Given a set of points S⊆R2, a subset X⊆S with |X|=k is called k-gon if all points of X lie on the boundary of the convex hull of X, and k-hole if, in addition, no point of S∖X lies in the convex hull of X. We use computer assistance to show that every set of 17 points in general position admits two disjoint 5-holes, that is, holes with disjoint respective convex hulls. This answers a question of Hosono

A beacon b∈Rd is a point-shaped object in d-dimensional space that can exert a magnetic pull on any other point-shaped object p∈Rd. This object p then moves greedily towards b. The motion stops when p gets stuck at an obstacle or when p reaches b. By placing beacons inside a d-dimensional polyhedron P, we can implement a scheme to route point-shaped objects between any two locations in P. We can also

An ε-approximate incidence between a point and some geometric object (line, circle, plane, sphere) occurs when the point and the object lie at distance at most ε from each other. Given a set of points and a set of objects, computing the approximate incidences between them is a major step in many database and web-based applications in computer vision and graphics, including robust model fitting, approximate

In this paper we study the computational complexity of the Upward Planarity Extension problem, which takes as input an upward planar drawing ΓH of a subgraph H of a directed graph G and asks whether ΓH can be extended to an upward planar drawing of G. Our study fits into the line of research on the extensibility of partial representations, which has recently become a mainstream in Graph Drawing. We

We consider defining the embedding of a triangle mesh into R3, up to translation, rotation, and scale, by its vector of dihedral angles. On the theoretical side, we show that locally the map from realizable vectors of dihedrals to mesh embeddings is one-to-one almost everywhere. On the implementation side, we are interested in using the dihedral parameterization in shape analysis. This demands a way

Let P be a set of n points in the plane in general position. We show that at least ⌊n/3⌋ plane spanning trees can be packed into the complete geometric graph on P. This improves the previous best known lower bound Ω(n). Towards our proof of this lower bound we show that the center of a set of points, in the d-dimensional space in general position, is of dimension either 0 or d.

We consider a new trajectory planning problem involving a simple articulated probe. The probe is modeled as two line segments ab and bc, with a joint at the common point b, where bc is of fixed length r and ab is of arbitrarily large length. Initially, ab and bc are collinear. Given a set of obstacles in the form of n line segments and a target point t, the probe is to first be inserted in straight

The persistence diagram (PD) is an increasingly popular topological descriptor. By encoding the size and prominence of topological features at varying scales, the PD provides important geometric and topological information about a space. Recent work has shown that well-chosen (finite) sets of PDs can differentiate between geometric simplicial complexes, providing a method for representing complex shapes

A witness set W in a polygon P is a subset of P such that any set G⊂P that guards W is guaranteed to guard P. We study the problem of finding a minimum witness set for an orthogonal polygon under three models of orthogonal visibility. It is known that not all simple polygons admit a finite witness set under the traditional line-segment visibility and, if a polygon admits a finite minimal witness set

The range closest-pair (RCP) problem, as a range-search version of the classical closest-pair problem, aims to store a dataset of points in some data structure such that whenever a query range Q is given, the closest-pair inside Q can be reported efficiently. In this paper, we study the approximate version of the RCP problem with two approximation criteria. The first criterion is in terms of the query

The alpha complex, a subset of the Delaunay triangulation, has been extensively used as the underlying representation for biomolecular structures. We propose a GPU-based parallel algorithm for the computation of the alpha complex, which exploits the knowledge of typical spatial distribution and sizes of atoms in a biomolecule. Unlike existing methods, this algorithm does not require prior construction

In the minimum constraint removal problem, we are given a set of overlapping geometric objects as obstacles in the plane, and we want to find the minimum number of obstacles that must be removed to reach a target point t from the source point s by an obstacle-free path. The problem is known to be intractable and no sub-linear approximations are known even for simple obstacles such as rectangles and

Erdős-Beck's theorem states that n points in the plane with at most n−x points collinear define at least cxn lines for some positive constant c. It implies n points in the plane define Θ(n2) lines unless most of the points (i.e. n−o(n) points) are collinear. In this paper, we will present two ways to extend this result to higher dimensions. Given a set S of n points in Rd, we want to estimate a lower

We show that, as a consequence of a new result of Pór on universal Tverberg partitions, any large-enough set P of points in Rd has a (d+2)-sized subset whose Radon point has half-space depth at least cd⋅|P|, where cd∈(0,1) depends only on d. We then give two applications of this result. The first is to computing weak ϵ-nets by random sampling. The second is to show that given any set P of points in

We study the complexity of symmetric assembly puzzles: given a collection of simple polygons, can we translate, rotate, and possibly flip them so that their interior-disjoint union is line symmetric? On the negative side, we show that the problem is strongly NP-complete even if the pieces are all polyominos. On the positive side, we show that the problem can be solved in polynomial time if the number

We consider a repulsion actuator located in an n-sided convex environment full of point particles. When the actuator is activated, all the particles move away from the actuator. We study the problem of gathering all the particles to a point. We give an O(n2) time algorithm to compute all the actuator locations that gather the particles to one point with one activation, and an O(n) time algorithm to

It is proved that every series-parallel digraph whose maximum vertex degree is Δ admits an upward planar drawing with at most one bend per edge such that each edge segment has one of Δ distinct slopes. The construction is worst-case optimal in terms of the number of slopes, and it gives rise to drawings with optimal angular resolution πΔ. A variant of the drawing algorithm is used to show that (non-directed)

We investigate weighted straight skeletons from a geometric, graph-theoretical, and combinatorial point of view. We start with a thorough definition and shed light on some ambiguity issues in the procedural definition. We investigate the geometry, combinatorics, and topology of faces and the roof model, and we discuss in which cases a weighted straight skeleton is connected. Finally, we show that the

In this paper, we study an interesting geometric partition problem, called optimal field splitting, which arises in Intensity-Modulated Radiation Therapy (IMRT). In current clinical practice, a multi-leaf collimator (MLC) with a maximum leaf spread constraint is used to deliver the prescribed intensity maps (IMs). However, the maximum leaf spread of an MLC may require to split a large intensity map

Many properties of finite point sets only depend on the relative position of the points, e.g., on the order type of the set. However, many fundamental algorithms in computational geometry rely on coordinate representations. This includes the straightforward algorithms for finding a halving line for a given planar point set, as well as finding a point on the convex hull, both in linear time. In his

Given a set B of n black points in general position, we say that a set of white points W blocks B if in the Delaunay triangulation of [Formula: see text] there is no edge connecting two black points. We give the following bounds for the size of the smallest set W blocking B: (i) [Formula: see text] white points are always sufficient to block a set of n black points, (ii) if B is in convex position

We study the problem how to draw a planar graph crossing-free such that every vertex is incident to an angle greater than π. In general a plane straight-line drawing cannot guarantee this property. We present algorithms which construct such drawings with either tangent-continuous biarcs or quadratic Bézier curves (parabolic arcs), even if the positions of the vertices are predefined by a given plane

In this paper we present a package for implementing exact kinetic data structures built on objects which move along polynomial trajectories. We discuss how the package design was influenced by various considerations, including extensibility, support for multiple kinetic data structures, access to existing data structures and algorithms in CGAL, as well as debugging. Due to the similarity between the

For a set S of points in ℝ(d), an s-spanner is a subgraph of the complete graph with node set S such that any pair of points is connected via some path in the spanner whose total length is at most s times the Euclidean distance between the points. In this paper we propose a new sparse (1 + ε)-spanner with O(n/ε(d)) edges, where ε is a specified parameter. The key property of this spanner is that it