scholarly journals Improved Total Least Square Algorithm

2015 ◽  
Vol 9 (1) ◽  
pp. 394-399 ◽  
Author(s):  
Deng Yonghe
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Least Square ◽  
Error Equation ◽  
Virtual Observation ◽  
Straight Line ◽  
Condition Equation ◽  
Total Least Square ◽  
Three Dimensional Space ◽  
Improved Algorithm

Aim to blemish of total least square algorithm based on error equation of virtual observation,this paper proposed a sort of improved algorithm which doesn’t neglect condition equation of virtual observation,and considers both error equation and condition equation of virtual observation.So,the improved algorithm is better.Finally,this paper has fitted a straight line in three-dimensional space based on the improved algorithm.The result showed that the improved algorithm is viable and valid.

scholarly journals A Sort of New Improved Algorithm For Total Least Square

2015 ◽  
Vol 9 (1) ◽  
pp. 238-247
Author(s):  
Deng Yonghe
Keyword(s):  
Least Square ◽  
New Method ◽  
Unknown Parameters ◽  
Error Equation ◽  
Virtual Observation ◽  
Condition Equation ◽  
Total Least Square ◽  
Improved Algorithm

Aim to blemish of total least square algorithm based on error equation of virtual observation, this paper put forward and deduced a sort of new improved algorithm which selects essential unknown parameters among designing matrix, and then, doesn’t consider condition equation of unknown parameters among designing matrix. So, this paper perfected and enriched algorithm, and sometimes, new method of this paper is better. Finally, the results of examples showed that new mothod is viable and valid.

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 ANY FIVE POINTS LIE IN A 3-SPACE

2020 ◽  
Author(s):  
Zh. Nikoghosyan ◽  
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Entire Space ◽  
Unique Line ◽  
Straight Line ◽  
Set Of Points ◽  
Three Dimensional Space

In axiomatic formulations, every two points lie in a (straight) line, every three points lie in a plane and every four points lie in a three-dimensional space (3-space). In this paper we show that every five points lie in a 3-space as well, implying that every set of points lie in a 3-space. In other words, the 3-space occupies the entire space. The proof is based on the following four axioms: 1) every two distinct points define a unique line, 2) every three distinct points, not lying on the line, define a unique plane, 3) if 𝐴 and 𝐵 are two distinct points in a 3-space, then the line defined by the points 𝐴, 𝐵, entirely lies in this 3-space, 4) if 𝐹1, 𝐹2, 𝐹3 are three distinct points in a 3-space, not lying in a line, then the plane defined by the points 𝐹1, 𝐹2, 𝐹3, lies entirely in this 3-space.

On Pedal Quadrics in Non-Euclidean Hyperspace

1925 ◽  
Vol 22 (5) ◽  
pp. 751-758
Author(s):  
J. P. Gabbatt
Keyword(s):  
Orthogonal Projection ◽  
Euclidean Plane ◽  
Dimensional Space ◽  
Three Dimensional ◽  
The Other ◽  
Elementary Geometry ◽  
Orthogonal Projections ◽  
Point Pair ◽  
Straight Line ◽  
Three Dimensional Space

1. The following are well-known theorems of elementary geometry: Given any euclidean plane triangle, A0 A1 A2, and any pair of points, X, Y, isogonally conjugate q. A0 A1 A2; then the orthogonal projections of X, Y on the sides of A0 A1 A2 lie on a circle, the pedal circle of the point-pair. If either of the points X,- Y describe a (straight) line, m, then the other describes a conic circumscribing A0 A1 A2, and the pedal circle remains orthogonal to a fixed circle, J; thus the pedal circles in question are members of an ∞2 linear system of circles of which the circle J and the line at infinity constitute the Jacobian. In particular, if the line m meet Aj Ak at Lt (i, j, k = 0, 1, 2), then the circles on Ai Li as diameter, which are the pedal circles of the point-pairs Ai, Li, are coaxial; the remaining circles of the coaxial system being the director circles of the conics, inscribed in the triangle A0 A1 A2, which touch the line m. If Mi denote the orthogonal projection on m of Ai, and Ni the orthogonal projection on Aj Ak of Mi, then the three lines Mi Ni meet at a point (Neuberg's theorem), viz. the centre of the circle J. Analogues for three-dimensional space of most of these theorems are also known ‖.

scholarly journals Piecewise Curve Fitting Based on Least Square Method in 3D Space

2020 ◽  
Vol 3 (1) ◽  
Author(s):  
Lihong Xue
Keyword(s):  
Least Squares ◽  
Curve Fitting ◽  
Dimensional Space ◽  
Experimental Studies ◽  
Three Dimensional ◽  
Least Square Method ◽  
Least Square ◽  
Least Square Fitting ◽  
Fitting In ◽  
Three Dimensional Space

Least squares curve fitting is widely used in various neighborhoods, such as industry, agriculture, and economy. The method of piecewise curve fitting is used to deal with some experimental studies with large amounts of data. It can not only be used for plane space points, but also for three-dimensional space points. Firstly, this article introduces the principles of curve fitting and least squares. Secondly, it clarifies the steps based on least square fitting and gives examples of improvements made based on the principle of least square fitting functions by some scientists. Finally, the piecewise curve fitting in the three-dimensional space is obtained using the least squares piecewise curve fitting. It lays a certain theoretical foundation for drawing the laws between data in three-dimensional space.

Reflection from Curved Mirrors in a Plane

2019 ◽  
Vol 7 (1) ◽  
pp. 46-54 ◽  
Author(s):  
Л. Жихарев ◽  
L. Zhikharev
Keyword(s):  
Dimensional Space ◽  
Spherical Surface ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Interesting Phenomenon ◽  
One Dimensional ◽  
Straight Line ◽  
The Family ◽  
Dimensional Object ◽  
Three Dimensional Space

Reflection from a certain mirror is one of the main types of transformations in geometry. On a plane a mirror represents a straight line. When reflecting, we obtain an object, each point of which is symmetric with respect to this straight line. In this paper have been considered examples of reflection from a circle – a general case of a straight line, if the latter is defined through a circle of infinite radius. While analyzing a simple reflection and generalization of this process to the cases of such curvature of the mirror, an interesting phenomenon was found – an increase in the reflection dimension by one, that is, under reflection of a one-dimensional object from the circle, a two-dimensional curve is obtained. Thus, under reflection of a point from the circle was obtained the family of Pascal's snails. The main cases, related to reflection from a circular mirror the simplest two-dimensional objects – a segment and a circle at their various arrangement, were also considered. In these examples, the reflections are two-dimensional objects – areas of bizarre shape, bounded by sections of curves – Pascal snails. The most interesting is the reflection of two-dimensional objects on a plane, because the reflection is too informative to fit in the appropriate space. To represent the models of obtained reflections, it was proposed to move into three-dimensional space, and also developed a general algorithm allowing obtain the object reflection from the curved mirror in the space of any dimension. Threedimensional models of the reflections obtained by this algorithm have been presented. This paper reveals the prospects for further research related to transition to three-dimensional space and reflection of objects from a spherical surface (possibility to obtain four-dimensional and five-dimensional reflections), as well as studies of reflections from geometric curves in the plane, and more complex surfaces in space.

Nonlinear secondary whirl of an overhung rotor

2009 ◽  
Vol 466 (2113) ◽  
pp. 283-301 ◽  
Author(s):  
K. Nandakumar ◽  
Anindya Chatterjee
Keyword(s):  
Multiple Scales ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Resonant Mode ◽  
Resonant Forcing ◽  
Straight Line ◽  
Finite Displacements ◽  
Jump Phenomena ◽  
Rotor Stiffness ◽  
Three Dimensional Space

Gravity critical speeds of rotors have hitherto been studied using linear analysis, and ascribed to rotor stiffness asymmetry. Here, we study an idealized asymmetric nonlinear overhung rotor model of Crandall and Brosens, spinning close to its gravity critical speed. Nonlinearities arise from finite displacements, and the rotor’s static lateral deflection under gravity is taken as small. Assuming small asymmetry and damping, slow modulations of whirl amplitudes are studied using the method of multiple scales. Inertia asymmetry appears only at second order. More interestingly, even without stiffness asymmetry, the gravity-induced resonance survives through geometric nonlinearities. The gravity resonant forcing does not influence the resonant mode at leading order, unlike the typical resonant oscillations. Nevertheless, the usual phenomena of resonances, namely saddle–node bifurcations, jump phenomena and hysteresis, are all observed. An unanticipated periodic solution branch is found. In the three-dimensional space of two modal coefficients and a detuning parameter, the full set of periodic solutions is found to be an imperfect version of three mutually intersecting curves: a straight line, a parabola and an ellipse.

scholarly journals Spatial Straight-Line Drawing Algorithm Based on Method of Discriminate Regions—A Control Algorithm of Motors

Energies ◽  
2020 ◽  
Vol 13 (19) ◽  
pp. 5002
Author(s):  
Jianping Wang ◽  
Shiguang Xiao ◽  
Tao Song ◽  
Junqi Yue ◽  
Pingyan Bian ◽  
...  
Keyword(s):  
Cooperative Control ◽  
Dimensional Space ◽  
Line Drawing ◽  
Three Dimensional ◽  
Step Length ◽  
Pulse Number ◽  
Straight Line ◽  
The Given ◽  
The Relationship ◽  
Three Dimensional Space

A novelty algorithm of spatial straight-line drawing based on a method of discriminate regions is proposed in this paper based on Bresenham’s algorithm. Three-dimensional space is divided into innumerable three-dimensional meshes according to the given rule; the distance between the start and the end points of the three coordinates is Δx, Δy, and Δz, respectively; the distribution types of spatial straight line and the position of the end point are determined by judging the relationship among Δx, Δy, and Δz; then, the active-passive directions can be determined. The plane of the ending point of the straight line in a three-dimensional mesh is divided into four regions; then, the discriminant is obtained; and this discriminant determine which region the point is located in The algorithm is verified and analyzed by the method of contrastive analysis; the results show that: the error of the algorithm is related to the step length L; the maximum theoretical error is 0.7071*L. The discriminants are all integers, so the problem of deviation from the theoretical straight line caused by the retention of decimals of significant digits can be avoided. Finally, the algorithm is applied to the cooperative control of multiple motors, and conversion between unit grid number and pulse number of motors is performed.

scholarly journals MEASURING IN IMAGES WITH PROJECTIVE GEOMETRY

2018 ◽  
Vol XLII-1 ◽  
pp. 141-148
Author(s):  
B. Erdnüß
Keyword(s):  
Projective Geometry ◽  
Dimensional Space ◽  
3D Structure ◽  
Three Dimensional ◽  
Imaging Geometry ◽  
Projective Transformations ◽  
Straight Line ◽  
Fundamental Relationship ◽  
Common Plane ◽  
Three Dimensional Space

<p><strong>Abstract.</strong> There is a fundamental relationship between projective geometry and the perspective imaging geometry of a pinhole camera. Projective scales have been used to measure within images from the beginnings of photogrammetry, mostly the cross-ratio on a straight line. However, there are also projective frames in the plane with interesting connections to affine and projective geometry in three dimensional space that can be utilized for photogrammetry. This article introduces an invariant on the projective plane, describes its relation to affine geometry, and how to use it to reduce the complexity of projective transformations. It describes how the invariant can be use to measure on projectively distorted planes in images and shows applications to this in 3D reconstruction. The article follows two central ideas. One is to measure coordinates in an image relatively to each other to gain as much invariance of the viewport as possible. The other is to use the remaining variance to determine the 3D structure of the scene and to locate the camera centers. For this, the images are projected onto a common plane in the scene. 3D structure not on the plane occludes different parts of the plane in the images. From this, the position of the cameras and the 3D structure are obtained.</p>

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