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.

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.

Three-dimensional extension of Bresenham's algorithm and its application in straight-line interpolation

2002 ◽  
Vol 216 (3) ◽  
pp. 459-463 ◽  
Author(s):  
X-W Liu ◽  
K Cheng
Keyword(s):  
Line Segment ◽  
Three Dimensional ◽  
Straight Line Segment ◽  
Numerical Control ◽  
Interpolation Algorithm ◽  
Straight Line ◽  
Actual Error ◽  
Potential Applications ◽  
Sample Position ◽  
Straight Lines

Conventional straight-line generating algorithms, such as the digital differential analyser (DDA), Bresenham's algorithm and the mid-point algorithm, are suitable only for planer straight lines on the coordinate planes, of which Bresenham's algorithm is the most efficient. In this paper, the authors have extended Bresenham's algorithm to spatial straight lines. Given a spatial straight-line segment with two end-points, the authors have applied Bresenham's algorithm to the projections of the line segment on two of the three coordinate planes, which is determined by the largest of the coordinate lengths of the line segment, thereby obtaining a three-dimensional extension of the algorithm. In a case study, the authors calculated the distance between each sample position and the given line segment. The result reveals that the actual error at each sample position is smaller than the maximum theoretical error, and the performance of the three-dimensional extension of Bresenham's algorithm is as good as that of Bresenham's original planer algorithm. One of its potential applications is the three-dimensional step straight-line interpolation used in computer numerical control (CNC) systems of machine tools and rapid prototyping machines. Application of the algorithm is contrasted with that of the traditional DDA step straight-line interpolation algorithm. The result confirms that the three-dimensional extension of Bresenham's algorithm is much better than the DDA straight-line interpolation algorithm.

A Geometrical treatment of the Correspondence between Lines in Threefold Space and Points of a Quadric in Fivefold space

1925 ◽  
Vol 22 (5) ◽  
pp. 694-699 ◽  
Author(s):  
H. W. Turnbull
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Quadratic Relation ◽  
Four Dimensions ◽  
Homogeneous Coordinates ◽  
Plücker Coordinates ◽  
Straight Line ◽  
Set Up ◽  
Image Position ◽  
Three Dimensional Space

§ 1. The six Plücker coordinates of a straight line in three dimensional space satisfy an identical quadratic relationwhich immediately shows that a one-one correspondence may be set up between lines in three dimensional space, λ, and points on a quadric manifold of four dimensions in five dimensional space, S5. For these six numbers pij may be considered to be six homogeneous coordinates of such a point.

scholarly journals Hydroacoustical regularities of food behavior of dolphins

2018 ◽  
Vol 3 (2) ◽  
pp. 81-97
Author(s):  
V. A. Ryabov
Keyword(s):  
Radial Velocity ◽  
Temporal Dynamics ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Acoustic Signals ◽  
Food Behavior ◽  
Individual Fish ◽  
Order Of Magnitude

Hydroacoustic regularities of food behavior of dolphins are determined by characteristics and functionality of their acoustic signals. All the acoustic signals of dolphins are classified depending on their physical characteristics by the theory of signals and echolocation as follows: sequences of ultrashort ultrawideband coherent pulses – ‘clicks’, frequency-modulated (FM) simultones with uniformly distributed tones – ‘whistles’, packets of mutually coherent pulses (CI), packets of mutually noncoherent pulses (NI) and packets of versatile pulses (VI). They play the role of probing signals of six dolphin sonars optimized for solving various echolocation tasks. The possibilities of using the signals by dolphins in searching and classifying food objects by echolocation have been studied in this work on the basis of both experimental data obtained by the researcher and those available in literature. The dolphins can use sequences of ‘clicks’ to detect food objects (individual fish at the distance up to 70–110 m, a school of fish at the distance up to 450–600 m) and conspecifics (dolphins) (at the distance up to 450–600 m) and to classify food objects. The dolphins can use ‘whistles’ to detect food objects (individual fish at the distance up to 2 km, a school of fish at the distance up to 9–13 km) and conspecifics (dolphins) (at the distance up to 9–13 km) and to determine their range and relative radial velocity. ‘Whistles’ provide higher accuracy and the range of echolocation of food objects and conspecifics compared to ‘clicks’ (by more than an order of magnitude). Furthermore, the FM sonar provides the measure of the radial velocity of approaching or distance removing of underwater object to or from a dolphin. However, an acute analysis of the amplitude-time regularities of the fish echo for the purpose of their classification is the advantage of sonar using ‘clicks’. The dolphins can use the packs of CI at the distances shorter than 2.5 m for tracing the position dynamics of the prey aiming at accurate capture. The high hearing resolution of the dolphin in time is about 0.02 ms; it allows processing the subtle temporal dynamics of the echoes. Packs of NI (signals of spoken language) can be used by dolphins to organize various types of association and complex cooperation between themselves when hunting and catching fish. Packets of VI can be used by dolphins to expand the echolocation of the survey area around the dolphin in order to improve the quality of food objects monitoring, regardless of the position of the dolphin’s head, as well as to determine the range, relative radial velocity and class of food objects, at short distances. Evolution and perfection of various types of acoustic signals, sonars and various methods of processing echo signals in dolphins is caused, first of all, by optimization of hydroacoustic regularities of their food behavior, along with the need for orientation in three-dimensional space. One can assume the presence of similar hydroacoustic regularities of food behavior in Odontoceti based on the similarity of their acoustic signals and morphology. The acoustic regularities of food behavior of dolphins and bats are similar, despite the fact that they have different habitats (water and terrestrial-air), and these mammals belong to different orders of the animal kingdom (Сetacea and Chiroptera).

Experience and design: Bringing in the brain

2021 ◽  
pp. 571-642
Author(s):  
Michael A. Arbib
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Memory Systems ◽  
External Memory ◽  
Collaborative Studies ◽  
Current Design ◽  
Long Term Working Memory ◽  
The Brain

The IBSEN model of Imagination in Brain Systems for Episodes and Navigation explores how the architect’s experience is brought to bear in the design of architecture by building on the VISIONS model of understanding a visual scene and the TAM-WGM model of navigation. IBSEN develops the idea that a building provides both views from various viewpoints and places where particular experiences can be felt, and actions can be performed. For this, the design must support a variety of scripts for both practical and contemplative action and the cognitive maps that relate places for them. Nodes from different maps may be combined as scripts are harmonized with respect to a specific embedding of places in three-dimensional space. The chapter examines the role of the hippocampus in episodic memory and imagination, and observes that memory and imagination, episodic or not, are construction processes. During design, long-term working memory links internal and external memory systems, providing priority access to (but not only to) memory fragments that have proved relevant to the current design process. The designer in some sense “inverts” imagined experiences and behaviors of users of the forthcoming building. As the book ends, the author notes that we are only at the beginning of new collaborative studies that take cog/neuroscience out of the lab and into the building and the street.

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.

scholarly journals Optimal Reliable Point-in-Polygon Test and Differential Coding Boolean Operations on Polygons

Symmetry ◽  
2018 ◽  
Vol 10 (10) ◽  
pp. 477
Author(s):  
Jianqiang Hao ◽  
Jianzhi Sun ◽  
Yi Chen ◽  
Qiang Cai ◽  
Li Tan
Keyword(s):  
Parallel Implementation ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Geometric Probability ◽  
Boolean Operations ◽  
Straight Line ◽  
Speed Up ◽  
The Stability ◽  
Serial Algorithm ◽  
Degenerate Cases

This paper provides a full theoretical and experimental analysis of a serial algorithm for the point-in-polygon test, which requires less running time than previous algorithms and can handle all degenerate cases. The serial algorithm can quickly determine whether a point is inside or outside a polygon and accurately determine the contours of input polygon. It describes all degenerate cases and simultaneously provides a corresponding solution to each degenerate case to ensure the stability and reliability. This also creates the prerequisites and basis for our novel boolean operations algorithm that inherits all the benefits of the serial algorithm. Using geometric probability and straight-line equation F ( P ) = ( y i − y i + 1 ) ( x p − x i ) − ( y i − y p ) ( x i + 1 − x i ) , it optimizes our two algorithms that avoid the division operation and do not need to compute any intersection points. Our algorithms are applicable to any polygon that may be self-intersecting or with holes nested to any level of depth. They do not have to sort the vertices clockwise or counterclockwise beforehand. Consequently, they process all edges one by one in any order for input polygons. This allows a parallel implementation of each algorithm to be made very easily. We also prove several theorems guaranteeing the correctness of algorithms. To speed up the operations, we assign each vector a number code and derive two iterative formulas using differential calculus. However, the experimental results as well as the theoretical proof show that our serial algorithm for the point-in-polygon test is optimal and the time complexities of all algorithms are linear. Our methods can be extended to three-dimensional space, in particular, they can be applied to 3D printing to improve its performance.

Role of division algebra in seven-dimensional gauge theory

2015 ◽  
Vol 30 (10) ◽  
pp. 1550047
Author(s):  
Pushpa Kalauni ◽  
J. C. A. Barata
Keyword(s):  
Gauge Theory ◽  
Division Algebra ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Dimensional Vector ◽  
Vector Product ◽  
Yang Mills ◽  
Rotational Transformation

The algebra of octonions 𝕆 forms the largest normed division algebra over the real numbers ℝ, complex numbers ℂ and quaternions ℍ. The usual three-dimensional vector product is given by quaternions, while octonions produce seven-dimensional vector product. Thus, octonionic algebra is closely related to the seven-dimensional algebra, therefore one can extend generalization of rotations in three dimensions to seven dimensions using octonions. An explicit algebraic description of octonions has been given to describe rotational transformation in seven-dimensional space. We have also constructed a gauge theory based on non-associative algebra to discuss Yang–Mills theory and field equation in seven-dimensional space.

Promoting Rapid Situation Awareness in Tactical Displays: The Role of 3-D Perspective Views and Realistic Symbols

2000 ◽  
Vol 44 (21) ◽  
pp. 3-394-3-397 ◽  
Author(s):  
Harvey S. Smallman ◽  
Elaine Schiller ◽  
Craig A. Mitchell
Keyword(s):  
Situation Awareness ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Online Questionnaire ◽  
Air Defense ◽  
Level 1 ◽  
Open Question ◽  
Symbol Display ◽  
Three Dimensional Space

3-D displays populated with realistic 3-D icons have been touted as making good “at a glance” displays. Do they promote more rapid Situation Awareness (SA) than comparable 2-D displays? If so, is it the display format (2-D vs. 3-D) or the nature of the symbols (realistic icons vs. non-realistic symbols) populating the displays that matters, or both? Three groups of 13 participants observed a 9 minute naval air defense scenario. The first group saw it depicted in 3-D with icons, the second group saw it depicted in 2-D with icons and the third group saw it in 2-D with symbols. In each condition, the scenario was stopped every 30 seconds and we assessed ability to recall the attributes of four random tracks with an online questionnaire. We measured Endlesy's (1995) level 1 SA: the perception of elements of the display. SA for the 3-D display increased fastest over the course of the scenario. However, it started from one third the level of that for the 2-D symbol display and it took 4 minutes to reach 2-D levels. The advantages the 3-D display did confer were for those attributes that were visually explicit in the 3-D icons but available only in pop-up text boxes in the 2-D conditions. Similarly, depicting heading explicitly with the 2-D icons was superior to that with the 2-D symbols. The benefits of 3-D displays may sometimes stem from indirect application of good design principles, such as making certain information visually explicit, rather than from depicting three-dimensional space, per se. It remains an open question whether 2-D displays can be designed with comparable explicit analog coding.

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