Efficient spherical surface integration of Gauss functions in three-dimensional spherical coordinates and the solution for the modified Bessel function of the first kind

2021 ◽  
Author(s):  
Yiting Wang ◽  
Jing Kong
Keyword(s):  
Bessel Function ◽  
Spherical Surface ◽  
Three Dimensional ◽  
Spherical Coordinates ◽  
Modified Bessel Function ◽  
Surface Integration

The experience of employment of the mathematical cartography methods in cosmology

2017 ◽  
Vol 923 (5) ◽  
pp. 7-16
Author(s):  
A.V. Kavrayskiy
Keyword(s):  
Background Radiation ◽  
Three Dimensional ◽  
Spherical Coordinates ◽  
Linear Element ◽  
Microwave Background ◽  
Euclidian Space ◽  
Microwave Background Radiation ◽  
Rectangular Coordinates ◽  
The Universe ◽  
Length Distortion

The experience of mathematical modeling of the 3D-sphere in the 4D-space and projecting it by mathematical cartography methods in the 3D-Euclidian space is presented. The problem is solved by introduction of spherical coordinates for the 3D-sphere and their transformation into the rectangular coordinates, using the mathematical cartography methods. The mathematical relationship for calculating the length distortion mp(s) of the ds linear element when projecting the 3D-sphere from the 4-dimensional Euclidian space into three-dimensional Euclidian space is derived. Numerical examples, containing the modeling of the ds small linear element by spherical coordinates of 3D-sphere, projecting this sphere into the 3D-Euclidian space and length of ds calculating by means of its projection dL and size of distortion mp(s) are solved. Based on the model of the Universe known in cosmology as the 3D-sphere, the hypothesis of connection between distortion mp(s) and the known observed effects Redshift and Microwave Background Radiation is considered.

A biomimetic 3D airflow sensor made of an array of two piezoelectric metal-core fibers

Sensor Review ◽  
2017 ◽  
Vol 37 (3) ◽  
pp. 312-321 ◽  
Author(s):  
Yixiang Bian ◽  
Can He ◽  
Kaixuan Sun ◽  
Longchao Dai ◽  
Hui Shen ◽  
...  
Keyword(s):  
Design Methodology ◽  
Spherical Surface ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Piezoceramic Cylinder ◽  
Metal Core ◽  
Content Type ◽  
3D Space ◽  
Highly Sensitive ◽  
Airflow Sensor

Purpose The purpose of this paper is to design and fabricate a three-dimensional (3D) bionic airflow sensing array made of two multi-electrode piezoelectric metal-core fibers (MPMFs), inspired by the structure of a cricket’s highly sensitive airflow receptor (consisting of two cerci). Design/methodology/approach A metal core was positioned at the center of an MPMF and surrounded by a hollow piezoceramic cylinder. Four thin metal films were spray-coated symmetrically on the surface of the fiber that could be used as two pairs of sensor electrodes. Findings In 3D space, four output signals of the two MPMFs arrays can form three “8”-shaped spheres. Similarly, the sensing signals for the same airflow are located on a spherical surface. Originality/value Two MPMF arrays are sufficient to detect the speed and direction of airflow in all three dimensions.

Solutions to the Vector Tetrahedron Equation

1965 ◽  
Vol 87 (2) ◽  
pp. 228-234 ◽  
Author(s):  
Milton A. Chace
Keyword(s):  
Closed Form ◽  
Digital Computer ◽  
Three Dimensional ◽  
Dimensional Vector ◽  
Spherical Coordinates ◽  
Tetrahedron Equation ◽  
Position Determination ◽  
Closed Form Solutions

A set of nine closed-form solutions are presented to the single, three-dimensional vector tetrahedron equation, sum of vectors equals zero. The set represents all possible combinations of unknown spherical coordinates among the vectors, assuming the coordinates are functionally independent. Optimum use is made of symmetry. The solutions are interpretable and can be evaluated reliably by digital computer in milliseconds. They have been successfully applied to position determination of many three-dimensional mechanisms.

Research on Transition Zones of Spherical Surface Parts in 3D Rolling Process

2014 ◽  
Vol 687-691 ◽  
pp. 3-6
Author(s):  
Da Ming Wang ◽  
Ming Zhe Li ◽  
Zhong Yi Cai
Keyword(s):  
Sheet Metal ◽  
Spherical Surface ◽  
Three Dimensional ◽  
Rolling Process ◽  
Experimental Results ◽  
Transition Zones ◽  
Spherical Surfaces ◽  
Sheet Width ◽  
Forming Precision ◽  
Roll Gap

3D rolling is a novel technology for three-dimensional surface parts. In this process, by controlling the gap between the upper and lower forming rolls, the sheet metal is non-uniformly thinned in thickness direction, and the longitudinal elongation of the sheet metal is different along the transverse direction, which makes the sheet metal generate three-dimensional deformation. In this paper, the transition zones of spherical surface parts in 3D rolling process are investigated. Spherical surface parts with the same widths but different lengths are simulated in condition of the same roll gap, and their experimental results are presented. The forming precision of forming parts and the causes of transition zones in the head and tail regions are analyzed through simulated results. The simulated and experimental results show that the lengths of transition zones of spherical surfaces in the head and tail regions are fixed values in condition of the same sheet width and roll gap.

Three-dimensional time-harmonic elastodynamic Green ’ s functions for anisotropic solids

1995 ◽  
Vol 449 (1937) ◽  
pp. 441-458 ◽  
Keyword(s):  
Differential Equations ◽  
Radon Transform ◽  
Transverse Isotropy ◽  
Three Dimensional ◽  
Surface Integral ◽  
Line Integral ◽  
Surface Integration ◽  
Time Harmonic ◽  
Inverse Radon Transform ◽  
Eigenvectors And Eigenvalues

A method based on the Radon transform is presented to determine the displacement field in a general anisotropic solid due to the application of a time-harmonic point force. The Radon transform reduces the system of coupled partial differential equations for the displacement components to a system of coupled ordinary differential equations. This system is reduced to an uncoupled form by the use of properties of eigenvectors and eigenvalues. The resulting simplified system can be solved easily. A back transformation to the original coordinate system and a subsequent application of the inverse Radon transform yields the displacements as a summation of a regular elastodynamic term and a singular static term. Both terms are integrals over a unit sphere. For the regular dynamic term, the surface integration can be evaluated numerically without difficulty. For the singular static term, the surface integral has been reduced to a line integral over half a unit circle. Reductions to the cases of isotropy and transverse isotropy have been worked out in detail. Examples illustrate applications of the method.

scholarly journals Bounds and algorithms for the -Bessel function of imaginary order

2013 ◽  
Vol 16 ◽  
pp. 78-108 ◽  
Author(s):  
Andrew R. Booker ◽  
Andreas Strömbergsson ◽  
Holger Then
Keyword(s):  
Bessel Function ◽  
Convergent Series ◽  
Steepest Descent ◽  
Modified Bessel Function ◽  
Software Library ◽  
Asymptotic Bounds ◽  
Mixed Derivatives ◽  
Rigorous Computation ◽  
Working Out ◽  
Precise Asymptotic

AbstractUsing the paths of steepest descent, we prove precise bounds with numerical implied constants for the modified Bessel function${K}_{ir} (x)$of imaginary order and its first two derivatives with respect to the order. We also prove precise asymptotic bounds on more general (mixed) derivatives without working out numerical implied constants. Moreover, we present an absolutely and rapidly convergent series for the computation of${K}_{ir} (x)$and its derivatives, as well as a formula based on Fourier interpolation for computing with many values of$r$. Finally, we have implemented a subset of these features in a software library for fast and rigorous computation of${K}_{ir} (x)$.

Absolute Measurement of Aspheric Surfaces Using White-Light Interferometry

2007 ◽  
Vol 364-366 ◽  
pp. 80-85
Author(s):  
Su Ping Chang ◽  
Tie Bang Xie ◽  
Xuang Ze Wang ◽  
Jun Guo
Keyword(s):  
High Precision ◽  
White Light ◽  
Spherical Surface ◽  
Three Dimensional ◽  
Influence Factors ◽  
Horizontal Displacement ◽  
Absolute Measurement ◽  
Aspheric Surfaces ◽  
Interferometric Technique ◽  
Main Influence

White-light interferometric technique has been widely applied in the measurement of three-dimensional profiles and roughness with high-precision. Based on the characteristic of interferometric technique, a new method combined with image location and a three-dimensional stage is proposed to achieve the non-contact absolute shape measurement for aspheric and spherical surface in a slarge range. The interference fringes vary with the horizontal displacement of the measured surface, the surface information was obtained by locating the transformation of the maximal intensity in the interferograms. Two main influence factors are discussed; they are performance of the inerferimetric microscope and the stage. Since the performance of the stage directly determines the measurement precision, a three-dimensional displacement stage with a large range and a high precision was developed. Some experiments were carried out to verify the performance of the three-dimensional displacement stage and the validity of the new measurement method with satisfactory results.

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