Related papers: Computing the Similarity Between Moving Curves
A surface embedded in space, in such a way that each point has a neighborhood within which the surface is a terrain, projects to an immersed surface in the plane, the boundary of which is a self-intersecting curve. Under what circumstances…
We consider the motion by mean curvature of an $n$-dimensional graph over a time-dependent domain in $\mathbb{R}^n$, intersecting $\mathbb{R}^n$ at a constant angle. In the general case, we prove local existence for the corresponding…
This paper explores the relationship between closed curves on surfaces and their intersections. Like Dehn-Thurston coordinates for simple curves, we explore how to determine closed curves using the number of times they intersect other…
We prove that the length difference between a closed periodic curve and its parallel curve at a sufficiently small distance is proportional to the rotation index. As an application, the rotation index of a curve could be estimated by means…
The problem of measuring similarity of graphs and their nodes is important in a range of practical problems. There is a number of proposed measures, some of them being based on iterative calculation of similarity between two graphs and the…
Numerous approximation algorithms for problems on unit disk graphs have been proposed in the literature, exhibiting a sharp trade-off between running times and approximation ratios. We introduce a variation of the known shifting strategy…
We investigate geometric properties of surfaces given by certain formulae. In particular, we calculate the singular curvature and the limiting normal curvature of such surfaces along the set of singular points consisting of singular points…
We establish convergence results for a spatial semidiscretization of Mean Curvature Flow (MCF) for surfaces with fixed boundaries. Our analysis is based on Huisken's evolution equations for the mean curvature and the normal vector, enabling…
We prove the first polynomial bound on the number of monotonic homotopy moves required to tighten a collection of closed curves on any compact orientable surface, where the number of crossings in the curve is not allowed to increase at any…
Partial differential equations (PDEs) on surfaces arise in a wide range of applications. The closest point method (Ruuth and Merriman, J. Comput. Phys. 227(3):1943-1961, [2008]) is a recent embedding method that has been used to solve a…
We fully generalize a previously-developed computational geometry tool [1] to perform large-scale simulations of arbitrary two-dimensional faceted surfaces $z = h(x,y)$. Our method uses a three-component facet/edge/junction storage model,…
The Closest Point Method for solving partial differential equations (PDEs) posed on surfaces was recently introduced by Ruuth and Merriman [J. Comput. Phys. 2008] and successfully applied to a variety of surface PDEs. In this paper we study…
How do social networks differ across platforms? How do information networks change over time? Answering questions like these requires us to compare two or more graphs. This task is commonly treated as a measurement problem, but numerical…
We analytically study proximity and distance properties of various kernels and similarity measures on graphs. This helps to understand the mathematical nature of such measures and can potentially be useful for recommending the adoption of…
Near a singular point of a surface or a curve, geometric invariants diverge in general, and the orders of diverge, in particular the boundedness about these invariants represent geometry of the surface and the curve. In this paper, we study…
We present variational approximations of boundary value problems for curvature flow (curve shortening flow) and elastic flow (curve straightening flow) in two-dimensional Riemannian manifolds that are conformally flat. For the evolving open…
Hydrophobic textured surfaces are studied for their low wettability and their capacity to create a 'slippery' fluid on the surface during lubrication. To this end, the flow between two parallel surfaces is numerically addressed by computing…
A significant problem with most functional data analyses is that of misaligned curves. Without adjustment, even an analysis as simple as estimation of the mean will fail. One common method to synchronize a set of curves involves equating…
In the classic polyline simplification problem we want to replace a given polygonal curve $P$, consisting of $n$ vertices, by a subsequence $P'$ of $k$ vertices from $P$ such that the polygonal curves $P$ and $P'$ are as close as possible.…
This communication describes a representation of images as a set of edges characterized by their position and orientation. This representation allows the comparison of two images and the computation of their similarity. The first step in…