We produce implicit equations for general biquadratic (order 2x2) Bézier triangle and quadrilateral surface patches and provide function evaluation code, using modern computing resources to exploit old algebraic construction techniques.
We present an algorithm for computing a bottleneck matching in a set of $n=2\ell$ points in the plane, which runs in $O(n^{ω/2}\log n)$ deterministic time, where $ω\approx 2.37$ is the exponent of matrix multiplication.
This paper provides an analytical solution to the Wolfram Alpha Rule 30 Problem 1. In this paper we discuss whether the central column of the Rule 30 structure is purely random and aperiodic.
In this extended abstract, we present a PTAS for guarding the vertices of a weakly-visible polygon $P$ from a subset of its vertices, or in other words, a PTAS for computing a minimum dominating set of the visibility graph of the vertices of $P$. We then show how to obtain a PTAS for vertex guarding $P$'s boundary.
The cyclic n-roots problem is an important benchmark problem for polynomial system solvers. We consider the pruning of cone intersections for a polyhedral method to compute series for the solution curves.
In this paper, we indicate a new way to define coordinates for the tiles of the tilings $\{p,3\}$ and $\{p$$-$$2,4\}$ where the natural number $p$ satisfies $p\geq 7$.
A terrain T is an x-monotone polygonal chain in the plane; T is orthogonal if each edge of T is either horizontal or vertical. In this paper, we give an exact algorithm for the problem of guarding the convex vertices of an orthogonal terrain with the minimum number of reflex vertices.
The Discretizable Molecular Distance Geometry Problem (DMDGP) consists in a subclass of the Molecular Distance Geometry Problem for which an embedding in ${\mathbb{R}^3}$ can be found using a Branch & Prune (BP) algorithm in a discrete search space. We propose a Clifford Algebra model of the DMDGP with an accompanying version of the BP algorithm.
There are numerous styles of planar graph drawings, notably straight-line drawings, poly-line drawings, orthogonal graph drawings and visibility representations. In this note, we show that many of these drawings can be transformed from one style to another without changing the height of the drawing. We then give some applications of these transformations.
Approximate counting is an algorithm that provides a count of a huge number of objects within an error tolerance. The first detailed analysis of this algorithm was given by Flajolet. In this paper, we propose a new analysis via the Poisson-Laplace-Mellin approach, a method devised for analyzing shape parameters of digital search trees in a recent paper of Hwang et al. Our approach yields a different and more compact expression for the periodic function from the asymptotic expansion of the variance. We show directly that our expression coincides with the one obtained by Flajolet. Moreover, we apply our method to variations of approximate counting, too.
In this paper, we review a method for computing and parameterizing the set of homotopy classes of chain maps between two chain complexes. This is then applied to finding topologically meaningful maps between simplicial complexes, which in the context of topological data analysis, can be viewed as an extension of conventional unsupervised learning methods to simplicial complexes.
The theory of zigzag persistence is a substantial extension of persistent homology, and its development has enabled the investigation of several unexplored avenues in the area of topological data analysis. In this paper, we discuss three applications of zigzag persistence: topological bootstrapping, parameter thresholding, and the comparison of witness complexes.
It is shown that there are examples of distinct polyhedra, each with a Hamiltonian path of edges, which when cut, unfolds the surfaces to a common net. In particular, it is established for infinite classes of triples of tetrahedra.
We show that the open problem presented in "Geometric Folding Algorithms: Linkages, Origami, Polyhedra" [DO07] is solved by a theorem of Burago and Zalgaller [BZ96] from more than a decade earlier.
We present an innovative algorithm that simplifies the topology of a cross-field. Our algorithm works through macro-operations that allow us editing the graph of separatrices, which is extracted from a cross-field, while maintaining it topologically consistent. We present preliminary results of our implementation.
This survey gives a brief overview of theoretically and practically relevant algorithms to compute geodesic paths and distances on three-dimensional surfaces. The survey focuses on polyhedral three-dimensional surfaces.
There exists a surface of a convex polyhedron P and a partition L of P into geodesic convex polygons such that there are no connected "edge" unfoldings of P without self-intersections (whose spanning tree is a subset of the edge skeleton of L).
In this paper we improve the approach of a previous paper about the domino problem in the hyperbolic plane, see arXiv.cs.CG/0603093. This time, we prove that the general problem of the hyperbolic plane with à la Wang tiles is undecidable.