计算几何

We investigate the relation between the convergence of a sequence of lattices and the set-theoretic convergence of their corresponding Voronoi cells sequence. We prove that if a sequence of full rank lattices converges to a full rank…

We present a linear-time algorithm for finding the quadrilateral of largest area contained in a convex polygon, and we show that it is closely related to an old algorithm for the smallest enclosing parallelogram of a convex polygon.

Random graphs are mathematical models that have applications in a wide range of domains. We study the following model where one adds Erd\H{o}s--R\'enyi (ER) type perturbation to a random geometric graph. More precisely, assume…

In this paper, we study the minimum number of unique tile types required for the self-assembly of thin rectangles in Winfree's abstract Tile Assembly Model (aTAM), restricted to temperature-1. Using Catalan numbers, planar self-assembly and…

We consider the problem of localizing relevant subsets of non-rigid geometric shapes given only a partial 3D query as the input. Such problems arise in several challenging tasks in 3D vision and graphics, including partial shape similarity,…

The design and implementation of theoretically-sound robot motion planning algorithms is challenging. Within the framework of resolution-exact algorithms, it is possible to exploit soft predicates for collision detection. The design of soft…

We develop a new class of distances for objects including lines, hyperplanes, and trajectories, based on the distance to a set of landmarks. These distances easily and interpretably map objects to a Euclidean space, are simple to compute,…

In this paper we study the problem of finding hotspots, i.e. regions in which a moving entity has spent a significant amount of time, for polygonal trajectories. The fastest optimal algorithm, due to Gudmundsson, van Kreveld, and Staals…

We investigate discrete spin transformations, a geometric framework to manipulate surface meshes by controlling mean curvature. Applications include surface fairing -- flowing a mesh onto say, a reference sphere -- and mesh extrusion --…

We show that the union of $n$ translates of a convex body in $\mathbb{R}^3$ can have $\Theta(n^3)$ holes in the worst case, where a hole in a set $X$ is a connected component of $\mathbb{R}^3 \setminus X$. This refutes a 20-year-old…

The fabrication flexibility of 3D printing has sparked a lot of interest in designing structures with spatially graded material properties. In this paper, we propose a new type of density graded structure that is particularly designed for…

We consider path planning for a rigid spatial robot moving amidst polyhedral obstacles. Our robot is either a rod or a ring. Being axially-symmetric, their configuration space is R^3 x S^2 with 5 degrees of freedom (DOF). Correct, complete…

In [3], algorithms to compute the density of the distance to a random variable uniformly distributed in (a) a ball, (b) a disk, (c) a line segment, or (d) a polygone were introduced. For case (d), the algorithm, based on Green's theorem,…

This paper studies the $r$-range search problem for curves under the continuous Fr\'echet distance: given a dataset $S$ of $n$ polygonal curves and a threshold $r>0$, construct a data structure that, for any query curve $q$, efficiently…

We introduce a variant of the watchman route problem, which we call the quickest pair-visibility problem. Given two persons standing at points $s$ and $t$ in a simple polygon $P$ with no holes, we want to minimize the distance they travel…

We develop a novel algorithm for feature extraction in time series data by leveraging tools from topological data analysis. Our algorithm provides a simple, efficient way to successfully harness topological features of the attractor of the…

Given a set of points in the plane, a \emph{crossing family} is a collection of segments, each joining two of the points, such that every two segments intersect internally. Aronov et al. [Combinatorica,~14(2):127-134,~1994] proved that any…

We present an optimal, linear-time algorithm for the following version of terrain guarding: given a 1.5D terrain and a horizontal line, place the minimum number of guards on the line to see all of the terrain. We prove that the cardinality…

Let S be a set of n points and let w be a function that assigns non-negative weights to points in S. The additive weighted distance d_w(p, q) between two points p,q belonging to S is defined as w(p) + d(p, q) + w(q) if p \ne q and it is…

We study randomized algorithms for constrained optimization, in abstract frameworks that include, in strictly increasing generality: convex programming; LP-type problems; violator spaces; and a setting we introduce, consistent spaces. Such…