arXiv subjects

Jack Snoeyink

Publications and source records attributed to Jack Snoeyink.

10 recordsLinked to original sources

arXiv2026

NP-Hardness of Non-Crossing Hamiltonian Path and Cycle in Non-Planar Graphs

We seek to disentangle the hardness of finding a Hamiltonian path or cycle from the hardness of finding a non-crossing path or cycle by giving a direct reduction from 3-SAT to the non-crossing Hamiltonian path and cycle problems on non-planar graphs. Prior hardness proofs proceed by reduction to planar graphs, where every path is automatically non-crossing; this conflates the two sources of difficulty and leaves unclear why forbidding crossings on the path alone makes the problem hard. Our reduction places the difficulty squarely in the non-crossing constraint, avoids planar gadget constructions, and yields a more transparent proof that may be easier to extend to related problems.

cs.CG↗

arXiv2019

Computation of Spatial Skyline Points

We discuss a method of finding skyline or non-dominated sites in a set $P$ of $n$ point sites with respect to a set $S$ of $m$ points. A site $p \in P$ is non-dominated if and only if for each $q \in P \setminus \{p\}$, there exists at least one point $s \in S$ that is closer to $p$ than to $q$. We reduce this problem of determining non-dominated sites to the problem of finding sites that have non-empty cells in an additively weighted Voronoi diagram under a convex distance function. The weights of said Voronoi diagram are derived from the coordinates of the sites of $P$, while the convex distance function is derived from $S$. In the two-dimensional plane, this reduction gives an $O((n + m) \log (n + m))$-time algorithm to find the non-dominated points.

cs.CG↗

arXiv2016

On the Precision to Sort Line-Quadric Intersections

To support exactly tracking a neutron moving along a given line segment through a CAD model with quadric surfaces, this paper considers the arithmetic precision required to compute the order of intersection points of two quadrics along the line segment. When the orders of all but one pair of intersections are known, we show that a resultant can resolve the order of the remaining pair using only half the precision that may be required to eliminate radicals by repeated squaring. We compare the time and accuracy of our technique with converting to extended precision to calculate roots.

cs.CG↗

arXiv2016

Visibility Graphs, Dismantlability, and the Cops and Robbers Game

We study versions of cop and robber pursuit-evasion games on the visibility graphs of polygons, and inside polygons with straight and curved sides. Each player has full information about the other player's location, players take turns, and the robber is captured when the cop arrives at the same point as the robber. In visibility graphs we show the cop can always win because visibility graphs are dismantlable, which is interesting as one of the few results relating visibility graphs to other known graph classes. We extend this to show that the cop wins games in which players move along straight line segments inside any polygon and, more generally, inside any simply connected planar region with a reasonable boundary. Essentially, our problem is a type of pursuit-evasion using the link metric rather than the Euclidean metric, and our result provides an interesting class of infinite cop-win graphs.

cs.CG↗

arXiv2014

Covering Folded Shapes

Can folding a piece of paper flat make it larger? We explore whether a shape $S$ must be scaled to cover a flat-folded copy of itself. We consider both single folds and arbitrary folds (continuous piecewise isometries $S\rightarrow R^2$). The underlying problem is motivated by computational origami, and is related to other covering and fixturing problems, such as Lebesgue's universal cover problem and force closure grasps. In addition to considering special shapes (squares, equilateral triangles, polygons and disks), we give upper and lower bounds on scale factors for single folds of convex objects and arbitrary folds of simply connected objects.

cs.CG↗

arXiv2010

The dilation of the Delaunay triangulation is greater than π/2

Consider the Delaunay triangulation T of a set P of points in the plane as a Euclidean graph, in which the weight of every edge is its length. It has long been conjectured that the dilation in T of any pair p, p \in P, which is the ratio of the length of the shortest path from p to p' in T over the Euclidean distance ||pp'||, can be at most π/2 \approx 1.5708. In this paper, we show how to construct point sets in convex position with dilation > 1.5810 and in general position with dilation > 1.5846. Furthermore, we show that a sufficiently large set of points drawn independently from any distribution will in the limit approach the worst-case dilation for that distribution.

cs.CG↗

arXiv2004

A 2-chain can interlock with a k-chain

One of the open problems posed in [3] is: what is the minimal number k such that an open, flexible k-chain can interlock with a flexible 2-chain? In this paper, we establish the assumption behind this problem, that there is indeed some k that achieves interlocking. We prove that a flexible 2-chain can interlock with a flexible, open 16-chain.

cs.CG↗

arXiv2002

The size of spanning disks for polygonal curves

Let $K$ be a closed polygonal curve in $\RR^3$ consisting of $n$ line segments. Assume that $K$ is unknotted, so that it is the boundary of an embedded disk in $\RR^3$. This paper considers the question: How many triangles are needed to triangulate a Piecewise-Linear (PL) spanning disk of $K$? The main result exhibits a family of unknotted polygons with $n$ edges, $n \to \infty$, such that the minimal number of triangles needed in any triangulated spanning disk grows exponentially with $n$. For each integer $n \ge 0$, there is a closed, unknotted, polygonal curve $K_n$ in $R^3$ having less than $10n+9$ edges, with the property that any Piecewise-Linear triangulated disk spanning the curve contains at least $2^{n-1}$ triangles.

math.GT↗

arXiv2001

Ununfoldable Polyhedra with Convex Faces

Unfolding a convex polyhedron into a simple planar polygon is a well-studied problem. In this paper, we study the limits of unfoldability by studying nonconvex polyhedra with the same combinatorial structure as convex polyhedra. In particular, we give two examples of polyhedra, one with 24 convex faces and one with 36 triangular faces, that cannot be unfolded by cutting along edges. We further show that such a polyhedron can indeed be unfolded if cuts are allowed to cross faces. Finally, we prove that ``open'' polyhedra with triangular faces may not be unfoldable no matter how they are cut.

cs.CG↗

arXiv1999

Emerging Challenges in Computational Topology

Here we present the results of the NSF-funded Workshop on Computational Topology, which met on June 11 and 12 in Miami Beach, Florida. This report identifies important problems involving both computation and topology.

cs.CG↗