Computational Geometry
Preserving the topology during a digitization process is a requirement of first importance. To this end, it is classical in Digital Geometry to assume the shape borders to be par-regular. Par-regularity was proved to be equivalent to having…
For a graph $G$, a function $\psi$ is called a \emph{bar visibility representation} of $G$ when for each vertex $v \in V(G)$, $\psi(v)$ is a horizontal line segment (\emph{bar}) and $uv \in E(G)$ iff there is an unobstructed, vertical,…
A well-established theoretical model for modular robots in two dimensions are edge-connected configurations of square modules, which can reconfigure through so-called sliding moves. Dumitrescu and Pach [Graphs and Combinatorics, 2006]…
Given a simple polygon $P$ on $n$ vertices, and a set $D$ of $m$ pairwise intersecting geodesic disks in $P$, we show that five points in $P$ are always sufficient to pierce all the disks in $D$. This improves the previous bound of 14,…
Given a set P of n points in the plane, the unit-disk graph G_{r}(P) with respect to a parameter r is an undirected graph whose vertex set is P such that an edge connects two points p, q \in P if the Euclidean distance between p and q is at…
We present a scalable approach for range and $k$ nearest neighbor queries under computationally expensive metrics, like the continuous Fr\'echet distance on trajectory data. Based on clustering for metric indexes, we obtain a dynamic tree…
Let $P$ be a set of points in $\mathbb{R}^d$, where each point $p\in P$ has an associated transmission range $\rho(p)$. The range assignment $\rho$ induces a directed communication graph $\mathcal{G}_{\rho}(P)$ on $P$, which contains an…
One of the most fundamental and well-studied problems in Tile Self-Assembly is the Unique Assembly Verification (UAV) problem. This algorithmic problem asks whether a given tile system uniquely assembles a specific assembly. The complexity…
We find a family of planar bipartite graphs all of whose Lombardi drawings (drawings with circular arcs for edges, meeting at equal angles at the vertices) are nonplanar. We also find families of embedded series-parallel graphs and…
The Hilbert metric is a distance function defined for points lying within a convex body. It generalizes the Cayley-Klein model of hyperbolic geometry to any convex set, and it has numerous applications in the analysis and processing of…
Readability criteria, such as distance or neighborhood preservation, are often used to optimize node-link representations of graphs to enable the comprehension of the underlying data. With few exceptions, graph drawing algorithms typically…
Three-dimensional $N^{th}$ order nodal Lagrangian tetrahedral finite elements ($P_N$ elements) can be generated using Pascal's tetrahedron $\mathcal{H}$ where each node in 3D element space corresponds to an entry in $\mathcal{H}$. For the…
Given a simple polygon $\cal P$, in the Art Gallery problem the goal is to find the minimum number of guards needed to cover the entire $\cal P$, where a guard is a point and can see another point $q$ when $\overline{pq}$ does not cross the…
In the Geometric Median problem with outliers, we are given a finite set of points in d-dimensional real space and an integer m, the goal is to locate a new point in space (center) and choose m of the input points to minimize the sum of the…
Given two points A,B in the plane, the locus of all points P for which the angles at A and B in the triangle A,B,P have a constant sum is a circular arc, by Thales' theorem. We show that the difference of these angles is kept a constant by…
We analyze human poses and motion by introducing three sequences of easily calculated surface descriptors that are invariant under reparametrizations and Euclidean transformations. These descriptors are obtained by associating to each…
We present an algorithm that enumerates and classifies all edge-to-edge gluings of unit squares that correspond to convex polyhedra. We show that the number of such gluings of $n$ squares is polynomial in $n$, and the algorithm runs in time…
A family of closed simple (i.e., Jordan) curves is $m$-intersecting if any pair of its curves have at most $m$ points of common intersection. We say that a pair of such curves touch if they intersect at a single point of common tangency. In…
We give a computer-based proof of the following fact: If a square is divided into seven or nine convex polygons, congruent among themselves, then the tiles are rectangles.
We introduce a fast and robust algorithm for finding a plane $\Gamma$ with given normal $\vec{n}_\Gamma$, which truncates an arbitrary polyhedron $\mathcal{P}$ such that the remaining sub-polyhedron admits a given volume…