Related papers: Length of parallel curves
The article analyzes similarity of closed polygonal curves with respect to the Frechet metric, which is stronger than the well-known Hausdorff metric and therefore is more appropriate in some applications. An algorithm is described that…
The article analyzes similarity of closed polygonal curves in Frechet metric, which is stronger than the well-known Hausdorff metric and therefore is more appropriate in some applications. An algorithm that determines whether the Frechet…
Computing the Fr\'echet distance between two polygonal curves takes roughly quadratic time. In this paper, we show that for a special class of curves the Fr\'echet distance computations become easier. Let $P$ and $Q$ be two polygonal curves…
For two closed curves on a plane (discrete version) and local criteria for similarity of points on the curves one gets a potential, which describes the similarity between curve points. This is the base for a global similarity measure of…
We give a new formula for the rotation number (or Whitney index) of a smooth closed plane curve. This formula is obtained from the winding numbers associated with the regions and the crossing points of the curve. One difference with the…
The classical Whitney formula relates the number of times an oriented plane curve cuts itself to its rotation number and the index of a base point. In this paper we generalize Whitney's formula to curves on an oriented punctured surface. To…
In this paper we study similarity measures for moving curves which can, for example, model changing coastlines or retreating glacier termini. Points on a moving curve have two parameters, namely the position along the curve as well as time.…
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…
When geometric structures on surfaces are determined by the lengths of curves, it is natural to ask: which curves' lengths do we really need to know? It is a result of Duchin--Leininger--Rafi that any flat metric induced by a unit-norm…
The paper presents a discrete variation of the Frechet distance between closed curves, which can be seen as an approximation of the continuous measure. A rather straightforward approach to compute the discrete Frechet distance between two…
We present simple and practical $(1+\eps)$-approximation algorithm for the Frechet distance between curves. To analyze this algorithm we introduce a new realistic family of curves, $c$-packed curves, that is closed under simplification. We…
One approach to studying the Fr\'echet distance is to consider curves that satisfy realistic assumptions. By now, the most popular realistic assumption for curves is $c$-packedness. Existing algorithms for computing the Fr\'echet distance…
A new isoperimetric estimate is proved for embedded closed curves evolving by curve shortening flow, normalized to have total length $2\pi$. The estimate bounds the length of any chord from below in terms of the arc length between its…
We obtain a coarse relationship between geometric intersection numbers of curves and the sum of their subsurface projection distances with explicit quasi-constants. By using this relationship, we give applications in the studies of the…
Given two simplicial complexes in R^d, and start and end vertices in each complex, we show how to compute curves (in each complex) between these vertices, such that the Fr\'echet distance between these curves is minimized. As a polygonal…
We propose sublinear algorithms for probabilistic testing of the discrete and continuous Fr\'echet distance - a standard similarity measure for curves. We assume the algorithm is given access to the input curves via a query oracle: a query…
A finite sequence of equidistant samples (a sample train) of a periodic signal can be identified with a point in a multi-dimensional space. Such a point depends on the sampled signal, the sampling period, and the starting time of the…
We generalize Pach and de Zeeuw's bound for distinct distances between points on two curves, from algebraic curves to Pfaffian curves. Pfaffian curves include those that can be defined by any combination of elementary functions, including…
In this paper we present a monotonicity which extends a classical theorem of A. Schur comparing the chord length of a convex plane curve with a space curve of smaller curvature. We also prove a Schur's Theorem for spherical curves, which…
In this paper we introduce a geometric quantity, the $r$-multiplicity, that controls the length of a smooth curve as it evolves by curve shortening flow. The length estimates we obtain are used to prove results about the level set flow in…