Consider an input consisting of a set of $n$ disjoint triangular obstacles in $\mathbb{R}^3$ and a target point $t$ in the free space, all enclosed by a large sphere $S$ of radius $R$ centered at $t$. An articulated probe is modeled as two line segments $ab$ and $bc$ connected at point $b$. The length of $ab$ can be equal to or greater than $R$, while $bc$ is of a given length $r \leq R$. The probe

An articulated probe is modeled in the plane as two line segments, $ab$ and $bc$, joined at $b$, with $ab$ being very long, and $bc$ of some small length $r$. We investigate a trajectory planning problem involving the articulated two-segment probe where the length $r$ of $bc$ can be customized. Consider a set $P$ of simple polygonal obstacles with a total of $n$ vertices, a target point $t$ located

We prove that for every $m$ there is a finite point set $\mathcal{P}$ in the plane such that no matter how $\mathcal{P}$ is three-colored, there is always a disk containing exactly $m$ points, all of the same color. This improves a result of Pach, Tardos and T\'oth who proved the same for two colors. The main ingredient of the construction is a subconstruction whose points are in convex position. Namely

Persistent Homology (PH) is a useful tool to study the underlying structure of a data set. Persistence Diagrams (PDs), which are 2D multisets of points, are a concise summary of the information found by studying the PH of a data set. However, PDs are difficult to incorporate into a typical machine learning workflow. To that end, two main methods for representing PDs have been developed: kernel methods

What functions, when applied to the pairwise Manhattan distances between any $n$ points, result in the Manhattan distances between another set of $n$ points? In this paper, we show that a function has this property if and only if it is Bernstein. This class of functions admits several classical analytic characterizations and includes $f(x) = x^s$ for $0 \leq s \leq 1$ as well as $f(x) = 1-e^{-xt}$

With the growth of machine learning algorithms with geometry primitives, a high-efficiency library with differentiable geometric operators are desired. We present an optimized Differentiable Geometry Algorithm Library (DGAL) loaded with implementations of differentiable operators for geometric primitives like lines and polygons. The library is a header-only templated C++ library with GPU support. We

We consider planar tiling and packing problems with polyomino pieces and a polyomino container $P$. A polyomino is a polygonal region with axis parallel edges and corners of integral coordinates, which may have holes. We give two polynomial time algorithms, one for deciding if $P$ can be tiled with $2\times 2$ squares (that is, deciding if $P$ is the union of a set of non-overlapping copies of the

We study harmonic-based algorithms for the $d$-dimensional ($d$D) generalizations of three classical geometric packing problems: geometric bin packing (BP), strip packing (SP), and geometric knapsack (KS). Caprara (MOR 2008) studied a harmonic-based algorithm $\mathtt{HDH}_k$, that has an asymptotic approximation ratio of $T_{\infty}^{d-1}$ (where $T_{\infty} \approx 1.691$) for $d$D BP and $d$D SP

The Escherization problem involves finding a closed figure that tiles the plane that is most similar to a given goal figure. In Koizumi and Sugihara's formulation of the Escherization problem, the tile and goal figures are represented as $n$-point polygons where the similarity between them is measured based on the difference in the positions between the corresponding points. This paper presents alternative

Designing a robot or structure that can fold itself into a target shape is a process that involves challenges originated from multiple sources. For example, the designer of rigid self-folding robots must consider foldability from geometric and kinematic aspects to avoid self-intersection and undesired deformations. Recent works have shown success in estimating foldability of a design using robot motion

We devise an algorithm for maintaining the visibility polygon of any query point in a dynamic polygonal domain, i.e., as the polygonal domain is modified with vertex insertions and deletions to its obstacles, we update the data structures that store the visibility polygon of the query point. After preprocessing the initial input polygonal domain to build a few data structures, our algorithm takes O(k(\lg{|VP_{\cal

We utilize symmetries of tori constructed from copies of given disk-type meshes in 3d, together with symmetries of corresponding tilings of fundamental domains of plane tori. We use these correspondences to prove optimality of the embedding of the mesh onto special types of triangles in the plane, and rectangles. The proof provides a certain framework for using symmetries of the image domain. The complexity

Motivated by recent work on Delaunay triangulations of hyperbolic surfaces, we consider the minimal number of vertices of such triangulations. First, we will show that every hyperbolic surface of genus $g$ has a simplicial Delaunay triangulation with $O(g)$ vertices, where edges are given by distance paths. Then, we will construct a class of hyperbolic surfaces for which the order of this bound is

Cartograms are map-based data visualizations in which the area of each map region is proportional to an associated numeric data value (e.g., population or gross domestic product). A cartogram is called contiguous if it conforms to this area principle while also keeping neighboring regions connected. Because of their distorted appearance, contiguous cartograms have been criticized as difficult to read

We consider the problem of computing the diameter of a unicycle graph (i.e., a graph with a unique cycle). We present an O(n) time algorithm for the problem, where n is the number of vertices of the graph. This improves the previous best O(n \log n) time solution [Oh and Ahn, ISAAC 2016]. Using this algorithm as a subroutine, we solve the problem of adding a shortcut to a tree so that the diameter

In optimization or machine learning problems we are given a set of items, usually points in some metric space, and the goal is to minimize or maximize an objective function over some space of candidate solutions. For example, in clustering problems, the input is a set of points in some metric space, and a common goal is to compute a set of centers in some other space (points, lines) that will minimize

Modeling 3D objects in domains like Computer Aided Design (CAD) is time-consuming and comes with a steep learning curve needed to master the design process as well as tool complexities. In order to simplify the modeling process, we designed and implemented a prototypical system that leverages the strengths of Virtual Reality (VR) hand gesture recognition in combination with the expressiveness of a

In this paper, we present a more efficient GJK algorithm to solve the collision detection and distance query problems in 2D. We contribute in two aspects: First, we propose a new barycode-based sub-distance algorithm that does not only provide a simple and unified condition to determine the minimum simplex but also improve the efficiency in distant, touching, and overlap cases in distance query. Second

Advances in natural language processing have resulted in increased capabilities with respect to multiple tasks. One of the possible causes of the observed performance gains is the introduction of increasingly sophisticated text representations. While many of the new word embedding techniques can be shown to capture particular notions of sentiment or associative structures, we explore the ability of

Both Reeb spaces and higher Morse functions induce natural stratifications. In the former, we show that the data of the Jacobi set of a function $f:X \to \mathbb{R}^k$ induces stratifications on $X,\mathbb{R}^k$, and the associated Reeb space, and give conditions under which maps between these three spaces are stratified maps. We then extend this type of construction to the codomain of higher Morse

In this paper, we propose a concept to design, track, and compare application-specific feature definitions expressed as sets of critical points. Our work has been inspired by the observation that in many applications a large variety of different feature definitions for the same concept are used. Often, these definitions compete with each other and it is unclear which definition should be used in which

Splines are one of the main methods of mathematically representing complicated shapes, which have become the primary technique in computer graphics for modeling complex surfaces. Among all, Bezier and Catmull-Rom splines are of the most used in the subfields of engineering. In this document, we focus on conversion of splines rather than going through the properties of them, i.e, converting the control

A new predictor-corrector type incremental algorithm is proposed for the exact construction of weighted straight skeletons of 2D general planar polygons of arbitrary complexity based on the notion of deforming polygon. In the proposed algorithm, the raw input provided by the polygon itself is enough to resolve edge collapse and edge split events. Neither the construction of a kinetic triangulation

We study more invariants in the elliptic billiard, including those manifested by self-intersected orbits and inversive polygons. We also derive expressions for some entries in "Eighty New Invariants of N-Periodics in the Elliptic Billiard" (2020), arXiv:2004.12497.

Graph Crossing Number is a fundamental problem with various applications. In this problem, the goal is to draw an input graph $G$ in the plane so as to minimize the number of crossings between the images of its edges. Despite extensive work, non-trivial approximation algorithms are only known for bounded-degree graphs. Even for this special case, the best current algorithm achieves a $\tilde O(\sqrt

In the past decade frame fields have emerged as a promising approach for generating hexahedral meshes for CFD and CAE applications. One important problem asks for construction of a boundary aligned frame field with prescribed singularity constraints that correspond to a valid hexahedral mesh. We give a necessary and sufficient condition in terms of solutions to a system of monomial equations whose

The contact goniometer is a commonly used tool in lithic and zooarchaeological analysis, despite suffering from a number of shortcomings due to the physical interaction between the measuring implement, the object being measured, and the individual taking the measurements. However, lacking a simple and efficient alternative, researchers in a variety of fields continue to use the contact goniometer to

We revisit the classic task of finding the shortest tour of $n$ points in $d$-dimensional Euclidean space, for any fixed constant $d \geq 2$. We determine the optimal dependence on $\varepsilon$ in the running time of an algorithm that computes a $(1+\varepsilon)$-approximate tour, under a plausible assumption. Specifically, we give an algorithm that runs in $2^{\mathcal{O}(1/\varepsilon^{d-1})} n\log

We present a simple algorithm for computing higher-order Delaunay mosaics that works in Euclidean spaces of any finite dimensions. The algorithm selects the vertices of the order-$k$ mosaic from incrementally constructed lower-order mosaics and uses an algorithm for weighted first-order Delaunay mosaics as a black-box to construct the order-$k$ mosaic from its vertices. Beyond this black-box, the algorithm

In recent years, the notion of rank metric in the context of coding theory has known many interesting developments in terms of applications such as space time coding, network coding or public key cryptography. These applications raised the interest of the community for theoretical properties of this type of codes, such as the hardness of decoding in rank metric or better decoding algorithms. Among

A total dominating set of a graph $G=(V,E)$ is a subset $D$ of $V$ such that every vertex in $V$ is adjacent to at least one vertex in $D$. The total domination number of $G$, denoted by $\gamma _t (G)$, is the minimum cardinality of a total dominating set of $G$. A near-triangulation is a biconnected planar graph that admits a plane embedding such that all of its faces are triangles except possibly

The Euclidean $k$-median problem is defined in the following manner: given a set $\mathcal{X}$ of $n$ points in $\mathbb{R}^{d}$, and an integer $k$, find a set $C \subset \mathbb{R}^{d}$ of $k$ points (called centers) such that the cost function $\Phi(C,\mathcal{X}) \equiv \sum_{x \in \mathcal{X}} \min_{c \in C} \|x-c\|_{2}$ is minimized. The Euclidean $k$-means problem is defined similarly by replacing

Mapper algorithm is a popular tool from topological data analysis for extracting topological summaries of high-dimensional datasets. It has enjoyed great success in data science, from shape classification to cancer research and sports analytics. In this paper, we present Mapper Interactive, a web-based framework for the interactive analysis and visualization of high-dimensional point cloud data built

Given a set S of n points, a weight function w to associate a non-negative weight to each point in S, a positive integer k \ge 1, and a real number \epsilon > 0, we present algorithms for computing a spanner network G(S, E) for the metric space (S, d_w) induced by the weighted points in S. The weighted distance function d_w on the set S of points is defined as follows: for any p, q \in S, d_w(p, q)

This paper studies a variant of the Art-gallery problem in which "walls" can be replaced by \emph{reflecting-edges}, which allows the guard to see further and thereby see a larger portion of the gallery. The art-gallery is a simple closed polygon $P$, a guard is a point $p$ in $P$, and a guard sees another point $q$ in $P$ if the segment $\overline{pq}$ is contained in the interior of $P$. The visibility

We study affine invariant 2D triangulation methods. That is, methods that produce the same triangulation for a point set $S$ for any (unknown) affine transformation of $S$. Our work is based on a method by Nielson [A characterization of an affine invariant triangulation. Geom. Mod, 191-210. Springer, 1993] that uses the inverse of the covariance matrix of $S$ to define an affine invariant norm, denoted

Given a bipartite graph $G=(V_b,V_r,E)$, the $2$-Level Quasi-Planarity problem asks for the existence of a drawing of $G$ in the plane such that the vertices in $V_b$ and in $V_r$ lie along two parallel lines $\ell_b$ and $\ell_r$, respectively, each edge in $E$ is drawn in the unbounded strip of the plane delimited by $\ell_b$ and $\ell_r$, and no three edges in $E$ pairwise cross. We prove that the

Intersection detection between three-dimensional bodies has various applications in computer graphics, video game development, robotics as well as military industries. In some respects, entities do not want to disclose sensitive information about themselves, including their location. In this paper, we present a secure two-party protocol to determine the existence of an intersection between entities

This paper introduces the problem of coresets for regression problems to panel data settings. We first define coresets for several variants of regression problems with panel data and then present efficient algorithms to construct coresets of size that depend polynomially on 1/$\varepsilon$ (where $\varepsilon$ is the error parameter) and the number of regression parameters - independent of the number

We propose a new kind of sliding-block puzzle, called Gourds, where the objective is to rearrange 1 x 2 pieces on a hexagonal grid board of 2n + 1 cells with n pieces, using sliding, turning and pivoting moves. This puzzle has a single empty cell on a board and forms a natural extension of the 15-puzzle to include rotational moves. We analyze the puzzle and completely characterize the cases when the

We investigate the limits of one of the fundamental ideas in data structures: fractional cascading. This is an important data structure technique to speed up repeated searches for the same key in multiple lists and it has numerous applications. Specifically, the input is a "catalog" graph, $G$, of constant degree together with a list of values assigned to every vertex of $G$. The goal is to preprocess

Cross field generation is often used as the basis for the construction of block-structured quadrangular meshes, and the field singularities have a key impact on the structure of the resulting meshes. In this paper, we extend Ginzburg-Landau cross field generation methods with a new formulation that allows a user to impose inner singularities. The cross field is computed via the optimization of a linear

An hieroglyph on n letters is a cyclic sequence of the letters 1,2, . . . , n of length 2n such that each letter appears in the sequence twice.Take an hieroglyph H. Take a convex polygon with 2n sides. Put the letters in the sequence of letters of the hieroglyph on the sides of the convexpolygon in the same order. For each letter i glue the ends of a ribbon to thepair of sides corresponding to the

With the advancement in 3D scanning technology, there has been a surge of interest in the use of point clouds in science and engineering. To facilitate the computations and analyses of point clouds, prior works have considered parameterizing them onto some simple planar domains with a fixed boundary shape such as a unit circle or a rectangle. However, the geometry of the fixed shape may lead to some

Motivated by the methods and results of manifold sampling based on Ricci curvature, we propose a similar approach for networks. To this end we make appeal to three types of discrete curvature, namely the graph Forman-, full Forman- and Haantjes-Ricci curvatures for edge-based and node-based sampling. We present the results of experiments on real life networks, as well as for square grids arising in

Studies on manufacturing cost prediction based on deep learning have begun in recent years, but the cost prediction rationale cannot be explained because the models are still used as a black box. This study aims to propose a manufacturing cost prediction process for 3D computer-aided design (CAD) models using explainable artificial intelligence. The proposed process can visualize the machining features

Quantum topological invariants have played an important role in computational topology, and they are at the heart of major modern mathematical conjectures. In this article, we study the experimental problem of computing large $r$ values of Turaev-Viro invariants $\mathrm{TV}_r$. We base our approach on an optimized backtracking algorithm, consisting of enumerating combinatorial data on a triangulation

We consider minimum-cardinality Manhattan connected sets with arbitrary demands: Given a collection of points $P$ in the plane, together with a subset of pairs of points in $P$ (which we call demands), find a minimum-cardinality superset of $P$ such that every demand pair is connected by a path whose length is the $\ell_1$-distance of the pair. This problem is a variant of three well-studied problems

In this short note, we prove that the degree-three dilation of the square lattice $\mathbb{Z}^2$ is $1+\sqrt{2}$. This disproves a conjecture of Dumitrescu and Ghosh. We give a computer-assisted proof of a local-global property for the uncountable set of geometric graphs achieving the optimal dilation.

We investigate the problem of drawing two posets of the same ground set so that one is drawn from left to right and the other one is drawn from the bottom up. The input to this problem is a directed graph $G = (V, E)$ and two sets $X, Y$ with $X \cup Y = E$, each of which can be interpreted as a partial order of $V$. The task is to find a planar drawing of $G$ such that each directed edge in $X$ is

We propose a cut-based algorithm for finding all vertices and all facets of the convex hull of all integer points of a polyhedron defined by a system of linear inequalities. Our algorithm DDM Cuts is based on the Gomory cuts and the dynamic version of the double description method. We describe the computer implementation of the algorithm and present the results of computational experiments comparing

In this paper, we propose a fully differentiable pipeline for estimating accurate dense correspondences between 3D point clouds. The proposed pipeline is an extension and a generalization of the functional maps framework. However, instead of using the Laplace-Beltrami eigenfunctions as done in virtually all previous works in this domain, we demonstrate that learning the basis from data can both improve

Given a bichromatic point set $P=\textbf{R}$ $ \cup$ $ \textbf{B}$ of red and blue points, a separator is an object of a certain type that separates $\textbf{R}$ and $\textbf{B}$. We study the geometric separability problem when the separator is a) rectangular annulus of fixed orientation b) rectangular annulus of arbitrary orientation c) square annulus of fixed orientation d) orthogonal convex polygon

Recent works in geometric deep learning have introduced neural networks that allow performing inference tasks on three-dimensional geometric data by defining convolution, and sometimes pooling, operations on triangle meshes. These methods, however, either consider the input mesh as a graph, and do not exploit specific geometric properties of meshes for feature aggregation and downsampling, or are specialized

Bounded-angle (minimum) spanning trees were first introduced in the context of wireless networks with directional antennas. They are reminiscent of bounded-degree spanning trees, which have received significant attention. Let $P = \{p_1,\ldots,p_n\}$ be a set of $n$ points in the plane, let $\Pi$ be the polygonal path $(p_1,\ldots,p_n)$, and let $0 < \alpha < 2\pi$ be an angle. An $\alpha$-spanning

Outer polyhedral representations of a given polynomial curve are extensively exploited in computer graphics rendering, computer gaming, path planning for robots, and finite element simulations. B\'ezier curves (which use the Bernstein basis) or B-Splines are a very common choice for these polyhedral representations because their non-negativity and partition-of-unity properties guarantee that each interval

We examine quadratic surfaces in 3-space that are tangent to nine given figures. These figures can be points, lines, planes or quadrics. The numbers of tangent quadrics were determined by Hermann Schubert in 1879. We study the associated systems of polynomial equations, also in the space of complete quadrics, and we solve them using certified numerical methods. Our aim is to show that Schubert's problems

Uniform distribution of the points has been of interest to researchers for a long time and has applications in different areas of Mathematics and Computer Science. One of the well-known measures to evaluate the uniformity of a given distribution is Discrepancy, which assesses the difference between the Uniform distribution and the empirical distribution given by putting mass points at the points of

We present a new meshing algorithm called guided and augmented meshing, GAMesh, which uses a mesh prior to generate a surface for the output points of a point network. By projecting the output points onto this prior and simplifying the resulting mesh, GAMesh ensures a surface with the same topology as the mesh prior but whose geometric fidelity is controlled by the point network. This makes GAMesh

The \v{C}ech and Rips constructions of persistent homology are stable with respect to perturbations of the input data. However, neither is robust to outliers, and both can be insensitive to topological structure of high-density regions of the data. A natural solution is to consider 2-parameter persistence. This paper studies the stability of 2-parameter persistent homology: We show that several related