Academic literature on the topic 'Apollonius diagrams'

This document shows an analysis of the second book Elementa Curvarum Linearum written by Jan De Witt, published for the first time in the second edition of Geometry [1]. This writing is considered the first analytical geometry textbook. The influence of the work carried out by Apollonius in his conics book is studied, the use and interpretation of diagrams is debated. The development of the analytical method and the generation of curves by means of movement are also studied. Some propositions were renewed versions in terms of eighteenth-century mathematics, they used symbology, algebraic techniques and curves were classified by means of their symbolic representations, in these propositions a work closer to Apollonius is seen, the conic is not generated, it is assumed its existence, its nature is geometric. The study concludes that, although the textbook was published in the second edition of Geometry, the genesis of the curves remains geometric. The conics appear as objects of study in action immersed in the symbolic and algebraic practice characteristic of the time.

Introduction. It is obvious that in the near future, the issues of equipping moving robotic systems with autonomous control elements will remain relevant. This requires the development of models of group pursuit. Note that optimization in pursuit tasks is reduced to the construction of optimal trajectories (shortest trajectories, trajectories with differential constraints, fuel consumption indicators). At the same time, the aspects of automated distribution by goals in group pursuit were not considered. To fill this gap, the presented piece of research has been carried out. Its result should be the construction of a model of automated distribution of pursuers by goals in group pursuit.Materials and Methods. A matrix was formed to study the multiple goal group pursuit. The control parameters for the movement of the pursuers were modified according to the minimum curvature of the trajectory. The methods of pursuit and approach were considered in detail. The possibilities of modifying the method of parallel approach were shown. Matrix simulation was used to build a scheme of multiple goal group pursuit. The listed processes were illustrated by functions in the given coordinate systems and animation. Block diagrams of the phase coordinates of the pursuer at the next step, the time and distance of the pursuer reaching the goal were constructed as a base of functions. In some cases, the location of targets and pursuers was defined as points on the circle of Apollonius. The matrix was formed by samples corresponding to the distribution of pursuers by goals.Results. Nine variants of the pursuit, parallel, proportional and three-point approach on the plane and in space were considered. The maximum value of the goal achievement time was calculated. There were cases when the speed vector of the pursuer was directed arbitrarily and to a point on the Apollonius circle. It was noted that the three-point approach method was convenient if the target was moving along a ballistic trajectory. To modify the method of parallel approach, a network of parallel lines was built on the plane. Here, the length of the arc of the line (which can be of any shape) and the array of reference points of the target trajectory were taken into account. An equation was compiled and solved with these elements. On an array of samples with corresponding time values, the minimum time was found, i.e., the optimal time for simultaneous group achievement of multiple goals was determined. For unified access to the library, the control vector was expressed through a one-parameter family of parallel planes. A library of calculations of control vectors was formed. An example of applying matrix simulation to group pursuit was shown. A scheme of group pursuit of multiple goals was presented. For two goals and three pursuers, six samples corresponding to the distribution of pursuers by goals were considered. The data was presented in the form of a matrix. Based on the research results, the computer program was created and registered – “Parallel Approach on Plane of Group of Pursuers with Simultaneous Achievement of the Goal”.Discussions and Conclusion. The methods of using matrices in modeling group pursuit were investigated. The possibility of modifying the method of parallel approach was shown. Matrix simulation of group pursuit enabled to build its scheme for a set of purposes. The matrix of the distribution of pursuers by goals would be generated at each moment of time. Methods of forming matrices of the distribution of pursuers and targets are of interest in the design of virtual reality systems, for tasks with simulating the process of group pursuit, escape, evasion. The dynamic programming method opens up the possibility of automating the distribution with optimization according to the specified parameters under the formation of the matrix of the distribution of pursuers by goals.

A preliminary study of an approach to internal structure generation used in lattice discrete particle models (LDPMs) [1]. The presented method used for particle generation and placement is intended to help realistically capture the internal structure of materials. First, a method for structure generation using LDPM is presented. Then, the method of particle generation using a Voronoi diagram [2] is described. The last part is the optimizations on the algorithm that use Apollonius circles to calculate the specific points of the Voronoi diagram.

A variety of advanced manufacturing processes have been developed based on the concept of rapid solidification processing (RSP), such as uniform droplet spraying (UDS) for the additive manufacturing of metals and alloys. This article introduces a morphological simulation of fractal dendric structures deposited by UDS of magnesium (Mg) alloys on two-dimensional (2D) planar sections. The fractal structure evolution is modeled as Apollonian packs of generalized ellipsoidal domains growing out of nuclei and dendrite arm fragments. The model employs descriptions of the dynamic thermal field based on superposed Green’s/Rosenthal functions with source images for initial/boundary effects, along with alloy phase diagrams and the classical solidification theory for nucleation and fragmentation rates. The initiation of grains is followed by their free and constrained growth by adjacent domains, represented via potential fields of level-set methods, for the effective mapping of the solidified topology and its metrics (grain size and fractal dimension of densely packed domains). The model is validated by comparing modeling results against micrographs of three UDS-deposited Mg–Zn–Y alloys. The further evolution of this real-time computational model and its application as a process observer for feedback control in 3D printing, as well as for off-line material design and optimization, is discussed.

Introduction. A kinematic model of group pursuit of a set of targets on a plane is considered. Pursuers use a technique similar to parallel approach method to achieve goals. Unlike the parallel approach method, the speed vectors of pursuers and targets are directed arbitrarily. In the parallel approach method, the instantaneous directions of movement of the pursuer and the target intersect at a point belonging to the circle of Apollonius. In the group model of pursuing multiple goals, the pursuers try to adhere to a network of predictable trajectories.Materials and Methods. The model sets the task of achieving goals by pursuers at designated points in time. This problem is solved by the methods of multidimensional descriptive geometry using the Radishchev diagram. The predicted trajectory is a composite line that moves parallel to itself when the target moves. On the projection plane “Radius of curvature — speed value”, the permissible speed range of the pursuer is displayed in the form of level lines (these are straight lines parallel to one of the projection planes). Images of speed level lines are displayed on the projection plane “Radius of curvature — time to reach the goal”. The search for points of intersection of the speed line images and the appointed time level line is being conducted. Along the communication lines, the values of the intersection points are lowered to the plane “Radius of curvature — speed value”. Using the obtained points, we construct an approximating curve and look for the intersection point with the line of the assigned speed. As a result, we get values of the radius of the circle at the predicted line of the trajectory of the pursuer.Results. Based on the results of the conducted research, test programs have been created, and animated images have been made in the computer mathematics system.Discussion and Conclusions. This method of constructing trajectories of pursuers to achieve a variety of goals at a given time values can be in demand by developers of autonomous unmanned aerial vehicles.

This article discusses a kinematic model of the problem of group pursuit of a set of goals. The article discusses a variant of the model when all goals are achieved simultaneously. In this model, the direction of the speeds by the pursuer can be arbitrary, in contrast to the method of parallel approach. In the method of parallel approach, the velocity vectors of the pursuer and the target are directed to a point on the Apollonius circle. The proposed pursuit model is based on the fact that the pursuer tries to follow the predicted trajectory of movement. The predicted trajectory is a compound curve. A compound curve consists of a circular arc and a straight line segment. The pursuer's velocity vector applied to the point where the pursuer is located touches the given circle. The straight line segment passes through the target point and touches the specified circle. The resulting compound line serves as an analogue of the line of sight in the parallel approach method. The iterative process of calculating the points of the pursuer's trajectory is that the next point of position is the point of intersection of the circle centered at the current point of the pursuer's position, with the line of sight corresponding to the point of the next position of the target. The radius of such a circle is equal to the product of the speed of the pursuer and the time interval corresponding to the time step of the iterative process. The time to reach the goal of each pursuer is a dependence on the speed of movement and the minimum radius of curvature of the trajectory. Multivariate analysis of the moduli of velocities and minimum radii of curvature of the trajectories of each of the pursuers for the simultaneous achievement of their goals is based on the methods of multidimensional descriptive geometry. To do this, the projection planes are entered on the Radishchev diagram: the radius of curvature of the trajectory and speed, the radius of curvature of the trajectory and the time to reach the goal. The optimizing factors are the set time for reaching the goal and the set value of the speed of the pursuer. This method of constructing the trajectories of pursuers to achieve a variety of goals at given time values may be in demand by the developers of autonomous unmanned aerial vehicles.

This article discusses a kinematic model of the problem of group pursuit of a set of goals. The article discusses a variant of the model when all goals are achieved simultaneously. And also the possibility is considered when the achievement of goals occurs at the appointed time. In this model, the direction of the speeds by the pursuer can be arbitrary, in contrast to the method of parallel approach. In the method of parallel approach, the velocity vectors of the pursuer and the target are directed to a point on the Apollonius circle. The proposed pursuit model is based on the fact that the pursuer tries to follow the predicted trajectory of movement. The predicted trajectory of movement is built at each moment of time. This path is a compound curve that respects curvature constraints. A compound curve consists of a circular arc and a straight line segment. The pursuer's velocity vector applied to the point where the pursuer is located touches the given circle. The straight line segment passes through the target point and touches the specified circle. The radius of the circle in the model is taken equal to the minimum radius of curvature of the trajectory. The resulting compound line serves as an analogue of the line of sight in the parallel approach method. The iterative process of calculating the points of the pursuer’s trajectory is that the next point of position is the point of intersection of the circle centered at the current point of the pursuer’s position, with the line of sight corresponding to the point of the next position of the target. The radius of such a circle is equal to the product of the speed of the pursuer and the time interval corresponding to the time step of the iterative process. The time to reach the goal of each pursuer is a dependence on the speed of movement and the minimum radius of curvature of the trajectory. Multivariate analysis of the moduli of velocities and minimum radii of curvature of the trajectories of each of the pursuers for the simultaneous achievement of their goals i based on the methods of multidimensional descriptive geometry. To do this, the projection planes are entered on the Radishchev diagram: the radius of curvature of the trajectory and speed, the radius of curvature of the trajectory and the time to reach the goal. On the first plane, the projection builds a one-parameter set of level lines corresponding to the range of velocities. In the second graph, corresponding to a given range of speeds, functions of the dependence of the time to reach the target on the radius of curvature. The preset time for reaching the target and the preset value of the speed of the pursuer are the optimizing factors. This method of constructing the trajectories of pursuers to achieve a variety of goals at given time values may be in demand by the developers of autonomous unmanned aerial vehicles.

L'étude structurelle de complexes moléculaires est nécessaire à la compréhension de leurs fonctionnements, mais aussi leur analyse à travers leur visualisation et illustration. La Surface Exclue au Solvant (SES) représente implicitement l'interaction entre un corps moléculaire et un solvant ce qui permet d'analyser géométriquement certaines de ses propriétés. Cette surface reste cependant complexe à construire notamment pour des structures de grandes tailles. Dans cette thèse, nous présentons ainsi une méthode de calcul sur GPU de la SES de grandes protéines. Les diagrammes d'Apollonius, ou diagrammes de Voronoï additivement pondérés, peuvent servir à étudier la structure des protéines, mais aussi construire la SES efficacement. Nous présentons une caractérisation mathématique de ces diagrammes permettant leur analyse et leur paramétrisation pour un calcul naïf, mais exhaustif, de leur géométrie. Enfin, sur la base de notre étude, nous proposons une méthode de calcul GPU de diagrammes d'Apollonius dans ℝ3 compatible avec des protéines de grandes tailles, mais aussi avec des distributions spatiales homogènes. Cette stratégie supporte les particularités des diagrammes d'Apollonius et permet le calcul exhaustif de leurs composantes
The study of the structure of large molecular systems through their visualization and illustration is needed to understand their features and the system they take part in. The Solvent Excluded Surface (SES) is the interaction surface between a protein and its environment. It then allows characterizing geometrically some of its properties. However, this surface is still complex to compute, especially for large molecular complex. In this thesis, we propose a dedicated GPU pipeline targeting fast and efficient computation of the SES of large proteins. Apollonius diagrams, or additively weighted Voronoi diagrams, can be used to analyze proteins structures and construct their surface. Then, we present a complete characterization of their facets which allows a parametrization supporting the exhaustive and naive computation of their geometry. Based on this study, we propose a method allowing the computation of Apollonius diagrams in ℝ3 which support the study of large proteins but is also compatible with uniform spatial distributions. Additionally, it allows an exhaustive computation of their facets