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Computing $k$-Crossing Visibility through $k$-levels

Frank Duque

TL;DR

This paper addresses computing $k$-crossing visibility for sets of geometric objects, notably lines in the plane and planes in $\mathbb{R}^3$. It introduces a linear-time reduction to $(\le k)$-level problems via a self-inverse transformation $\mathcal{T}$ that maps lines/planes to lines/planes and preserves incidences, enabling computation of $\V_k(p,\mathcal{A})$ through $(\le k)$-level queries. The main results are $O(n\log n + kn)$ time for planar line sets and $O(n\log n + k^2n)$ time for 3D plane arrangements, with corresponding arrangement complexities of $\Theta(kn)$ and $\Theta(k^2n)$, respectively; these bounds are optimal for certain regimes of $k$. The approach also yields a linear-time solution for polygon visibility and can be adapted to other settings where $(\le k)$-levels are known, illustrating broad applicability of level-based reductions in visibility problems.

Abstract

Let $\mathcal{A}$ be a set of straight lines in the plane (or planes in $\mathbb{R}^3$). The $k$-crossing visibility of a point $p$ on $\mathcal{A}$ is the set $Q$ of points in the elements of $\mathcal{A}$ such that the segment $pq$, where $q\in Q$, intersects at most $k$ elements of $\mathcal{A}$. In this paper, we present algorithms for computing the $k$-crossing visibility. Specifically, we provide $O(n\log n + kn)$ and $O(n\log n + k^2n)$ time algorithms for sets of $n$ lines in the plane and arrangements of $n$ planes in $\mathbb{R}^3$, which are optimal for $k=Ω(\log n)$ and $k=Ω(\sqrt{\log n})$, respectively. We also introduce an algorithm for computing $k$-crossing visibilities on polygons, which achieves the same asymptotic time complexity as the one presented by Bahoo et al. The techniques proposed in this paper can be easily adapted for computing $k$-crossing visibilities on other instances where the $(\leq k)$-level is known.

Computing $k$-Crossing Visibility through $k$-levels

TL;DR

This paper addresses computing -crossing visibility for sets of geometric objects, notably lines in the plane and planes in . It introduces a linear-time reduction to -level problems via a self-inverse transformation that maps lines/planes to lines/planes and preserves incidences, enabling computation of through -level queries. The main results are time for planar line sets and time for 3D plane arrangements, with corresponding arrangement complexities of and , respectively; these bounds are optimal for certain regimes of . The approach also yields a linear-time solution for polygon visibility and can be adapted to other settings where -levels are known, illustrating broad applicability of level-based reductions in visibility problems.

Abstract

Let be a set of straight lines in the plane (or planes in ). The -crossing visibility of a point on is the set of points in the elements of such that the segment , where , intersects at most elements of . In this paper, we present algorithms for computing the -crossing visibility. Specifically, we provide and time algorithms for sets of lines in the plane and arrangements of planes in , which are optimal for and , respectively. We also introduce an algorithm for computing -crossing visibilities on polygons, which achieves the same asymptotic time complexity as the one presented by Bahoo et al. The techniques proposed in this paper can be easily adapted for computing -crossing visibilities on other instances where the -level is known.
Paper Structure (5 sections, 24 theorems, 9 equations, 3 figures)

This paper contains 5 sections, 24 theorems, 9 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Results in $\mathbb{R}^2$
  3. Properties of $\mathcal{T}$
  4. Proofs of results in $\mathbb{R}^2$
  5. Results in $\mathbb{R}^3$

Key Result

Theorem 1.1

Given a simple polygon $P$ with $n$ vertices and a query point $p$ in $P$, the region of $P$ that is $k$-crossing visible from $p$, can be computed in $O(kn)$ time.

Figures (3)

  • Figure 1: Illustration of the $2$-crossing visibility region (yellow) and the $2$-crossing visibility (blue) of the red point on a set of lines.
  • Figure 2: Illustration of a polygon partition in proof of Theorem \ref{['thm:polygon_visibility']}.
  • Figure 3: Illustration for proof of Theorem \ref{['thm:polygon_visibility']}.

Theorems & Definitions (41)

  • Theorem 1.1: Bahoo et al. Computing_k_Visibility_Polygon
  • Theorem 1.2
  • Proposition 1.3
  • Theorem 1.4
  • Proposition 1.5
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • ...and 31 more