible from a query point $q$. We present several data structures for the two query problems. The structures build upon a result by O'Rourke and Suri (1984) who showed that the subset, $V_S(s)$, of $\R^2$ that is weakly visible from a segment $s$ can be represented as the union of a set, $C_S(s)$, of $O(n^2)$ triangles, even though the complexity of $V_S(s)$ can be $\Omega(n^4)$. We define a variant of their covering, give efficient output-sensitive algorithms for computing it, and prove additional properties needed to obtain approximation bounds. Some of our bounds rely on a new combinatorial result that relates the number of segments of $S$ visible from a point $p$ to the number of triangles in $\bigcup_{s\in S} C_S(s)$ that contain $p$.Comment: 22 page

Segment Visibility Counting Queries in Polygons

Let $P$ be a simple polygon with $n$ vertices, and let $A$ be a set of $m$ points or line segments inside $P$. We develop data structures that can efficiently count the number of objects from $A$ that are visible to a query point or a query segment. Our main aim is to obtain fast, $O(\mathop{\textrm{polylog}} nm$), query times, while using as little space as possible. In case the query is a single point, a simple visibility-polygon-based solution achieves $O(\log nm)$ query time using $O(nm^2)$ space. In case $A$ also contains only points, we present a smaller, $O(n + m^{2 + \varepsilon}\log n)$-space, data structure based on a hierarchical decomposition of the polygon. Building on these results, we tackle the case where the query is a line segment and $A$ contains only points. The main complication here is that the segment may intersect multiple regions of the polygon decomposition, and that a point may see multiple such pieces. Despite these issues, we show how to achieve $O(\log n\log nm)$ query time using only $O(nm^{2 + \varepsilon} + n^2)$ space. Finally, we show that we can even handle the case where the objects in $A$ are segments with the same bounds.Comment: 27 pages, 13 figure

Minimizing Visible Edges in Polyhedra

We prove that, given a polyhedron $\mathcal P$ in $\mathbb{R}^3$, every point in $\mathbb R^3$ that does not see any vertex of $\mathcal P$ must see eight or more edges of $\mathcal P$, and this bound is tight. More generally, this remains true if $\mathcal P$ is any finite arrangement of internally disjoint polygons in $\mathbb{R}^3$. We also prove that every point in $\mathbb{R}^3$ can see six or more edges of $\mathcal{P}$ (possibly only the endpoints of some these edges) and every point in the interior of $\mathcal{P}$ can see a positive portion of at least six edges of $\mathcal{P}$. These bounds are also tight.Comment: 19 pages, 9 figure

LIPIcs, Volume 248, ISAAC 2022, Complete Volume

LIPIcs, Volume 248, ISAAC 2022, Complete Volum