The flow complex: a data structure for geometric modeling

Overview maps are a highly demanded feature in Massively Multiplayer Online Games. Current server-based techniques will be replaced by distributed networks due to scalability and investment costs. In a P2P MMOG it is more complex to calculate a local histogram map for every player in the virtual environment than with a centralized approach. In this work a process is proposed which aggregates local information to a 2-dimensional histogram for all players. The aggregation performs on a dynamic Delaunay triangulation network with compass routing in a clustered virtual space. It is shown that the aggregation is consistent and the Delaunay triangulation can be maintained with the proposed dislocation limitation algorithm. Furthermore, it was mathematically proved that the localized histogram aggregation process is asymptotically average optimal in message complexity.

Formation control of multiple autonomous agents or a swarm of robots have become popular in robotics. Formation control is to maintain specific connections among multiple autonomous robots while performing tasks such as traversing trajectories, exploring environments, or covering spaces. We assume that robots are virtually connected by Delaunay triangulation, which is a general form of acute angle connection. This virtual connection forms a spring network, and we find tight upper bounds of the swarm size when we apply quadratic potential fields to the group of robots for the position and formation control. We present a method of Delaunay triangulation using circle bounds. By assuming hexagonal shape of each cell of swarm connection, we present a method of predicting a tight size of an ellipse that encircles the swarm.

Computing the shape of a planar points set

Given a triangulation of a point set in the plane, a flip deletes an edge e whose removal leaves a convex quadrilateral, and replaces e by the opposite diagonal of the quadrilateral. It is well known that any triangulation of a point set can be reconfigured to any other triangulation by some sequence of flips. We explore this question in the setting where each edge of a triangulation has a label, and a flip transfers the label of the removed edge to the new edge. It is not true that every labelled triangulation of a point set can be reconfigured to every other labelled triangulation via a sequence of flips, but we characterize when this is possible. There is an obvious necessary condition: for each label l, if edge e has label l in the first triangulation and edge f has label l in the second triangulation, then there must be some sequence of flips that moves label l from e to f, ignoring all other labels. Bose, Lubiw, Pathak and Verdonschot formulated the Orbit Conjecture, which states that this necessary condition is also sufficient, i.e. that all labels can be simultaneously mapped to their destination if and only if each label individually can be mapped to its destination. We prove this conjecture. Furthermore, we give a polynomial-time algorithm to find a sequence of flips to reconfigure one labelled triangulation to another, if such a sequence exists, and we prove an upper bound of O(n^7) on the length of the flip sequence. Our proof uses the topological result that the sets of pairwise non-crossing edges on a planar point set form a simplicial complex that is homeomorphic to a high-dimensional ball (this follows from a result of Orden and Santos; we give a different proof based on a shelling argument). The dual cell complex of this simplicial ball, called the flip complex, has the usual flip graph as its 1-skeleton. We use properties of the 2-skeleton of the flip complex to prove the Orbit Conjecture.

Sylvester [7] proposed the following question in 1893. If a finite set of points in a plane is such that on the line determined by any two points of the set there is always a third point of the set, is the set collinear? Equivalently, given a finite planar set of non-collinear points, does there exist a line containing exactly two of the points?

In this paper we present new optimality results for the Delaunay triangulation of a set of points in ℝ d . These new results are true in all dimensionsd. In particular, we define a power function for a triangulation and show that the Delaunay triangulation minimizes the power function over all triangulations of a point set. We use this result to show that (a) the maximum min-containment radius (the radius of the smallest sphere containing the simplex) of the Delaunay triangulation of a point set in ℝ d is less than or equal to the maximum min-containment radius of any other triangulation of the point set, (b) the union of circumballs of triangles incident on an interior point in the Delaunay triangulation of a point set lies inside the union of the circumballs of triangles incident on the same point in any other triangulation of the point set, and (c) the weighted sum of squares of the edge lengths is the smallest for Delaunay triangulation, where the weight is the sum of volumes of the triangles incident on the edge. In addition we show that if a triangulation consists of only self-centered triangles (a simplex whose circumcenter falls inside the simplex), then it is the Delaunay triangulation.

The notion of a centerpoint of a finite set of points in two and higher dimensions is a generalisation of the concept of the median of a (finite) set of points on the real line. In this paper, we present an algorithm for computing a centerpoint of a set of n points in the plane. The algorithm has complexity O(n) which significantly improves the O(n log3 n) complexity of the previously best known algorithm. We use suitable modifications of the ham-sandwich-cut algorithm and the prune-and-search technique to achieve this improvement.

Determination of minimum number of sensors and their locations for an automated facility: An algorithmic approach

Paramagnetic effects are valuable structural restraints in study of structures, dynamics and interactions of proteins by means of NMR spectroscopy. Site-specific tagging of proteins with paramagnetic ions is generally required to achieve these paramagnetic effects. In this review, we first introduce the concept of paramagnetic effects including paramagnetic relaxation enhancement (PRE), pseudocontact shift (PCS) and residual dipolar couplings (RDCs), which are generally used in biomolecular NMR. Thereafter, strategies of site-specific paramagnetic tagging of proteins are outlined, and the paramagnetic effects are evaluated. On the basis of paramagnetic tagging methods described above, the applications of paramagnetic NMR in characterization of multi-domain protein replacement, 3D structural determination of unstable and low-abundance enzyme intermediate complex in real time condition, and 3D structural determination of proteins in live cells using PCSs are introduced. NMR optimized paramagnetic tags suitable for achieving high spatial resolution of distances determined by double electron-electron resonance (DEER) are reviewed. In the end, the applications of DEER measurement in characterization of protein structures in cells are demonstrated. Paramagnetic NMR and electron paramagnetic resonance (EPR) in characterization of proteins in cells is shown as a powerful technique in structural biology and chemical biology, and the development of biological compatible methods of protein modifications suitable for NMR and EPR applications are in high demand. In summary, paramagnetic tagging proteins are a powerful tool in elucidation of protein structures, dynamics and interactions in chemical and structural biology and it is surely to find wide applications in cells.

If $$\mathbb{X}$$ is a finite set of points in a multiprojective space $$\mathbb{P}^{n_1 } \times \cdots \times \mathbb{P}^{n_r } $$ withr ≥ 2, then $$\mathbb{X}$$ may or may not be arithmetically CohenMacaulay (ACM). For sets of points in ℙ1 × ℙ1 there are several classifications of the ACM sets of points. In this paper we investigate the natural generalizations of these classifications to an arbitrary multiprojective space.We show that each classification for ACM points in ℙ1 × ℙ1 fails to extend to the general case. We also give some new necessary and sufficient conditions for a set of points to be ACM

It is difficult to extract the boundary of complex planar points with nonuniform distribution of point density, concave envelopes, and holes. To solve this problem, an algorithm is proposed in this paper. Based on Delaunay triangulation, the maximum boundary angle threshold is introduced as the parameter in the extraction of the rough boundary. Then, the point looseness threshold is introduced, and the fine boundary extraction is conducted for the local areas such as concave envelopes and holes. Finally, the complete boundary result of the whole point set is obtained. The effectiveness of the proposed algorithm is verified by experiments on the simulated point set and practical measured point set. The experimental results indicate that it has wider applicability and more effectiveness in engineering applications than the state‐of‐the‐art boundary construction algorithms based on Delaunay triangulation.

On the boundary of a finite set of points in the plane

The discrete multi-agent rendezvous problem we consider in this paper is concerned with a specified set of points in the plane, called dwell-points, and a set of mobile autonomous agents with limited sensing range. Each agent is initially positioned at some dwell-point, and is able to determine its distances to dwell-points within sensing range, and also the bearings of those dwell-points with respect to a local reference of the agent. An agent can also determine if each dwell-point within sensing range is occupied by at least one agent. We do not, however, assume that an agent can count the number of agents occupying a dwell-point. We say the agents have rendezvoused if all the agents have moved to a set of dwell-points all within sensing radius of each other. The goal is to devise distributed motion strategies for each agent which will cause all the agents to rendezvous in the sense just defined, without any active communication among the agents. We propose a distributed motion rule and use graphs to characterize a class of dwell-points in the plane for which rendezvous is guaranteed under the rule.

We consider the problem of surface reconstruction of a geometric object from a finite set of sample points with normals. Our contribution is to present a new scheme for implicit surface reconstruction. Similarly to the multilevel partition of unity (MPU) method we hierarchically divide the domain obtaining local approximation for the object on each part, and then patch all together obtaining a global description of the object. Our new scheme uses ridge regression and weighted gradient one fitting techniques to get better stability on local approximations. The method behaves reasonably on sparse set of points and data with holes as those which comes from 3D scanning of real objects.

Given a finite set of points in R3, polyhedronization deals with constructing a simple polyhedron such that the vertices of the polyhedron are precisely the given points. In this paper, we present randomized approximation algorithms for minimal volume polyhedronization (MINVP) and maximal volume polyhedronization (MAXVP) of three dimensional point sets in general position. Both, MINVP and MAXVP, problems have been shown to be NP-hard and to the best of our knowledge, no practical algorithms exist to solve these problems. It has been shown that for any point set S in R3, there always exists a tetrahedralizable polyhedronization of S. We exploit this fact to develop a greedy heuristic for MINVP and MAXVP constructions. Further, we present an empirical analysis on the quality of the approximation results of some well defined point sets. The algorithms have been validated by comparing the results with the optimal results generated by an exhaustive searching (brute force) method for MINVP and MAXVP for some well chosen point sets of smaller sizes. Finally, potential applications of minimum and maximum volume polyhedra in 4D printing and surface lofting, respectively, have been discussed.