Computational Geometry
Vertex splitting replaces a vertex by two copies and partitions its incident edges amongst the copies. This problem has been studied as a graph editing operation to achieve desired properties with as few splits as possible, most often…
Two important classes of three-dimensional elements in computational meshes are hexahedra and tetrahedra. While several efficient methods exist that convert a hexahedral element to a tetrahedral elements, the existing algorithm for…
Arkin et al. in 2002 introduced a scheduling-like problem called Freeze-Tag Problem (FTP) motivated by robot swarm activation. The input consists of the locations of n mobile punctual robots in some metric space or graph. Only one begins…
A graph G is a (Euclidean) unit disk graph if it is the intersection graph of unit disks in the Euclidean plane $\mathbb{R}^2$. Recognizing them is known to be $\exists\mathbb{R}$-complete, i.e., as hard as solving a system of polynomial…
Topological data analysis (TDA) is an expanding field that leverages principles and tools from algebraic topology to quantify structural features of data sets or transform them into more manageable forms. As its theoretical foundations have…
We study the concept of the continuous mean distance of a weighted graph. For connected unweighted graphs, the mean distance can be defined as the arithmetic mean of the distances between all pairs of vertices. This parameter provides a…
Periodic Geometry studies isometry invariants of periodic point sets that are also continuous under perturbations. The motivations come from periodic crystals whose structures are determined in a rigid form but any minimal cells can…
Given an $n$-vertex 1.5D terrain $\T$ and a set $\A$ of $m<n$ viewpoints, the Voronoi visibility map $\vorvis(\T,\A)$ is a partitioning of $\T$ into regions such that each region is assigned to the closest (in Euclidean distance) visible…
Let $P=(p_1, p_2, \dots, p_n)$ be a polygonal chain in $\mathbb{R}^d$. The stretch factor of $P$ is the ratio between the total length of $P$ and the distance of its endpoints, $\sum_{i = 1}^{n-1} |p_i p_{i+1}|/|p_1 p_n|$. For a parameter…
The \emph{strong collapse} of a simplicial complex, proposed by Barmak and Minian (\emph{Disc. Comp. Geom. 2012}), is a combinatorial collapse of a complex onto its sub-complex. Recently, it has received attention from computational…
We investigate the combinatorial discrepancy of geometric set systems having bounded shallow cell complexity in the \emph{Beck-Fiala} setting, where each point belongs to at most $t$ ranges. For set systems with shallow cell complexity…
Computing persistent homology using Gaussian kernels is useful in the domains of topological data analysis and machine learning as shown by Phillips, Wang and Zheng [SoCG 2015]. However, contrary to the case of computing persistent homology…
In machining feature recognition, geometric elements generated in a three-dimensional computer-aided design model are identified. This technique is used in manufacturability evaluation, process planning, and tool path generation. Here, we…
Consider the regular $n$-simplex $\Delta_n$ - it is formed by the convex-hull of $n+1$ points in Euclidean space, with each pair of points being in distance exactly one from each other. We prove an exact bound on the width of $\Delta_n$…
We produce implicit equations for general biquadratic (order 2x2) B\'ezier triangle and quadrilateral surface patches and provide function evaluation code, using modern computing resources to exploit old algebraic construction techniques.
This thesis consists of two topics related to computational geometry and one topic related to topological data analysis (TDA), which combines fields of computational geometry and algebraic topology for analyzing data. The first part studies…
Persistent homology is a tool that can be employed to summarize the shape of data by quantifying homological features. When the data is an object in $\mathbb{R}^d$, the (augmented) persistent homology transform ((A)PHT) is a family of…
Planar/flat configurations of fixed-angle chains and trees are well studied in the context of polymer science, molecular biology, and puzzles. In this paper, we focus on a simple type of fixed-angle linkage: every edge has unit length…
We present a new topological connection method for the local bilinear computation of Jacobi sets that improves the visual representation while preserving the topological structure and geometric configuration. To this end, the topological…
Porous structures are widely used in various industries because of their excellent properties. Porous surfaces have no thickness and should be thickened to sheet structures for further fabrication. However, conventional methods for…