Related papers: Can we Compute the Similarity Between Surfaces?
The classical measure of similarity between two polygonal chains in Euclidean space is the Fr\'echet distance, which corresponds to the coordinated motion of two mobile agents along the chains while minimizing their maximum distance. As…
We introduce an arbitrary order, computationally efficient method to smooth corners on curves in the plane, as well as edges and vertices on surfaces in $\mathbb R^3$. The method is local, only modifying the original surface in a…
The discrete Fr\'echet distance is a useful similarity measure for comparing two sequences of points $P=(p_1,\ldots, p_m)$ and $Q=(q_1,\ldots,q_n)$. In many applications, the quality of the matching can be improved if we let $Q$ undergo…
A classical inequality which is due to Lickorish and Hempel says that the distance between two curves in the curve complex can be measured by their intersection number. In this paper, we show a converse version; the intersection number of…
Notions of graph similarity provide alternative perspective on the graph isomorphism problem and vice-versa. In this paper, we consider measures of similarity arising from mismatch norms as studied in Gervens and Grohe: the edit distance…
Riemann surfaces are among the simplest and most basic geometric objects. They appear as key players in many branches of physics, mathematics, and other sciences. Despite their widespread significance, how to compute distances between pairs…
Approximating complex curves with simple parametric curves is widely used in CAGD, CG, and CNC. This paper presents an algorithm to compute a certified approximation to a given parametric space curve with cubic B-spline curves. By…
We study the problem of polygonal curve simplification under uncertainty, where instead of a sequence of exact points, each uncertain point is represented by a region, which contains the (unknown) true location of the vertex. The regions we…
Traditional problems in computational geometry involve aspects that are both discrete and continuous. One such example is nearest-neighbor searching, where the input is discrete, but the result depends on distances, which vary continuously.…
Approximate near-neighbors search (\textsc{ANNS}) is a long-studied problem in computational geometry. %that has received considerable attention by researchers in the community. In this paper, we revisit the problem and propose the first…
The Fr\'echet distance is a similarity measure between two curves $A$ and $B$: Informally, it is the minimum length of a leash required to connect a dog, constrained to be on $A$, and its owner, constrained to be on $B$, as they walk…
Let $P$ be a polygonal curve in $\mathbb{R}^D$ of length $n$, and $S$ be a point set of size $k$. The Curve/Point Set Matching problem consists of finding a polygonal curve $Q$ on $S$ such that its Fr\'echet distance from $P$ is less than a…
We consider the problem of approximating a two-dimensional shape contour (or curve segment) using discrete assembly systems, which allow to build geometric structures based on limited sets of node and edge types subject to edge length and…
A sphere is a fundamental geometric object widely used in (computer aided) geometric design. It possesses rational parameterizations but no parametric polynomial parameterization exists. The present study provides an approach to the optimal…
We give necessary and sufficient topological conditions for a simple closed curve on a real rational surface to be approximable by smooth rational curves. We also study approximation by smooth rational curves with given complex…
We provide criteria for deciding whether a given planar curve is an image of a given spatial curve, obtained by a central or a parallel projection with unknown parameters. These criteria reduce the projection problem to a certain…
We study the problem of sub-trajectory nearest-neighbor queries on polygonal curves under the continuous Fr\'echet distance. Given an $n$ vertex trajectory $P$ and an $m$ vertex query trajectory $Q$, we seek to report a vertex-aligned…
Map matching is a common preprocessing step for analysing vehicle trajectories. In the theory community, the most popular approach for map matching is to compute a path on the road network that is the most spatially similar to the…
The (combinatorial) diameter of a polytope $P \subseteq \mathbb R^d$ is the maximum value of a shortest path between a pair of vertices on the 1-skeleton of $P$, that is the graph where the nodes are given by the $0$-dimensional faces of…
Starting with a similarity function between objects, it is possible to define a distance metric on pairs of objects, and more generally on probability distributions over them. These distance metrics have a deep basis in functional analysis,…