Related papers: How to Reconstruct a Planar Map From its Branching…
This work studies path planning in two-dimensional space, in the presence of polygonal obstacles. We specifically address the problem of building a roadmap graph, that is, an abstract representation of all the paths that can potentially be…
We provide a natural smooth projective compactification of the space of algebraic maps from the projective line to the projective space of dimension n by adding a divisor with simple normal crossings.
In this paper we show how, under surprisingly weak assumptions, one can split a planar curve into three arcs and rearrange them (matching tangent directions) to obtain a closed curve. We also generalize this construction to curves split…
A $\textit{polygonal curve}$ is a collection of $m$ connected line segments specified as the linear interpolation of a list of points $\{p_0, p_1, \ldots, p_m\}$. These curves may be obtained by sampling points from an oriented curve in…
We give an algorithm for approximating a given plane curve segment by a planar elastic curve. The method depends on an analytic representation of the space of elastic curve segments, together with a geometric method for obtaining a good…
Grafting is a method of obtaining new projective structures from a hyperbolic structure, basically by gluing a flat cylinder into a surface along a closed geodesic in the hyperbolic structure, or by limits of that procedure. This induces a…
A plane curve is a knot diagram in which each crossing is replaced by a 4-valent vertex, and so are dual to a subset of planar quadrangulations. The aim of this paper is to introduce a new tool for sampling diagrams via sampling of plane…
We derive an efficient way to obtain generating functions of bipartite maps of arbitrary genus and boundary length using a spectral curve as initial data for the framework of topological recursion. Based on an earlier result of Chapuy and…
We develop in this article an algorithm that, given a projective curve $C$, computes a \textit{gonal map}, that is, a finite morphism from $C$ to the projective line of minimal degree. Our method is based on the computation of scrollar…
We propose a novel tree-like curvilinear structure reconstruction algorithm based on supervised learning and graph theory. In this work we analyze image patches to obtain the local major orientations and the rankings that correspond to the…
For a given spectral curve, we construct a family of symplectic dual spectral curves for which we prove an explicit formula expressing the $n$-point functions produced by the topological recursion on these curves via the $n$-point functions…
A well-known and difficult problem in computational number theory and algebraic geometry is to write down equations for branched covers of algebraic curves with specified monodromy type. In this article, we present a technique for computing…
Designing mechanical devices, called linkages, that draw a given plane curve has been a topic that interested engineers and mathematicians for hundreds of years, and recently also computer scientists. Already in 1876, Kempe proposed a…
For a finite graph, a spectral curve is constructed as the zero set of a two-variate polynomial with integer coefficients coming from p-adic diffusion on the graph. It is shown that certain spectral curves can distinguish non-isomorphic…
The persistence diagram (PD) is an increasingly popular topological descriptor. By encoding the size and prominence of topological features at varying scales, the PD provides important geometric and topological information about a space.…
We present a matrix-based algorithm for deciding if the parametrization of a curve or a surface is invertible or not, and for computing the inverse of the parametrization if it exists.
We consider the problem of computing the topology and describing the geometry of a parametric curve in $\mathbb{R}^n$. We present an algorithm, PTOPO, that constructs an abstract graph that is isotopic to the curve in the embedding space.…
We introduce a procedure for constructing a generalized elliptic curve from a genus-one twisted curve, and we use this procedure to define an explicit, modular isomorphism between two compactifications of moduli stacks of elliptic curves…
We describe germs of mappings $(\mathbb{C}^2,0) \to (\mathbb{C}^2,0)$ ramified along a germ of irreducible curve whose image is of the form $x^p=y^q$.
In a random linear graph, vertices are points on a line, and pairs of vertices are connected, independently, with a link probability that decreases with distance. We study the problem of reconstructing the linear embedding from the graph,…