Computational Geometry
Persistent (co)homology is a central construction in topological data analysis, where it is used to quantify prominence of features in data to produce stable descriptors suitable for downstream analysis. Persistence is challenging to…
We study the problem of partitioning a given simple polygon $P$ into a minimum number of connected polygonal pieces, each of bounded size. We describe a general technique for constructing such partitions that works for several notions of…
The problem Level Planarity asks for a crossing-free drawing of a graph in the plane such that vertices are placed at prescribed y-coordinates (called levels) and such that every edge is realized as a y-monotone curve. In the variant…
In engineering design, one of the most daunting problems in the design-through-analysis workflow is to deal with trimmed NURBS (Non-Uniform Rational B-Splines), which often involve topological/geometric issues and lead to inevitable gaps…
Let $m$ and $n$ be the numbers of vertices of two polygonal curves in $\mathbb{R}^d$ for any fixed $d$ such that $m \leq n$. Since it was known in 1995 how to compute the Fr\'{e}chet distance of these two curves in $O(mn\log (mn))$ time, it…
In the Partially Embedded Planarity problem, we are given a graph $G$ together with a topological drawing of a subgraph $H$ of $G$. The task is to decide whether the drawing can be extended to a drawing of the whole graph such that no two…
We study algorithms for drawing planar graphs and 1-planar graphs using cubic B\'ezier curves with bounded curvature. We show that any n-vertex 1-planar graph has a 1-planar RAC drawing using a single cubic B\'ezier curve per edge, and this…
In this work, we study the perception problem for sampled surfaces (possibly with boundary) using tools from computational topology, specifically, how to identify their underlying topology starting from point-cloud samples in space, such as…
A Reeb graph is a graphical representation of a scalar function on a topological space that encodes the topology of the level sets. A Reeb space is a generalization of the Reeb graph to a multiparameter function. In this paper, we propose…
Methodology is provided towards the solution of the minimum enclosing ball problem. This problem concerns the determination of the unique spherical surface of smallest radius enclosing a given bounded set in the d-dimensional Euclidean…
Generalizing pseudospherical drawings, we introduce a new class of simple drawings, which we call separable drawings. In a separable drawing, every edge can be closed to a simple curve that intersects each other edge at most once. Curves of…
This paper answers a long-standing open question in tile-assembly theory, namely that it is possible to strictly assemble discrete self-similar fractals (DSSFs) in the abstract Tile-Assembly Model (aTAM). We prove this in 2 separate ways,…
Let $\mathcal{P}$ be a set of $n=2m+1$ points in the plane in general position. We define the graph $GM_\mathcal{P}$ whose vertex set is the set of all plane matchings on $\mathcal{P}$ with exactly $m$ edges. Two vertices in…
In this paper, an integrated sensing and communication (ISAC) system is investigated. Initially, we introduce a design criterion wherein sensing data acquired from the preceding time slot is employed for instantaneous optimal beamforming in…
We present a novel method for analyzing geospatial trajectory data using topological data analysis (TDA) to identify a specific class of anomalies, commonly referred to as crop circles, in AIS data. This approach is the first of its kind to…
Recent advances in artificial intelligence (AI) have impacted various fields, including mechanical engineering. However, the development of diverse, high-quality datasets for structural analysis remains a challenge. Traditional datasets,…
Reconstructing 2D curves from sample points has long been a critical challenge in computer graphics, finding essential applications in vector graphics. The design and editing of curves on surfaces has only recently begun to receive…
We study a geometric facility location problem under imprecision. Given $n$ unit intervals in the real line, each with one of $k$ colors, the goal is to place one point in each interval such that the resulting \emph{minimum color-spanning…
For a hypergraph $\mathcal{H}=(X,\mathcal{E})$ a \emph{support} is a graph $G$ on $X$ such that for each $E\in\mathcal{E}$, the induced subgraph of $G$ on the elements in $E$ is connected. If $G$ is planar, we call it a planar support. A…
In this paper, general conversion equations are derived between Absolute Nodal Coordinates Formulation (ANCF) finite surface elements and B-spline surfaces, an extension of our previous work on the conversion between ANCF cable elements and…