计算几何

We cluster a set of trajectories T using subtrajectories of T. Clustering quality may be measured by the number of clusters, the number of vertices of T that are absent from the clustering, and by the Fr\'{e}chet distance between…

Let $\tau$ and $\sigma$ be two polygonal curves in $\mathbb{R}^d$ for any fixed $d$. Suppose that $\tau$ and $\sigma$ have $n$ and $m$ vertices, respectively, and $m\le n$. While conditional lower bounds prevent approximating the Fr\'echet…

The $(k, z)$-Clustering problem in Euclidean space $\mathbb{R}^d$ has been extensively studied. Given the scale of data involved, compression methods for the Euclidean $(k, z)$-Clustering problem, such as data compression and dimension…

Let $\mathcal{T}$ be a set of $n$ flat (planar) semi-algebraic regions in $\mathbb{R}^3$ of constant complexity (e.g., triangles, disks), which we call plates. We wish to preprocess $\mathcal{T}$ into a data structure so that for a query…

We study the problem of perfect tiling in the plane and exploring the possibility of tiling a rectangle using integral distinct squares. Assume a set of distinguishable squares (or equivalently a set of distinct natural numbers) is given,…

In the problem "Localization and trilateration with the minimum number of landmarks", we faced the 3-Guard and classic Art Gallery Problems. The goal of the art gallery problem is to find the minimum number of guards within a simple polygon…

In this study, a geometric version of an NP-hard problem ("Almost $2-SAT$" problem) is introduced which has potential applications in clustering, separation axis, binary sensor networks, shape separation, image processing, etc. Furthermore,…

We study the problem of computing the diameter and the mean distance of a continuous graph, i.e., a connected graph where all points along the edges, instead of only the vertices, must be taken into account. It is known that for continuous…

The algorithm of Tutte for constructing convex planar straight-line drawings and the algorithm of Floater and Gotsman for constructing planar straight-line morphs are among the most popular graph drawing algorithms. In this paper, focusing…

A dichotomous ordinal graph consists of an undirected graph with a partition of the edges into short and long edges. A geometric realization of a dichotomous ordinal graph $G$ in a metric space $X$ is a drawing of $G$ in $X$ in which every…

Collision detection is a critical functionality for robotics. The degree to which objects collide cannot be represented as a continuously differentiable function for any shapes other than spheres. This paper proposes a framework for…

Assume that you have lost your puppy on an embedded graph. You can walk around on the graph and the puppy will run towards you at infinite speed, always locally minimizing the distance to your current position. Is it always possible for you…

We address the problem of computing the minimum number of triangles to separate a set of blue points from a set of red points in $\mathbb{R}^2$. A set of triangles is a \emph{separator} of one color from the other if every point of that…

We address the problem of object arrangement and scheduling for sequential 3D printing. Unlike the standard 3D printing, where all objects are printed slice by slice at once, in sequential 3D printing, objects are completed one after other.…

Pseudoline arrangements are fundamental objects in discrete and computational geometry, and different works have tackled the problem of improving the known bounds on the number of simple arrangements of $n$ pseudolines over the past…

In the field of computer graphics, conformal surface flattening has been widely studied for tasks such as texture mapping, geometry processing, and mesh generation. Typically, existing methods aim to flatten a given input geometry while…

We investigate saturated geometric drawings of graphs with geometric thickness $k$, where no edge can be added without increasing $k$. We establish lower and upper bounds on the number of edges in such drawings if the vertices lie in convex…

In the nearest neighbor problem, we are given a set $S$ of point sites that we want to store such that we can find the nearest neighbor of a (new) query point efficiently. In the dynamic version of the problem, the goal is to design a data…

The congestion of a curve is a measure of how much it zigzags around locally. More precisely, a curve $\pi$ is $c$-packed if the length of the curve lying inside any ball is at most $c$ times the radius of the ball, and its congestion is…

Given a point set $\mathcal{P}$ and a plane perfect matching $\mathcal{M}$ on $\mathcal{P}$, a flip is an operation that replaces two edges of $\mathcal{M}$ such that another plane perfect matching on $\mathcal{P}$ is obtained. Given two…