Computational Geometry

While there are software systems that simplify trajectory streams on the fly, few curve simplification algorithms with quality guarantees fit the streaming requirements. We present streaming algorithms for two such problems under the…

In 1907, Henry Ernest Dudeney posed a puzzle: ``cut any equilateral triangle \dots\ into as few pieces as possible that will fit together and form a perfect square'' (without overlap, via translation and rotation). Four weeks later, Dudeney…

Constructing cell developmental trajectories is a critical task in single-cell RNA sequencing (scRNA-seq) analysis, enabling the inference of potential cellular progression paths. However, current automated methods are limited to…

We study the problem of sweeping a pseudoline arrangement with $n$ $x$-monotone curves with a rope (an $x$-monotone curve that connects the points at infinity). The rope can move by flipping over a face of the arrangement, replacing parts…

Given a collection of points in R^3, KD-Tree and R-Tree are well-known nearest neighbor search (NNS) algorithms that rely on space partitioning and spatial indexing techniques. However, when the query point is far from the data points or…

In this thesis, a new approach for constructing subdivision algorithms for generalized quadratic and cubic B-spline subdivision for subdivision surfaces and volumes is presented. First, a catalog of quality criteria for these subdivision…

The Mumford-Shah (MS) model is an important technique for mesh segmentation. Many existing researches focus on piecewise constant MS mesh segmentation model with total variation regularization, which pursue the shortest length of…

Large language models (LLMs) produce high-dimensional embeddings that capture rich semantic and syntactic relationships between words, sentences, and concepts. Investigating the topological structures of LLM embedding spaces via mapper…

The network homology Hk-core decomposition proposed in this article is similar to the k-core decomposition based on node degrees of the network. The C. elegans neural network and the cat cortical network are used as examples to reveal the…

The Freeze-Tag Problem (FTP) involves activating a set of initially asleep robots as quickly as possible, starting from a single awake robot. Once activated, a robot can assist in waking up other robots. Each active robot moves at unit…

This paper presents a novel algorithmic framework for the computational design, simulation, and fabrication of a hexagonal grid-based double-curvature structure with planar hexagonal panels. The journey begins with constructing a robust…

Extracting a quad mesh from a grid preserving map is straightforward in theory, but typical inputs are not exactly grid preserving maps. Previous works can manage minor deviations from grid preserving maps, but without a clear specification…

Given a weighted, undirected graph $G$ cellularly embedded on a topological surface $S$, we describe algorithms to compute the second shortest and third shortest closed walks of $G$ that are neither homotopically trivial in $S$ nor…

An important problem in topological data analysis (TDA)$\unicode{x2014}$of both theoretical and practical interest$\unicode{x2014}$is to reconstruct the topology and geometry of an underlying (usually unknown) metric graph from possibly…

Consider a graph drawn on a surface (for example, the plane minus a finite set of obstacle points), possibly with crossings. We provide an algorithm to decide whether such a drawing can be untangled, namely, if one can slide the vertices…

We study the problem of aggregating polygons by covering them with disjoint representative regions, thereby inducing a clustering of the polygons. Our objective is to minimize a weighted sum of the total area and the total perimeter of the…

Halfspace (or Tukey) depth is a fundamental and robust measure of centrality of data points in multivariate datasets. Computing the depth of a point with respect to the uniform distribution on an open convex body in $\mathbb{R}^d$ is a…

Analyzing singular patterns in vector fields is a fundamental problem in theoretical and practical domains due to the ability of such patterns to detect the intrinsic characteristics of vector fields. In this study, we propose an approach…

Data consisting of a graph with a function mapping into $\mathbb{R}^d$ arise in many data applications, encompassing structures such as Reeb graphs, geometric graphs, and knot embeddings. As such, the ability to compare and cluster such…

Here we study the multipacking problems for geometric point sets with respect to their Euclidean distances. We consider a set of $n$ points $P$ and define $N_s[v]$ as the subset of $P$ that includes the $s$ nearest points of $v \in P$ and…