Use of Mechanisms Marking Centers of Simplexes in Their 2-Dimensional Projections as Axonographs of Multidimensional Spaces

2021 ◽  
Vol 8 (4) ◽  
pp. 13-23
Author(s):  
Sherzod Abdurahmanov
Keyword(s):  
Dimensional Space ◽  
Spatial Location ◽  
Straight Line Segment ◽  
Barycentric Coordinates ◽  
Parallel Projection ◽  
Problem Solution ◽  
Similarity Coefficients ◽  
Translational Movement ◽  
Straight Line ◽  
Multidimensional Objects

A brief historical excursion into the graphics of geometry of multidimensional spaces at the paper beginning clarifies the problem – the necessary to reduce the number of geometric actions performed when depicting multidimensional objects. The problem solution is based on the properties of geometric figures called N- simplexes, whose number of vertices is equal to N + 1, where N expresses their dimensionality. The barycenter (centroid) of the N-simplex is located at the point that divides the straight-line segment connecting the centroid of the (N–1)-simplex contained in it with the opposite vertex by 1: N. This property is preserved in the parallel projection (axonometry) of the simplex on the drawing plane, that allows the solution of the problem of determining the centroid of the simplex in its axonometry to be assigned to a mechanism which is a special Assembly of pantographs (the author's invention) with similarity coefficients 1:1, 1:2, 1:3, 1:4,...1:N. Next, it is established, that the spatial location of a point in N-dimensional space coincides with the centroid of the simplex, whose vertices are located on the point’s N-fold (barycentric) coordinates. In axonometry, the ends of both first pantograph’s links and the ends of only long links of the remaining ones are inserted into points indicating the projections of its barycentric coordinates and the mechanism node, which serves as a determinator, graphically marks the axonometric location of the point defined by its coordinates along the axes х1, х2, х3 … хN.. The translational movement of the support rods independently of each other can approximate or remote the barycentric coordinates of a point relative to the origin of coordinates, thereby assigning the corresponding axonometric places to the simplex barycenter, which changes its shape in accordance with its points’ occupied places in the coordinate axes. This is an axonograph of N-dimensional space, controlled by a numerical program. The last position indicates the possibility for using the equations of multidimensional spaces’ geometric objects given in the corresponding literature for automatic drawing when compiling such programs.

scholarly journals Extrapolation of Functions of Many Variables by Means of Metric Analysis

2018 ◽  
Vol 173 ◽  
pp. 03014 ◽  
Author(s):  
Alexandr Kryanev ◽  
Victor Ivanov ◽  
Anastasiya Romanova ◽  
Leonid Sevastianov ◽  
David Udumyan
Keyword(s):  
Dimensional Space ◽  
Straight Line Segment ◽  
Interpolation Scheme ◽  
Second Stage ◽  
Metric Analysis ◽  
Several Variables ◽  
Straight Line ◽  
Extrapolation Problem ◽  
Two Stages ◽  
Auto Regression

The paper considers a problem of extrapolating functions of several variables. It is assumed that the values of the function of m variables at a finite number of points in some domain D of the m-dimensional space are given. It is required to restore the value of the function at points outside the domain D. The paper proposes a fundamentally new method for functions of several variables extrapolation. In the presented paper, the method of extrapolating a function of many variables developed by us uses the interpolation scheme of metric analysis. To solve the extrapolation problem, a scheme based on metric analysis methods is proposed. This scheme consists of two stages. In the first stage, using the metric analysis, the function is interpolated to the points of the domain D belonging to the segment of the straight line connecting the center of the domain D with the point M, in which it is necessary to restore the value of the function. In the second stage, based on the auto regression model and metric analysis, the function values are predicted along the above straight-line segment beyond the domain D up to the point M. The presented numerical example demonstrates the efficiency of the method under consideration.

Method of Rotation of Geometrical Objects Around the Curvilinear Axis

2017 ◽  
Vol 5 (3) ◽  
pp. 45-50 ◽  
Author(s):  
И. Беглов ◽  
I. Beglov ◽  
Вячеслав Рустамян ◽  
Vyacheslav Rustamyan
Keyword(s):  
Rotation Axis ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Straight Line Segment ◽  
Finite Radius ◽  
Straight Line ◽  
Nontrivial Center ◽  
Dupin Cyclides ◽  
Geometric Techniques

Rotation is the motion of geometric objects along a circle. This is one of geometric techniques used to form lines and surfaces. In this paper has been considered the rotation of objects in a three-dimensional space around a straight axis. It is known that a straight line can be considered as a particular case of a circle with a radius equal to infinity. Such circle’s center is at infinite distance from the considered straight line segment. Then in the general case, the rotation axis is a closed curve, for example, a circle with a radius of finite magnitude. Rotation of a point around a straight axis now splits into two trajectories. One of them is a circle with a radius, the second is a straight line crossing with the axis, and the center of this trajectory is at an infinite distance from the point. The method of point rotation about an axis of finite radius was considered. Note that a circle is a special case of an ellipse. When the actual focus of the circle is stratified into two, the line itself loses its curvature constancy, and is called an ellipse. The point, rotating around the elliptical axis, is stratified into four ones, forming four circles (trajectories). Axis foci appearing in turn in the role of the main one determine two trajectories by each with a trivial and nontrivial center of rotation. We have considered the variant for arrangement of the generating circle so that its center coincided with one of the elliptic axis’s foci. The obtained surfaces are a pair of co-axial Dupin cyclides, since they have identical properties. Changing the circle generatrix radius, other things being equal, we get different types of closed cyclides.

scholarly journals EPCO-31. EPIGENOMIC INTRATUMORAL HETEROGENEITY OF GLIOBLASTOMA IN THREE-DIMENSIONAL SPACE

2020 ◽  
Vol 22 (Supplement_2) ◽  
pp. ii76-ii76
Author(s):  
Radhika Mathur ◽  
Sriranga Iyyanki ◽  
Stephanie Hilz ◽  
Chibo Hong ◽  
Joanna Phillips ◽  
...  
Keyword(s):  
Spatial Organization ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Spatial Location ◽  
Therapy Resistance ◽  
Regulatory Elements ◽  
Chromatin Accessibility ◽  
Intratumoral Heterogeneity ◽  
Tumor Evolution ◽  
Open Chromatin

Abstract Treatment failure in glioblastoma is often attributed to intratumoral heterogeneity (ITH), which fosters tumor evolution and generation of therapy-resistant clones. While ITH in glioblastoma has been well-characterized at the genomic and transcriptomic levels, the extent of ITH at the epigenomic level and its biological and clinical significance are not well understood. In collaboration with neurosurgeons, neuropathologists, and biomedical imaging experts, we have established a novel topographical approach towards characterizing epigenomic ITH in three-dimensional (3-D) space. We utilize pre-operative MRI scans to define tumor volume and then utilize 3-D surgical neuro-navigation to intra-operatively acquire 10+ samples representing maximal anatomical diversity. The precise spatial location of each sample is mapped by 3-D coordinates, enabling tumors to be visualized in 360-degrees and providing unprecedented insight into their spatial organization and patterning. For each sample, we conduct assay for transposase-accessible chromatin using sequencing (ATAC-Seq), which provides information on the genomic locations of open chromatin, DNA-binding proteins, and individual nucleosomes at nucleotide resolution. We additionally conduct whole-exome sequencing and RNA sequencing for each spatially mapped sample. Integrative analysis of these datasets reveals distinct patterns of chromatin accessibility within glioblastoma tumors, as well as their associations with genetically defined clonal expansions. Our analysis further reveals how differences in chromatin accessibility within tumors reflect underlying transcription factor activity at gene regulatory elements, including both promoters and enhancers, and drive expression of particular gene expression sets, including neuronal and immune programs. Collectively, this work provides the most comprehensive characterization of epigenomic ITH to date, establishing its importance for driving tumor evolution and therapy resistance in glioblastoma. As a resource for further investigation, we have provided our datasets on an interactive data sharing platform – The 3D Glioma Atlas – that enables 360-degree visualization of both genomic and epigenomic ITH.

scholarly journals Constant-Time Complete Visibility for Robots with Lights: The Asynchronous Case

Algorithms ◽  
2021 ◽  
Vol 14 (2) ◽  
pp. 56
Author(s):  
Gokarna Sharma ◽  
Ramachandran Vaidyanathan ◽  
Jerry L. Trahan
Keyword(s):  
Mobile Robots ◽  
Line Segment ◽  
Autonomous Robots ◽  
Straight Line Segment ◽  
Autonomous Mobile Robots ◽  
Asymptotically Optimal ◽  
Time Translation ◽  
Straight Line ◽  
A New Technique ◽  
Colored Lights

We consider the distributed setting of N autonomous mobile robots that operate in Look-Compute-Move (LCM) cycles and use colored lights (the robots with lights model). We assume obstructed visibility where a robot cannot see another robot if a third robot is positioned between them on the straight line segment connecting them. In this paper, we consider the problem of positioning N autonomous robots on a plane so that every robot is visible to all others (this is called the Complete Visibility problem). This problem is fundamental, as it provides a basis to solve many other problems under obstructed visibility. In this paper, we provide the first, asymptotically optimal, O(1) time, O(1) color algorithm for Complete Visibility in the asynchronous setting. This significantly improves on an O(N)-time translation of the existing O(1) time, O(1) color semi-synchronous algorithm to the asynchronous setting. The proposed algorithm is collision-free, i.e., robots do not share positions, and their paths do not cross. We also introduce a new technique for moving robots in an asynchronous setting that may be of independent interest, called Beacon-Directed Curve Positioning.

An Approach to Solving Spatial-Allocation Models with Constraints

1979 ◽  
Vol 11 (1) ◽  
pp. 3-22 ◽  
Author(s):  
I N Williams
Keyword(s):  
Urban Economics ◽  
Dimensional Space ◽  
Spatial Interaction ◽  
Spatial Location ◽  
Type Model ◽  
Model Framework ◽  
Allocation Model ◽  
Spatial Allocation ◽  
Practical Applications ◽  
Wide Range

This paper introduces a loose-knit family of spatial-allocation models, which locate entities in two-dimensional space, based on a general framework which merges an input—output type model with a spatial-interaction type model. Explicit attention is paid to the solution and interpretation of constraints on the subtotals generated within these models. In this way a link is forged between the fields of land-use modelling and urban economics. One efficient method of solving a particular form of spatial-allocation model is described in detail and some characteristics of this and alternative approaches are discussed. Four practical applications of the spatial-allocation model framework are outlined to demonstrate its wide range of usefulness in representing spatial-location processes.

RADISHCHEV PLOT IN MULTIVARIATE ANALYSIS OF THE PROBLEM OF MULTIPLE TARGETS GROUP PURSUIT

2021 ◽  
pp. 22-31
Author(s):  
A. A. Dubanov
Keyword(s):  
Multivariate Analysis ◽  
Line Segment ◽  
Iterative Process ◽  
Straight Line Segment ◽  
Line Of Sight ◽  
Radius Of Curvature ◽  
Time Step ◽  
Minimum Radius ◽  
Group Pursuit ◽  
Straight Line

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.

A Direct Least Squares Method for Camera Pose Estimation Based on Straight Line Segment Correspondences

2015 ◽  
Vol 35 (6) ◽  
pp. 0615003
Author(s):  
李鑫 Li Xin ◽  
张跃强 Zhang Yueqiang ◽  
刘进博 Liu Jinbo ◽  
张小虎 Zhang Xiaohu ◽  
于起峰 Yu Qifeng
Keyword(s):  
Least Squares ◽  
Pose Estimation ◽  
Line Segment ◽  
Least Squares Method ◽  
Straight Line Segment ◽  
Camera Pose Estimation ◽  
Straight Line ◽  
Camera Pose

Estimation of Cumulative Fatigue Damage Under Random Loading

1976 ◽  
Vol 98 (1) ◽  
pp. 348-353 ◽  
Author(s):  
A. K. Abu-Akeel
Keyword(s):  
Fatigue Damage ◽  
Computation Time ◽  
Straight Line Segment ◽  
Computer Applications ◽  
Accurate Estimation ◽  
Random Loading ◽  
Structural Element ◽  
Load Curve ◽  
Straight Line ◽  
Cumulative Fatigue Damage

A method is presented that leads to accurate estimation of the cumulative fatigue damage incurred in a randomly loaded structural element when loading is given in the form of spectral density load, or stress, plots. The load plots are here approximated by a series of straight lines and a closed formula is obtained to yield the damage incurred by the load within each straight line segment. The method avoids the errors that result from human misjudgment in the commonly used curve-stepping approach. It is also adaptable for computer applications and can be incorporated in a stress calculation program to save on computation time. In comparison to curve stepping, five straight-line segments may give the same accuracy as a hundred curve steps. This contrast, however, depends on the degree of irregularity of the load curve.

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