Spherical kinematics in 3-dimensional generalized space

2020 ◽  
pp. 2150033
Author(s):  
Erhan Ata ◽  
Ümi̇t Zi̇ya Savci
Keyword(s):  
Matrix Equation ◽  
Orthogonal Matrix ◽  
Matrix Form ◽  
Euler Parameters ◽  
Lorentzian Space ◽  
Split Quaternions ◽  
3 Dimensional ◽  
Unit Quaternions ◽  
Special Cases ◽  
Generalized Quaternion

In this study, we obtained generalized Cayley formula, Rodrigues equation and Euler parameters of an orthogonal matrix in 3-dimensional generalized space [Formula: see text]. It is shown that unit generalized quaternion, which is defined by the generalized Euler parameters, corresponds to a rotation in [Formula: see text] space.We found that the rotation in matrix equation forms using matrix form of the generalized quaternion product. Besides, in [Formula: see text] space, we obtained the rotations determined by the unit quaternions and unit split quaternions, which are special cases of generalized quaternions for [Formula: see text] in 3-dimensional Eulidean space [Formula: see text] in 3-dimensional Lorentzian space [Formula: see text] respectively.

scholarly journals New Results on the Motions of Asteroids in Resonances

1992 ◽  
Vol 152 ◽  
pp. 145-152 ◽  
Author(s):  
R. Dvorak
Keyword(s):  
Initial Conditions ◽  
Numerical Study ◽  
Equations Of Motion ◽  
Close Approach ◽  
Integration Time ◽  
Time Interval ◽  
Time Intervals ◽  
3 Dimensional ◽  
Special Cases ◽  
Good Agreement

In this article we present a numerical study of the motion of asteroids in the 2:1 and 3:1 resonance with Jupiter. We integrated the equations of motion of the elliptic restricted 3-body problem for a great number of initial conditions within this 2 resonances for a time interval of 104 periods and for special cases even longer (which corresponds in the the Sun-Jupiter system to time intervals up to 106 years). We present our results in the form of 3-dimensional diagrams (initial a versus initial e, and in the z-axes the highest value of the eccentricity during the whole integration time). In the 3:1 resonance an eccentricity higher than 0.3 can lead to a close approach to Mars and hence to an escape from the resonance. Asteroids in the 2:1 resonance with Jupiter with eccentricities higher than 0.5 suffer from possible close approaches to Jupiter itself and then again this leads in general to an escape from the resonance. In both resonances we found possible regions of escape (chaotic regions), but only for initial eccentricities e > 0.15. The comparison with recent results show quite a good agreement for the structure of the 3:1 resonance. For motions in the 2:1 resonance our numeric results are in contradiction to others: high eccentric orbits are also found which may lead to escapes and consequently to a depletion of this resonant regions.

scholarly journals The Algebraic Riccati Matrix Equation for Eigendecomposition of Canonical Forms

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
M. Nouri ◽  
S. Talatahari
Keyword(s):  
Matrix Equation ◽  
Present Method ◽  
Similarity Transformation ◽  
Structured Matrices ◽  
Canonical Forms ◽  
Low Dimensions ◽  
Special Cases ◽  
Riccati Matrix Equation ◽  
Riccati Matrix

The algebraic Riccati matrix equation is used for eigendecomposition of special structured matrices. This is achieved by similarity transformation and then using the algebraic Riccati matrix equation to the triangulation of matrices. The process is the decomposition of matrices into small and specially structured submatrices with low dimensions for easy finding of eigenpairs. Here, we show that previous canonical forms I, II, III, and so on are special cases of the presented method. Numerical and structural examples are included to show the efficiency of the present method.

Surface Colour Matching under Conditions of Multiple Illuminants

Perception ◽  
1997 ◽  
Vol 26 (1_suppl) ◽  
pp. 190-190 ◽  
Author(s):  
H Irtel
Keyword(s):  
Light Source ◽  
Green Light ◽  
Light Sources ◽  
Local Contrast ◽  
Colour Constancy ◽  
Lighting Condition ◽  
3 Dimensional ◽  
Special Cases ◽  
Type Pattern ◽  
Light And Shadow

Most theories of colour constancy assume a flat coloured surface and a single homogenous light source. Natural situations, however, are 3-dimensional (3-D), are hardly ever restricted to a single light source, and object illumination is never homogenous. Here, two special cases of secondary light sources with sharp boundaries were simulated on a computer screen: a house-like 3-D object with colour patches in sunlight and shadow, and a Mondrian-type pattern with a coloured transparency covering some of the colour patches. Subjects made ‘paper’-matches between colour patches in light and shadow and between patches under the transparency and without the transparency. Matching did not depend on whether the simulated lighting condition was natural (yellow light, blue shadow) or artificial (green light, magenta shadow). Patches under a coloured transparency produced lightness constancy but subjects could not discount chromaticity shifts induced by the transparency. The number of context patches (2 vs 6) made no difference, and it made no difference whether the transparency covered the Mondrian completely or only partially. These results indicate that subjects were not able to use local contrast cues at sharp illumination boundaries to discount for the illuminant.

Solutions to the matrix equation X − AXB = CY+R and its application

2016 ◽  
Vol 40 (3) ◽  
pp. 995-1004 ◽  
Author(s):  
Caiqin Song ◽  
Guoliang Chen
Keyword(s):  
Linear Systems ◽  
Matrix Equation ◽  
Controller Design ◽  
Equivalent Form ◽  
Closed Form Solutions ◽  
Lyapunov Matrix Equation ◽  
Special Cases ◽  
The Matrix ◽  
Lyapunov Matrix ◽  
Low Dimensional

The solution of the nonhomogeneous Yakubovich matrix equation [Formula: see text] is important in stability analysis and controller design in linear systems. The nonhomogeneous Yakubovich matrix equation [Formula: see text], which contains the well-known Kalman–Yakubovich matrix equation and the general discrete Lyapunov matrix equation as special cases, is investigated in this paper. Closed-form solutions to the nonhomogeneous Yakubovich matrix equation are presented using the Smith normal form reduction. Its equivalent form is provided. Compared with the existing method, the method presented in this paper has no limit to the dimensions of an unknown matrix. The present method is suitable for any unknown matrix, not only low-dimensional unknown matrices, but also high-dimensional unknown matrices. As an application, parametric pole assignment for descriptor linear systems by PD feedback is considered.

Rotations and Screw Motion with Timelike Vector in 3-Dimensional Lorentzian Space

2011 ◽  
Vol 22 (4) ◽  
pp. 1081-1091 ◽  
Author(s):  
Osman Keçilioğlu ◽  
Siddika Özkaldi ◽  
Halit Gündoğan
Keyword(s):  
Timelike Vector ◽  
Lorentzian Space ◽  
3 Dimensional ◽  
Screw Motion

On the explicit characterization of spherical curves in 3-dimensional Lorentzian space L3

2003 ◽  
Vol 11 (4) ◽  
pp. 389-397
Author(s):  
K. Ilarslan ◽  
C. Camci ◽  
H. Kocayigit ◽  
H. H. Hacisalihoglu
Keyword(s):  
Lorentzian Space ◽  
3 Dimensional ◽  
Explicit Characterization

On the explicit characterization of spherical curves in 3-dimensional Lorentzian space

2003 ◽  
Vol 11 (4) ◽  
pp. 389-397
Author(s):  
K. Ilarslan ◽  
C. Camci ◽  
H. Kocayigit ◽  
H. H. Hacisalihoglu
Keyword(s):  
Lorentzian Space ◽  
3 Dimensional ◽  
Explicit Characterization
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