Computational Geometry
Indexing data is a fundamental problem in computer science. Recently, various papers apply machine learning to this problem. For a fixed integer $\varepsilon$, a \emph{learned index} is a function $h : \mathcal{U} \rightarrow [0, n]$ where…
Efficiently solving the Shortest Vector Problem (SVP) in two-dimensional lattices holds practical significance in cryptography and computational geometry. While simpler than its high-dimensional counterpart, two-dimensional SVP motivates…
Given a set $P$ of $n$ points in the plane, the unit-disk graph $G(P)$ is a graph with $P$ as its vertex set such that two points of $P$ have an edge if their Euclidean distance is at most $1$. We consider the problem of computing a maximum…
For a planar point set $P$, its convex hull is the smallest convex polygon that encloses all points in $P$. The construction of the convex hull from an array $I_P$ containing $P$ is a fundamental problem in computational geometry. By…
We consider the problem of reconfiguring a two-dimensional connected grid arrangement of passive building blocks from a start configuration to a goal configuration, using a single active robot that can move on the tiles, remove individual…
In this paper, we study the Contiguous Art Gallery Problem, introduced by Thomas C. Shermer at the 2024 Canadian Conference on Computational Geometry, a variant of the classical art gallery problem from 1973 by Victor Klee. In the…
We investigate the Dispersive Art Gallery Problem with vertex guards and rectangular visibility ($r$-visibility) for a class of orthogonal polygons that reflect the properties of real-world floor plans: these office-like polygons consist of…
Extracting high-fidelity mesh surfaces from Signed Distance Fields has become a fundamental operation in geometry processing. Despite significant progress over the past decades, key challenges remain namely, how to automatically capture the…
We present NeCGS, the first neural compression paradigm, which can compress a geometry set encompassing thousands of detailed and diverse 3D mesh models by up to 900 times with high accuracy and preservation of detailed geometric…
We address the problem of covering a target segment $\overline{uv}$ using a finite set of guards $\mathcal{S}$ placed on a source segment $\overline{xy}$ within a simple polygon $\mathcal{P}$, assuming weak visibility between the target and…
A geometric graph is a drawing of a graph in the plane where the vertices are drawn as points in general position and the edges as straight-line segments connecting their endpoints. It is plane if it contains no crossing edges. We study…
The Euler Characteristic Transform (ECT) is a robust method for shape classification. It takes an embedded shape and, for each direction, computes a piecewise constant function representing the Euler Characteristic of the shape's sublevel…
On a convex polyhedron P, the cut locus C(x) with respect to a point x is a tree of geodesic segments (shortest paths) on P that includes every vertex. We say that P has a skeletal cut locus if there is some x in P such that C(x) subset…
Word cloud use is a popular text visualization technique that scales font sizes based on word frequencies within a defined spatial layout. However, traditional word clouds disregard semantic relationships between words, arranging them…
In this paper, we study the problem of map matching with travel time constraints. Given a sequence of $k$ spatio-temporal measurements and an embedded path graph with travel time costs, the goal is to snap each measurement to a close-by…
We show the following problems are in $\textsf{P}$: 1. The contiguous art gallery problem -- a variation of the art gallery problem where each guard can protect a contiguous interval along the boundary of a simple polygon. This was posed at…
This note establishes that every polyhedron that has a Hamiltonian quasigeodesic can be edge-unfolded to a net, and shows that the class of such polyhedra is infinite.
With advances in technology, there has been growing interest in developing effective mapping methods for 3-dimensional objects in recent years. Volumetric parameterization for 3D solid manifolds plays an important role in processing 3D…
Given a set $P$ of $n$ weighted points and a set $H$ of $n$ half-planes in the plane, the hitting set problem is to compute a subset $P'$ of points from $P$ such that each half-plane contains at least one point from $P'$ and the total…
Unoriented surface reconstruction is an important task in computer graphics and has extensive applications. Based on the compact support of wavelet and orthogonality properties, classic wavelet surface reconstruction achieves good and fast…