Geometry Graphics Research Articles

Determining the coordinates of unmanned aerial vehicles

The purpose of the work was to develop a method for determining unmanned aerial vehicles’ (UAVs) coordinates, trajectory and parameters of their spatial movements by means of kinematic projection. The paper analyses the use of technologies for determining coordinates and spatial movements of aircraft trajectories. The principle of the kinematic scheme of fixing moving objects by means of orthogonal and kinematic design has been developed. The principle scheme of determining UAV coordinates by kinematic projection was studied. The theoretical mathematical dependencies of the determination and calculations of the coordinates of spatial movements of unmanned aerial vehicles have been practically verified. The main object of research in this work was the theory of kinematic design as a means of graphically displaying the patterns of spatial movements of objects. The subject of the study was the specific features of searching and recording the trajectories of spatial movements of unmanned aerial vehicles in order to determine the coordinates of their instantaneous location in space. In the process of conducting theoretical and experimental research, methods and techniques of physical and mathematical modelling of fast-moving processes and mathematical statistics of analysis and classification of their results were used. The basis of the experimental study was the theory of mapping coordinates and trajectories of spatial movements of moving objects by means of graphic geometry when combining classical orthogonal design with dynamic features of kinematic design. For an objective assessment of the results of the theoretical and experimental study of the dynamics of objects moving in space, the classical theory of research planning with a mathematical apparatus for processing their results was used. Graphical models of UAV coordinate fixation were made with the use of computer technology and the AutoCAD graphic editor software.

CDGT: Constructing diverse graph transformers for emotion recognition from facial videos

Recognizing expressions from dynamic facial videos can find more natural affect states of humans, and it becomes a more challenging task in real-world scenes due to pose variations of face, partial occlusions and subtle dynamic changes of emotion sequences. Existing transformer-based methods often focus on self-attention to model the global relations among spatial features or temporal features, which cannot well focus on important expression-related locality structures from both spatial and temporal features for the in-the-wild expression videos. To this end, we incorporate diverse graph structures into transformers and propose a CDGT method to construct diverse graph transformers for efficient emotion recognition from in-the-wild videos. Specifically, our method contains a spatial dual-graphs transformer and a temporal hyperbolic-graph transformer. The former deploys a dual-graph constrained attention to capture latent emotion-related graph geometry structures among local spatial tokens for efficient feature representation, especially for the video frames with pose variations and partial occlusions. The latter adopts a hyperbolic-graph constrained self-attention that explores important temporal graph structure information under hyperbolic space to model more subtle changes of dynamic emotion. Extensive experimental results on in-the-wild video-based facial expression databases show that our proposed CDGT outperforms other state-of-the-art methods.

The impact of graph theory and geometry based on some topics to develop critical thinking of secondary school mathematics’ students: A case of Mbulu district

The impact of graph theory and geometry based on some topics to develop critical thinking of secondary school mathematics’ students: A case of Mbulu district

Combinatorial Gray Codes — an Updated Survey

A combinatorial Gray code for a class of objects is a listing that contains each object from the class exactly once such that any two consecutive objects in the list differ only by a `small change'. Such listings are known for many different combinatorial objects, including bitstrings, combinations, permutations, partitions, triangulations, but also for objects defined with respect to a fixed graph, such as spanning trees, perfect matchings or vertex colorings. This survey provides a comprehensive picture of the state-of-the-art of the research on combinatorial Gray codes. In particular, it gives an update on Savage's influential survey [C. D. Savage. A survey of combinatorial Gray codes. SIAM Rev., 39(4):605--629, 1997.], incorporating many more recent developments. We also elaborate on the connections to closely related problems in graph theory, algebra, order theory, geometry and algorithms, which embeds this research area into a broader context. Lastly, we collect and propose a number of challenging research problems, thus stimulating new research endeavors.

Increasing the accuracy of finding the coordinates of sea mines by means of kinematic design

The issue of demining reservoirs, rivers and sea raids and ports, which is relevant for Europe during the period of active deployment of military confrontations on its territory, is considered. Only in the maritime territorial waters of Ukraine and in the water areas of its freshwater reservoirs and rivers, during the year of the war, the enemy laid up to a thousand different mines. Demining of the Ukrainian water spaces of the sea, reservoirs and rivers is an important social and economic problem, the solution of which does not tolerate delay and waiting for the full end of hostilities.
 The purpose of the work is the development of principle schemes for optimizing the spatial movements of search and mine clearance vessels and the creation of algorithms for calculating the precise coordinates of the spatial location of floating, anchor and bottom mines by means of kinematic design.
 The research methodology was the application of the theory of mapping coordinates and trajectories of spatial movements of moving objects by means of graphic geometry, taking into account the specifics and dynamic features of kinematic design.
 A schematic diagram of the spatial location of search trawlers and UAVs or auxiliary search floats when searching for mines has been developed. Mathematical dependencies have also been established for the precise calculation of the coordinates of detected mines based on the data of the applied theory and kinematic design schemes. For the search for floating mines, the optimal trajectory of search movements of unmanned aerial vehicles along the Archimedean spiral is proposed. The optimal number of aircraft that search for floating mines at the same time has been determined. As one of the most effective options for the disposal of mines, their detonation with warheads dropped from an aerial liquidator drone has been proposed.
 The main result of the study is the creation of a methodology for calculating the refined coordinates of the search for floating, anchor and bottom mines in combination with the optimization of the trajectories of the search movements of aerial and floating search vehicles along the Archimedean spiral. This ensures not only a 25-30% increase in surveyed and demined areas, but also a proportional saving of fuel due to a reduction in the number of crossings of search trawlers. It has been established that at speeds of spatial movements of search aircraft or auxiliary floating means of 5÷5.5 m/s, using the proposed search scheme, it is possible to survey up to 6 square kilometers of sea water area every hour. At the same time, it is possible to save up to a third of the cost of expensive fuel due to the reduction of movements of the trawler boat.

Increasing the accuracy of finding the coordinates of sea mines by means of kinematic design

The issue of demining reservoirs, rivers and sea raids and ports, which is relevant for Europe during the period of active deployment of military confrontations on its territory, is considered. Only in the maritime territorial waters of Ukraine and in the water areas of its freshwater reservoirs and rivers, during the year of the war, the enemy laid up to a thousand different mines. Demining of the Ukrainian water spaces of the sea, reservoirs and rivers is an important social and economic problem, the solution of which does not tolerate delay and waiting for the full end of hostilities. The purpose of the work is the development of principle schemes for optimizing the spatial movements of search and mine clearance vessels and the creation of algorithms for calculating the precise coordinates of the spatial location of floating, anchor and bottom mines by means of kinematic design. The research methodology was the application of the theory of mapping coordinates and trajectories of spatial movements of moving objects by means of graphic geometry, taking into account the specifics and dynamic features of kinematic design. A schematic diagram of the spatial location of search trawlers and UAVs or auxiliary search floats when searching for mines has been developed. Mathematical dependencies have also been established for the precise calculation of the coordinates of detected mines based on the data of the applied theory and kinematic design schemes. For the search for floating mines, the optimal trajectory of search movements of unmanned aerial vehicles along the Archimedean spiral is proposed. The optimal number of aircraft that search for floating mines at the same time has been determined. As one of the most effective options for the disposal of mines, their detonation with warheads dropped from an aerial liquidator drone has been proposed. The main result of the study is the creation of a methodology for calculating the refined coordinates of the search for floating, anchor and bottom mines in combination with the optimization of the trajectories of the search movements of aerial and floating search vehicles along the Archimedean spiral. This ensures not only a 25–30 % increase in surveyed and demined areas, but also a proportional saving of fuel due to a reduction in the number of crossings of search trawlers. It has been established that at speeds of spatial movements of search aircraft or auxiliary floating means of 5÷5.5 m/s, using the proposed search scheme, it is possible to survey up to 6 square kilometers of sea water area every hour. At the same time, it is possible to save up to a third of the cost of expensive fuel due to the reduction of movements of the trawler boat.

Symmetric Tangling of Honeycomb Networks

Symmetric, elegantly entangled structures are a curious mathematical construction that has found their way into the heart of the chemistry lab and the toolbox of constructive geometry. Of particular interest are those structures—knots, links and weavings—which are composed locally of simple twisted strands and are globally symmetric. This paper considers the symmetric tangling of multiple 2-periodic honeycomb networks. We do this using a constructive methodology borrowing elements of graph theory, low-dimensional topology and geometry. The result is a wide-ranging enumeration of symmetric tangled honeycomb networks, providing a foundation for their exploration in both the chemistry lab and the geometers toolbox.

Geometry of nontransitive graphs

In this note, we study non-transitive graphs and prove a number of results when they satisfy a coarse version of transitivity. Also, for each finitely generated group $G$, we produce continuum many pairwise non-quasi-isometric regular graphs that have the same growth rate, number of ends, and asymptotic dimension as $G$.

Key graph properties affecting transport efficiency of flip-flop Grover percolated quantum walks

Quantum walks exhibit properties without classical analogues. One of those is the phenomenon of asymptotic trapping---there can be nonzero probability of the quantum walker being localized in a finite part of the underlying graph indefinitely even though locally all directions of movement are assigned nonzero amplitudes at each step. We study quantum walks with the flip-flop shift operator and the Grover coin, where this effect has been identified previously. For the version of the walk further modified by a random dynamical disruption of the graph (percolated quantum walks) we provide a recipe for the construction of a complete basis of the subspace of trapped states allowing to determine the asymptotic probability of trapping for arbitrary finite connected simple graphs, thus significantly generalizing the previously known result restricted to planar 3-regular graphs. We show how the position of the source and sink together with the graph geometry and its modifications affect the excitation transport. This gives us a deep insight into processes where elongation or addition of dead-end subgraphs may surprisingly result in enhanced transport and we design graphs exhibiting this pronounced behavior. In some cases this even provides closed-form formulas for the asymptotic transport probability in dependence on some structure parameters of the graphs.

Research on Multi-Precision Fabric Modeling Method Based on Machine Learning

In order to solve the shortcomings of complex calculation and time consumption of fabric physical simulation methods, many single-precision fabric simulation technologies based on machine learning methods have emerged and the amount of calculation is increased for regions with small changes. Aiming at this problem, this paper proposes two multi-precision fabric modeling method based on machine learning. Firstly, the fabric mesh is simulated by the physical method, the initial position of the vertex is calculated, and the deformation of each region of the fabric is measured by Rayleigh quotient curvature, and the multi-precision fabric mesh is constructed. Secondly, the multi-precision fabric graph structure and geometry image are extracted from the multiprecision fabric mesh. Finally, the subgraph convolutional neural network and super-resolution network are trained to model the multi-precision fabric, and we compared the two different multi-precision fabric machine learning modeling methods. Through the experimental verification, in the garment modeling, the garment modeled by the subgraph convolutional neural network is no longer only dependent on the change of human joints, resulting in a more realistic effect. At the same time, the efficiency of the subgraph convolutional neural network is 25.3% higher than that of the single-precision garment modeling based on the machine learning method. In the cloth simulation, speed of the super-resolution network is nearly 16 times faster than that of the physical simulation, which supplements the imperfection of insufficient flexibility of the subgraph convolutional neural network modeling.

Tensor Networks and the Enumerative Geometry of Graphs

Tensor Networks and the Enumerative Geometry of Graphs

A comparison between analogic and digital media in Engineering teaching

Form representation is essential for any area that involves the creation of artefacts, since without it the project is just an idea in the mental field. In general terms, it is the ability to perceive and understand shapes and represent them through perspectives and orthographic views, as well as through models and geometric models. Given the issues raised, it is possible to understand the importance of knowledge related to the graphic representation of architecture for professionals involved with the Aeco Industry (Architecture, Engineering, Construction and Operation) and, also the relevance of graphic geometry disciplines on the development of Visiographic Three-dimensional Capacity. So, this paper aims to investigate the impact caused by the use of traditional and digital media in the teaching of graphic representation. To this end, two groups of students were compared, one exposed to the use of both media - analogic and digital - and another that used just digital media. From the application of questionnaires, it was possible to identify the student’s profile, their preferences of media and difficulties encountered. The work presumes that students exposed to more disciplines of graphic geometry will have less difficulties compared to those who had less contact. The main results indicated that: there is a preference, in both groups, for digital media to perform practical activities; the need to adapt the content to each specific engineering and that the most significant difficulties were related to work management, regardless of the media used.

HOMOTOPY CONTINUATION FOR THE SPECTRA OF PERSISTENT LAPLACIANS.

The p-persistent q-combinatorial Laplacian defined for a pair of simplicial complexes is a generalization of the q-combinatorial Laplacian. Given a filtration, the spectra of persistent combinatorial Laplacians not only recover the persistent Betti numbers of persistent homology but also provide extra multiscale geometrical information of the data. Paired with machine learning algorithms, the persistent Laplacian has many potential applications in data science. Seeking different ways to find the spectrum of an operator is an active research topic, becoming interesting when ideas are originated from multiple fields. In this work, we explore an alternative approach for the spectrum of persistent Laplacians. As the eigenvalues of a persistent Laplacian matrix are the roots of its characteristic polynomial, one may attempt to find the roots of the characteristic polynomial by homotopy continuation, and thus resolving the spectrum of the corresponding persistent Laplacian. We consider a set of simple polytopes and small molecules to prove the principle that algebraic topology, combinatorial graph, and algebraic geometry can be integrated to understand the shape of data.

Two-phase Stokes flow by capillarity in full 2D space: an approach via hydrodynamic potentials

We study the two-phase Stokes flow driven by surface tension with two fluids of equal viscosity, separated by an asymptotically flat interface with graph geometry. The flow is assumed to be two-dimensional with the fluids filling the entire space. We prove well-posedness and parabolic smoothing in Sobolev spaces up to critical regularity. The main technical tools are an analysis of nonlinear singular integral operators arising from the hydrodynamic single-layer potential and abstract results on nonlinear parabolic evolution equations.

A novel path planning methodology for automated valet parking based on directional graph search and geometry curve

The paper presents a novel path planning methodology based on the directional graph search and the geometry curve for the Automated Valet Parking (AVP) system. The whole path planning methodology is divided into three parts including the global path planning, the path coordination strategy and the parking path planning. Firstly, the global path planning is triggered to find a path from the parking slot entrance to the rough location of the assigned parking spot. A novel directional Hybrid A* algorithm is proposed to generate the global path efficiently without redundant searches, such as the dead end. Afterwards, the path coordination strategy gives a transitional path to connect the end node of the global path to the parking planning start node. The transitional path is composed of geometry curves including arcs and line segments based on the optimal parking start node. Finally, the parking path planning generates a parking path to guide the vehicle from parking start node to the parking space. A modified C-type vertical parking path planning algorithm is utilized to generate the parking path, offering flexibility for choosing the parking start node. Simulation results based on Matlab and PreScan show that it takes less time for the proposed path planning algorithm to generate a feasible path for the AVP system compared with the general planning algorithm. The novel AVP path planning algorithm also has the potential for practical use.

Geometry of the graphs of nonseparating curves: covers and boundaries

We investigate the geometry of the graphs of nonseparating curves for surfaces of finite positive genus with potentially infinitely many punctures. This graph has infinite diameter and is known to be Gromov hyperbolic by work of the author. We study finite covers between such surfaces and show that lifts of nonseparating curves to the nonseparating curve graph of the cover span quasiconvex subgraphs which are infinite diameter and not coarsely equal to the nonseparating curve graph of the cover. In the finite type case, we also reprove a theorem of Hamenstädt identifying the Gromov boundary with the space of ending laminations on full genus subsurfaces. We introduce several tools based around the analysis of bicorn curves and laminations which may be of independent interest for studying the geometry of nonseparating curve graphs of infinite type surfaces and their boundaries.

Terminal Triangles Centroid Algorithms for Quality Delaunay Triangulation

Two Lepp algorithms for quality Delaunay triangulation are discussed. Firstly a terminal triangles centroid Delaunay algorithm is studied. For each bad quality triangle t, the algorithm uses the longest edge propagating path (Lepp(t)) to find a couple of Delaunay terminal triangles (with largest angles less than or equal to 120 degrees) sharing a common longest (terminal) edge. Then the centroid of the terminal quadrilateral is Delaunay inserted in the mesh. Insertion of the midpoints of some constrained edges are also performed to assure convergence close to the constrained edges. We prove algorithm termination and that a graded, optimal size, 30 degrees triangulation is obtained, for any planar straight line graph (PSLG) geometry with constrained angles greater than or equal to 30 degrees. We also prove that the size of the final triangulation is optimal and that this size is independent of the processing order of the bad triangles in the mesh. Next, by introducing the concept of non-improvable triangles (with constrained angle < 30 degrees), we generalize the algorithm to deal with PSLG geometries with N small constrained angles. Thus given a triangle size parameter δ for non-improvable triangles, the generalized algorithm constructs a quality triangulation with non constrained angles ≥ 30 degrees and at most N non-improvable triangles of size δ (longest edge ≤δ). In practice the algorithms behave as predicted by the theory.

On the Bi-Lipschitz Geometry of Lamplighter Graphs

In this article we start a systematic study of the bi-Lipschitz geometry of lamplighter graphs. We prove that lamplighter graphs over trees bi-Lipschitzly embed into Hamming cubes with distortion at most~$6$. It follows that lamplighter graphs over countable trees bi-Lipschitzly embed into $\ell_1$. We study the metric behaviour of the operation of taking the lamplighter graph over the vertex-coalescence of two graphs. Based on this analysis, we provide metric characterizations of superreflexivity in terms of lamplighter graphs over star graphs or rose graphs. Finally, we show that the presence of a clique in a graph implies the presence of a Hamming cube in the lamplighter graph over it. An application is a characterization in terms of a sequence of graphs with uniformly bounded degree of the notion of trivial Bourgain-Milman-Wolfson type for arbitrary metric spaces, similar to Ostrovskii's characterization previously obtained in \cite{ostrovskii:11}.

1-Safe Petri Nets and Special Cube Complexes

Nielsen et al. [35] proved that every 1-safe Petri net N unfolds into an event structure E N . By a result of Thiagarajan [46], these unfoldings are exactly the trace-regular event structures. Thiagarajan [46] conjectured that regular event structures correspond exactly to trace-regular event structures. In a recent paper (Chalopin and Chepoi [12]), we disproved this conjecture, based on the striking bijection between domains of event structures, median graphs, and CAT(0) cube complexes. However, we proved that Thiagarajan’s conjecture is true for regular event structures whose domains are principal filters of universal covers of finite special cube complexes. In the current article, we prove the converse: To any finite 1-safe Petri net N , one can associate a finite special cube complex X N such that the domain of the event structure E N (obtained as the unfolding of N ) is a principal filter of the universal cover X̃ N of X N . This establishes a bijection between 1-safe Petri nets and finite special cube complexes and provides a combinatorial characterization of trace-regular event structures. Using this bijection and techniques from graph theory and geometry (MSO theory of graphs, bounded treewidth, and bounded hyperbolicity), we disprove yet another conjecture by Thiagarajan (from the paper with Yang [48]) that the monadic second-order logic of a 1-safe Petri net (i.e., of its event structure unfolding) is decidable if and only if its unfolding is grid-free. It was proven by Thiagarajan and Yang [48] that the MSO logic is undecidable if the unfolding is not grid-free. Our counterexample is the trace-regular event structure that arises from a virtually special square complex Z. The domain of this event structure Ė Z is the principal filter of the universal cover Z̃ of Z in which to each vertex we added a pendant edge. The graph of the domain of Ė Z has bounded hyperbolicity (and, thus, the event structure Ė Z is grid-free) but has infinite treewidth. Using results of Seese, Courcelle, and Müller and Schupp, we show that this implies that the MSO theory of the event structure Ė Z is undecidable.

On the local geometry of graphs in terms of their spectra

In this paper we consider the relation between the spectrum and the number of short cycles in large graphs. Suppose G1,G2,G3,… is a sequence of finite and connected graphs that share a common universal cover T and such that the proportion of eigenvalues of Gn that lie within the support of the spectrum of T tends to 1 in the large n limit. This is a weak notion of being Ramanujan. We prove such a sequence of graphs is asymptotically locally tree-like. This is deduced by way of an analogous theorem proved for certain infinite sofic graphs and unimodular networks, which extends results for regular graphs and certain infinite Cayley graphs.