计算几何

The manuscript presents a theoretical proof in conglomeration with new definitions on Inaccessibility and Inside for a point S related to a simple or self intersecting polygon P. The proposed analytical solution depicts a novel way of…

A technique is described for constructing three-dimensional vector graphics representations of planar regions bounded by cubic B\'ezier curves, such as smooth glyphs. It relies on a novel algorithm for compactly partitioning planar B\'ezier…

The problem of finding "small" sets that meet every straight-line which intersects a given convex region was initiated by Mazurkiewicz in 1916. We call such a set an {\em opaque set} or a {\em barrier} for that region. We consider the…

We consider the following problem: Given a finite set of straight line segments in the plane, determine the positions of a minimal number of points on the segments, from which guards can see all segments. This problem can be interpreted as…

We define the Reflexive Tile Assembly Model (RTAM), which is obtained from the abstract Tile Assembly Model (aTAM) by allowing tiles to reflect across their horizontal and/or vertical axes. We show that the class of directed temperature-1…

We prove that for any convex polygon $S$ with at least four sides, or a concave one with no parallel sides, and any $m>0$, there is an $m$-fold covering of the plane with homothetic copies of $S$ that cannot be decomposed into two…

The Discrete Morse Theory of Forman appeared to be useful for providing filtration-preserving reductions of complexes in the study of persistent homology. So far, the algorithms computing discrete Morse matchings have only been used for…

We present three new approximation algorithms with improved constant ratios for selecting $n$ points in $n$ disks such that the minimum pairwise distance among the points is maximized. (1) A very simple $O(n\log n)$-time algorithm with…

We define a notion for unfolding smooth, ruled surfaces, and prove that every smooth prismatoid (the convex hull of two smooth curves lying in parallel planes), has a nonoverlapping "volcano unfolding." These unfoldings keep the base…

The linear-time ham-sandwich cut algorithm of Lo, Matou\v{s}ek, and Steiger for bi-chromatic finite point sets in the plane works by appropriately selecting crossings of the lines in the dual line arrangement with a set of well-chosen…

Given a hypergraph $H$ with $m$ hyperedges and a set $X$ of $m$ \emph{pins}, i.e.\ globally fixed subspaces in Euclidean space $\mathbb{R}^d$, a \emph{pinned subspace-incidence system} is the pair $(H, X)$, with the constraint that each pin…

We derive tight expressions for the maximum number of $k$-faces, $0\le{}k\le{}d-1$, of the Minkowski sum, $P_1+...+P_r$, of $r$ convex $d$-polytopes $P_1,...,P_r$ in $\mathbb{R}^d$, where $d\ge{}2$ and $r<d$, as a (recursively defined)…

We consider the problem of covering the boundary of a simple polygon on n vertices using the minimum number of geodesic unit disks. We present an O(n \log^2 n+k) time 2-approximation algorithm for finding the centers of the disks, with k…

Several researchers proposed using non-Euclidean metrics on point sets in Euclidean space for clustering noisy data. Almost always, a distance function is desired that recognizes the closeness of the points in the same cluster, even if the…

Let $P$ be a set of $n$ points in $\mathbb{R}^d$ and $\mathcal{F}$ be a family of geometric objects. We call a point $x \in P$ a strong centerpoint of $P$ w.r.t $\mathcal{F}$ if $x$ is contained in all $F \in \mathcal{F}$ that contains more…

For $\cal C$ a collection of $n$ objects in $R^d$, let the packing and piercing numbers of $\cal C$, denoted by $Pack({\cal C})$, and $Pierce({\cal C})$, respectively, be the largest number of pairwise disjoint objects in ${\cal C}$, and…

Given an initial placement of a set of rectangles in the plane, we consider the problem of finding a disjoint placement of the rectangles that minimizes the area of the bounding box and preserves the orthogonal order i.e.\ maintains the…

We present a strict separation between the class of "mismatch free" self-assembly systems and general aTAM systems. Mismatch free systems are those systems in which concurrently grown parts must always agree with each other. Tile…

It is well-known that the number of non-crossing perfect matchings of $2k$ points in convex position in the plane is $C_k$, the $k$th Catalan number. Garc\'ia, Noy, and Tejel proved in 2000 that for any set of $2k$ points in general…

We extend linkage unfolding results from the well-studied case of polygonal linkages to the more general case of linkages of polygons. More precisely, we consider chains of nonoverlapping rigid planar shapes (Jordan regions) that are hinged…