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Статті в журналах з теми "Ellipsoid of revolution"

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Автор: Grafiati

Опубліковано: 30 травня 2022

Оновлено: 31 травня 2022

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1

Nyrtsov, M. V., and M. E. Fleis. "Classification of the triaxial ellipsoid projections." Geodesy and Cartography 972, no. 6 (July 20, 2021): 17–25. http://dx.doi.org/10.22389/0016-7126-2021-972-6-17-25.

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Анотація:

There are generally accepted classifications of cartographic projections of a sphere and an ellipsoid of revolution according to various criteria. The projections of a triaxial ellipsoid have a number of differences from those of a sphere and an ellipsoid of revolution; therefore, the existing classifications need to be clarified. The definitions of the main classes of cartographic projections of a sphere and an ellipsoid of revolution by the type of cartographic grid cannot be extended to those of a triaxial ellipsoid. At the same time, the traditional approach with the auxiliary surface is maintained. To obtain projections of a triaxial ellipsoid in transverse orientation, there is no need to recalculate through polar spherical coordinates as is done for those of a sphere and an ellipsoid of revolution. The transition is carried out by rotating the ellipsoid around the axes, which is much easier. In the classification of the projections of a triaxial ellipsoid according to the distortions, it is necessary to distinguish conformal, quasiconformal, equal-area projections and the ones which preserve lengths along the meridians.

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2

Grafarend, Erik W. "Geophysical models of the surface global vorticity vector." Symposium - International Astronomical Union 128 (1988): 411. http://dx.doi.org/10.1017/s0074180900119813.

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Within the framework of Newtonian kinematics the local vorticity vector is introduced and averaged with respect to global earth geometry, namely the ellipsoid of revolution. For a deformable body like the earth the global vorticity vector is defined as the earth rotation. A decomposition of the Lagrangean displacement and of the Lagrangean vorticity vector into vector spherical harmonics, namely into spheroidal and toroidal parts, proves that the global vorticity vector only contains toroidal coefficients of degree and order one (polar motion) and toroidal coefficients of degree one and order zero (length of the day) in the case of an ellipsoidal earth. Once we assume an earth model of type ellipsoid of revolution the earth rotation is also slightly dependent on the ellipsoidal flattening and the radial derivative of the spheroidal coefficients of degree two and order one.

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3

Williams, Roy. "Middle Latitude Sailing Revisited." Journal of Navigation 51, no. 1 (January 1998): 132–40. http://dx.doi.org/10.1017/s0373463397257553.

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Анотація:

Many problems in navigation can be best viewed and solved as problems in analytical geometry. We only need to understand the geometry of two ‘navigable’ surfaces; the sphere and the ellipsoid of revolution. The ellipsoidal model is generated by revolving an ellipse about its minor axis and this model is used as a global model for the surface of the Earth. The eccentricity of the meridian ellipse is small (≈0·082) so we sometimes refer to this surface as a ‘spheroid’ since the surface is still ‘sphere-like’. The physical Earth is, in fact, referred to as a ‘geoid’ whose surface is that which approximates global mean sea level. The mathematical representation of the geoid is not trivial and the ellipsoid of revolution is an extremely good approximation to it.

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4

Egorov, Nickolay V., and Ekaterina M. Vinogradova. "Mathematical modeling of triode system on the basis of field emitter with ellipsoid shape." Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes 17, no. 2 (2021): 131–36. http://dx.doi.org/10.21638/11701/spbu10.2021.203.

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Анотація:

In this paper the mathematical modeling of the triode emission axially symmetric system on the basis of field emitter is considered. Emitter is an ellipsoid of revolution, anode is a confocal ellipsoidal surface of revolution. Modulator is a part of the ellipsoidal surface of revolution, confocal with the cathode and anode surfaces. The boundary-value problem for the Laplace's equation in the prolate spheroidal coordinates with the boundary conditions of the first kind is solved. The variable separation method is applied to calculate the axisymmetrical electrostatic potential distribution. The potential distribution is represented as the Legendre functions expansion. The expansion coefficients are the solution of the system of linear equations. All geometrical dimensions of the system are the parameters of the problem.

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5

Frenkel, D., and B. M. Mulder. "The hard ellipsoid-of-revolution fluid." Molecular Physics 55, no. 5 (August 10, 1985): 1171–92. http://dx.doi.org/10.1080/00268978500101971.

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6

Mulder, B. M., and D. Frenkel. "The hard ellipsoid-of-revolution fluid." Molecular Physics 55, no. 5 (August 10, 1985): 1193–215. http://dx.doi.org/10.1080/00268978500101981.

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7

Kochiev, A. A., A. D. Tikhonov, and E. V. Kalinova. "New proof of the inhomogeneous density distribution of matter inside the Earth (PZ-90.11)." Zemleustrojstvo, kadastr i monitoring zemel' (Land management, cadastre and land monitoring), no. 2 (January 17, 2022): 142–47. http://dx.doi.org/10.33920/sel-04-2202-10.

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Анотація:

Based on the revealed significant difference between the Stokes kernel and a uniform homogenous ellipsoid of revolution, the authors calculated that the density distribution of matter inside the Earth is not uniform. At the same time, the exact density distribution of the matter in a uniform ellipsoid of revolution is not specified.

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8

Vyshnyepolskiy, Vladimir, E. Zavarihina, and D. Peh. "Geometric Locations of Points Equally Distance from Two Given Geometric Figures. Part 4: Geometric Locations of Points Equally Remote from Two Spheres." Geometry & Graphics 9, no. 3 (December 9, 2021): 12–29. http://dx.doi.org/10.12737/2308-4898-2021-9-3-12-29.

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Анотація:

The article deals with the geometric locations of points equidistant from two spheres. In all variants of the mutual position of the spheres, the geometric places of the points are two surfaces. When the centers of the spheres coincide with the locus of points equidistant from the spheres, there will be spheres equal to the half-sum and half-difference of the diameters of the original spheres. In three variants of the relative position of the initial spheres, one of the two surfaces of the geometric places of the points is a two-sheet hyperboloid of revolution. It is obtained when: 1) the spheres intersect, 2) the spheres touch, 3) the outer surfaces of the spheres are removed from each other. In the case of equal spheres, a two-sheeted hyperboloid of revolution degenerates into a two-sheeted plane, more precisely, it is a second-order degenerate surface with a second infinitely distant branch. The spheres intersect - the second locus of the points will be the ellipsoid of revolution. Spheres touch - the second locus of points - an ellipsoid of revolution, degenerated into a straight line, more precisely into a zero-quadric of the second order - a cylindrical surface with zero radius. The outer surfaces of the spheres are distant from each other - the second locus of points will be a two-sheet hyperboloid of revolution. The small sphere is located inside the large one - two coaxial confocal ellipsoids of revolution. In all variants of the mutual position of spheres of the same diameters, the common geometrical place of equidistant points is a plane (degenerate surface of the second order) passing through the middle of the segment perpendicular to it, connecting the centers of the original spheres. The second locus of points equidistant from two spheres of the same diameter can be either an ellipsoid of revolution (if the original spheres intersect), or a straight (cylindrical surface with zero radius) connecting the centers of the original spheres when the original spheres touch each other, or a two-sheet hyperboloid of revolution (if continue to increase the distance between the centers of the original spheres).

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9

Müller, J. J., and H. Schrauber. "The inertia-equivalent ellipsoid: a link between atomic structure and low-resolution models of small globular proteins determined by small-angle X-ray scattering." Journal of Applied Crystallography 25, no. 2 (April 1, 1992): 181–91. http://dx.doi.org/10.1107/s0021889891011421.

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Анотація:

Low-resolution three-parameter models of the shape of a biopolymer in solution can be determined by a new indirect method from small-angle X-ray scattering without contrast-variation experiments. The basic low-resolution model employed is a triaxial ellipsoid – the inertia-equivalent ellipsoid (IEE). The IEE is related to the tensor of inertia of a body and the eigenvalues and eigenvectors of this tensor can be calculated directly from the atomic coordinates and from the homogeneous solvent-excluded body of a molecule. The IEE defines a mean molecular surface (like the sea level on earth) which models the molecular shape adequately if the IEE volume is not more than 30% larger than the dry volume of the molecule. Approximately 10 to 15% of the solvent-excluded volume is outside the ellipsoid; the radii of gyration of the IEE and of the homogeneous molecular body are identical. The largest diameter of the IEE is about 5 to 15% (~0.2–0.8 nm) smaller than the maximum dimension of globular molecules with molecular masses smaller than 65000 daltons. From the scattering curve of a molecule in solution the IEE can be determined by a calibration procedure. 29 proteins of known crystal structure have been used as a random sample. Systematic differences between the axes of the IEE, calculated directly from the structure, and the axes of the scattering-equivalent ellipsoids of revolution, estimated from the scattering curve of the molecule in solution, are used to derive correction factors for the axial dimensions. Distortions of model dimensions of 20 to 40% (up to 1 nm), caused by misinterpretation of scattering contributions from electron density fluctuations within the molecule, are reduced to a quarter by applying these correction factors to the axes of the scattering-equivalent ellipsoids of revolution. In a computer experiment the axes of the inertia-equivalent ellipsoids have been determined for a further nine proteins with the same accuracy. The automated estimation of the IEE from the scattering curve of a molecule in solution is realized by the Fortran77 program AUTOIEE.

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10

Vyshnepolsky, V. I., N. S. Kadykova, and D. S. Peh. "Geometric modeling and study of properties of surfaces equidistant to two spheres." Journal of Physics: Conference Series 2182, no. 1 (March 1, 2022): 012013. http://dx.doi.org/10.1088/1742-6596/2182/1/012013.

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Анотація:

Abstract The paper considers the geometric locus of points equidistant to two spheres of different diameters. If these spheres are concentric, the sought multitude constitutes a single surface – a sphere of diameter equal to arithmetic mean of the diameters of the given spheres. In other cases the geometric locus of points equidistant to two spheres of different diameters constitutes two surfaces. In case the spheres intersect, are tangent or distant to each other, the first of these surfaces is a two-sheet hyperboloid of revolution that degenerates into a plane in case the spheres are equal. In case the spheres intersect, the second of the surfaces is an ellipsoid of revolution that degenerates into a straight line if the spheres are tangent to each other. In the case of distant spheres, the second of the surfaces is a two-sheet hyperboloid of revolution. In case the spheres contain one another, the sough geometric locus constitutes two co-axial co-focused ellipsoids of revolution. The equations defining the mentioned surfaces are presented. The regularities in shape and location of these surfaces were studied; the formulas for the major and the minor axes of the ellipsoids and the vertices of the two-sheet hyperboloids of revolution were derived.

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11

Davis, W. E., and J. Craig Yacoe. "A New Polyhedral Approximation to an Ellipsoid of Revolution." International Journal of Space Structures 5, no. 3-4 (September 1990): 187–95. http://dx.doi.org/10.1177/026635119000500304.

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12

Grafarend, E. W., and R. Syffus. "Mixed cylindric map projections of the ellipsoid of revolution." Journal of Geodesy 71, no. 11 (October 16, 1997): 685–94. http://dx.doi.org/10.1007/s001900050136.

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13

Harding, S. E., and H. Colfen. "Inversion Formulas for Ellipsoid of Revolution Macromolecular Shape Functions." Analytical Biochemistry 228, no. 1 (June 1995): 131–42. http://dx.doi.org/10.1006/abio.1995.1324.

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14

Bezuglyi, M. A., N. V. Bezuglaya, A. V. Ventsuryk, and K. P. Vonsevych. "Angular Photometry of Biological Tissue by Ellipsoidal Reflector Method." Devices and Methods of Measurements 10, no. 2 (June 24, 2019): 160–68. http://dx.doi.org/10.21122/2220-9506-2019-10-2-160-168.

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Анотація:

Angular measurements in optics of biological tissues are used for different applied spectroscopic task for roughness surface control, define of refractive index and for research of optical properties. Purpose of the research is investigation of the reflectance of biologic tissues by the ellipsoidal reflector method under the variable angle of the incident radiation.The research investigates functional features of improved photometry method by ellipsoidal reflectors. The photometric setup with mirror ellipsoid of revolution in reflected light was developed. Theoretical foundations of the design of an ellipsoidal reflector with a specific slot to ensure the input of laser radiation into the object area were presented. Analytical solution for calculating the angles range of incident radiation depending on the eccentricity and focal parameter of the ellipsoid are obtained. Also created the scheme of image processing at angular photometry by ellipsoidal reflector.The research represents results of experimental series for samples of muscle tissues at wavelengths 405 nm, 532 nm, 650 nm. During experiment there were received photometric images on the equipment with such parameters: laser beam incident angles range 12.5–62.5°, ellipsoidal reflector eccentricity 0.6, focal parameter 18 mm, slot width 8 mm.The nature of light scattering by muscle tissues at different wavelengths was represented by graphs for the collimated reflection area. The investigated method allows qualitative estimation of influence of internal or surface layers of biologic tissues optical properties on the light scattering under variable angles of incident radiation by the shape of zone of incident light.

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15

Gevorgyan, Eva, Armen Nersessian, Vadim Ohanyan, and Evgeny Tolkachev. "Landau problem on the ellipsoid, hyperboloid and paraboloid of revolution." Modern Physics Letters A 29, no. 29 (September 21, 2014): 1450148. http://dx.doi.org/10.1142/s021773231450148x.

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We define the Landau problem on two-dimensional ellipsoid, hyperboloid and paraboloid of revolution. Starting from the two-center McIntosh–Cisneros–Zwanziger (MICZ)–Kepler system and making the reduction into these two-dimensional surfaces, we obtain the Hamiltonians of the charged particle moving on the corresponding surface of revolution in the magnetic field conserving the symmetry of the two-dimensional surface (Landau problem). For each case we figure out the values of parameter for which the qualitative character of the motion coincides with that of a free particle moving on the same two-dimensional surface. For the case of finite trajectories we construct the action-angle variables.

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16

Williams, Roy. "Gnomonic Projection of the Surface of an Ellipsoid." Journal of Navigation 50, no. 2 (May 1997): 314–20. http://dx.doi.org/10.1017/s0373463300023936.

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When a surface is mapped onto a plane so that the image of a geodesic arc is a straight line on the plane then the mapping is known as a geodesic mapping. It is only possible to perform a geodesic mapping of a surface onto a plane when the surface has constant normal curvature. The normal curvature of a sphere of radius r at all points on the surface is I/r hence it is possible to map the surface of a sphere onto a plane using a geodesic mapping. The geodesic mapping of the surface of a sphere onto a plane is achieved by a gnomonic projection which is the projection of the surface of the sphere from its centre onto a tangent plane. There is no geodesic mapping of the ellipsoid of revolution or the spheroid onto a plane because the ellipsoid of revolution or the spheroid are not surfaces whose curvature is constant at all points. We can, however, still construct a projection of the surface of the ellipsoid from the centre of the body onto a tangent plane and we call this projection a gnomonic projection also.

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17

FRENKEL, D., and B. M. MULDER. "The hard ellipsoid-of-revolution fluid I. Monte Carlo simulations." Molecular Physics 100, no. 1 (January 10, 2002): 201–17. http://dx.doi.org/10.1080/00268970110088992.

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18

Gavrilenko, V. V. "Transient loading as an ellipsoid of revolution penetrates a fluid." Soviet Applied Mechanics 22, no. 8 (August 1986): 797–802. http://dx.doi.org/10.1007/bf00911335.

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19

Zurzycki, J., and H. Gabryś. "Changes in light absorption by the chloroplast, related to its structural transformations." Acta Societatis Botanicorum Poloniae 46, no. 2 (2015): 369–80. http://dx.doi.org/10.5586/asbp.1977.029.

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Анотація:

The changes in light absorption of single chloroplasts and one layer of chloroplasts related to their structural transformations were considered. Theoretical calculations of light absorption (E<sub>A</sub>) and transmission (E<sub>T</sub>) as well as for the mean intensity of absorption (I<sub>A</sub>) for the ellipsoid of revolution were given by the formulas 3,2 and l respectively. It was shown that the true shape of <i>Funaria</i> chloroplasts can be considered as ellipsoid of revolution. From the four conformational states of chloroplasts the most flattened one (corresponding to the low intensity of illumination) absorbs maximal amount of light energy. For the one layer of chloroplasts the changes in light absorption connected with structural transformations were estimated as ca. 4%.

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20

Williams, Roy. "The Great Ellipse on the Surface of the Spheroid." Journal of Navigation 49, no. 2 (May 1996): 229–34. http://dx.doi.org/10.1017/s0373463300013333.

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On any surface which fulfils the required continuity conditions, the shortest path between two points on the surface is along the are of a geodesic curve. On the surface of a sphere the geodesic curves are the great circles and the shortest path between any two points on this surface is along the arc of a great circle, but on the surface of an ellipsoid of revolution, the geodesic curves are not so easily defined except that the equator of this ellipsoid is a circle and its meridians are ellipses.

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21

Kim, Sun-Chul. "THE MOTION OF POINT VORTEX DIPOLE ON THE ELLIPSOID OF REVOLUTION." Bulletin of the Korean Mathematical Society 47, no. 1 (January 31, 2010): 73–79. http://dx.doi.org/10.4134/bkms.2010.47.1.073.

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22

Bulaev, D. V., V. A. Geyler, and V. A. Margulis. "Magnetic response for an ellipsoid of revolution in a magnetic field." Physical Review B 62, no. 17 (November 1, 2000): 11517–26. http://dx.doi.org/10.1103/physrevb.62.11517.

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23

Nyrtsov, M. V., M. E. Fleis, and A. I. Sokolov. "Meridian section projections: a new class of the triaxial ellipsoid projections." Geodesy and Cartography 968, no. 2 (March 20, 2021): 11–22. http://dx.doi.org/10.22389/0016-7126-2021-968-2-11-22.

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Historically the conformal projections have been used for mapping not only the Earth, but other celestial bodies as well. Their application enables preserving the shape of the relief features on the maps, which is extremely important for various analyses of celestial bodies’ surfaces. For many small bodies of the Solar system the International Astronomical Union recommends to apply a triaxial ellipsoid as a reference surface. But if the conformal projections for the reference surfaces of a sphere and an ellipsoid of revolution already exist, obtaining these projections for a triaxial ellipsoid will be significantly complicated, and the task of preserving the shape of relief features still actual. In general, the article deals with cylindrical and azimuthal projections of the meridian section for global mapping the celestial body surface in accordance with the idea formulated by prof. L. M. Bugaevsky. The projections are implemented for mapping of Phobos, moon of Mars.

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24

Fleis, M. E., M. V. Nyrtsov, M. M. Borisov, and A. I. Sokolov. "The accurate calculation of the geodetic heights of the celestial body’s surface points relative to the triaxial ellipsoid." Доклады Академии наук 486, no. 4 (June 10, 2019): 489–93. http://dx.doi.org/10.31857/s0869-56524864489-493.

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Анотація:

A sphere or ellipsoid of revolution are usually used for approximation of the physical surface of the Earth. In some cases, a triaxial ellipsoid is used. The calculation of the geodetic height of points on the Earth’s surface is carried out mainly by approximate methods using the formulas for the dependence of spatial rectangular coordinates x, y, z on geodesics B, L, H. However, there are small bodies of the Solar system, for example, Eros 433 asteroid, for which such variants of the first approximation are incorrect, since in this case both first approximations are not small quantities. This article proposes a fundamentally new approach to calculating the geodesic height relative to the triaxial ellipsoid, based on the joint use of the normal equation for a surface passing through a given point and the equations of the surface itself. The method is reduced to solving the sixth-degree equation by the Sturm method and the fourth-degree equation by the Ferrari method.

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25

Sjöberg, L. E. "Solutions to the ellipsoidal Clairaut constant and the inverse geodetic problem by numerical integration." Journal of Geodetic Science 2, no. 3 (November 1, 2012): 162–71. http://dx.doi.org/10.2478/v10156-011-0037-4.

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AbstractWe derive computational formulas for determining the Clairaut constant, i.e. the cosine of the maximum latitude of the geodesic arc, from two given points on the oblate ellipsoid of revolution. In all cases the Clairaut constant is unique. The inverse geodetic problem on the ellipsoid is to determine the geodesic arc between and the azimuths of the arc at the given points. We present the solution for the fixed Clairaut constant. If the given points are not(nearly) antipodal, each azimuth and location of the geodesic is unique, while for the fixed points in the ”antipodal region”, roughly within 36”.2 from the antipode, there are two geodesics mirrored in the equator and with complementary azimuths at each point. In the special case with the given points located at the poles of the ellipsoid, all meridians are geodesics. The special role played by the Clairaut constant and the numerical integration make this method different from others available in the literature.

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26

Hansen, Steen. "Update for BayesApp: a web site for analysis of small-angle scattering data." Journal of Applied Crystallography 47, no. 4 (July 19, 2014): 1469–71. http://dx.doi.org/10.1107/s1600576714013156.

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Анотація:

An update for BayesApp, a web site for analysis of small-angle scattering data, is presented. The indirect transformation of the scattering data now includes an option for a maximum-entropy constraint in addition to the conventional smoothness constraint. The maximum-entropy constraint uses an ellipsoid of revolution as a prior, and the dimensions of the ellipsoid as well as the overall noise level of the experimental data are estimated using Bayesian methods. Furthermore, a correction for slit smearing has been added. The web site also includes options for calculation of the scattering intensity from simple models as well as the estimation of structure factors for polydisperse spheres and nonspherical objects of axial ratios between 0.4 and 2.5.

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27

Fatyanov, S. O., A. P. Pustovalov, V. M. Pashchenko, A. S. Morozov, and E. S. Semina. "Determination of the parameters of an ellipsoidal electrode tip for treating agricultural animals using UHF – therapy methods." BIO Web of Conferences 37 (2021): 00046. http://dx.doi.org/10.1051/bioconf/20213700046.

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Анотація:

The use of non-medicamentous means of treating farm animals presupposes the presence of not only an ultra-high frequency electromagnetic radiation generator but also a rectal emitter, which is directly inserted into the animal's rectum. The effectiveness of the treatment carried out using UHF therapy methods largely depends on the shape of the emitter tip. When choosing the external shape of the emitter tip in the form of half of an ellipsoid of revolution, it becomes necessary to optimize the parameters of this ellipsoid. The goal of optimization is to minimize the resistance force of the living tissue of the animal when a cylindrical emitter with a tip is inserted into the rectum.

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28

Modenini, D. "Five-Degree-of-Freedom Pose Estimation from an Imaged Ellipsoid of Revolution." Journal of Spacecraft and Rockets 56, no. 3 (May 2019): 952–58. http://dx.doi.org/10.2514/1.a34340.

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29

Savchenko, A. O., and O. Ya Savchenko. "Flow around an ellipsoid of revolution in a harmonic coaxial vector field." Journal of Applied and Industrial Mathematics 6, no. 2 (April 2012): 224–28. http://dx.doi.org/10.1134/s1990478912020111.

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30

Bonnard, Bernard, Jean-Baptiste Caillau, and Ludovic Rifford. "Convexity of injectivity domains on the ellipsoid of revolution: The oblate case." Comptes Rendus Mathematique 348, no. 23-24 (December 2010): 1315–18. http://dx.doi.org/10.1016/j.crma.2010.10.036.

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31

Grafarend, E. W., and R. Syffus. "The Hammer projection of the ellipsoid of revolution (azimuthal, transverse, rescaled equiareal)." Journal of Geodesy 71, no. 12 (November 19, 1997): 736–48. http://dx.doi.org/10.1007/s001900050140.

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32

Li, Xiong, and Hans‐Jürgen Götze. "Ellipsoid, geoid, gravity, geodesy, and geophysics." GEOPHYSICS 66, no. 6 (November 2001): 1660–68. http://dx.doi.org/10.1190/1.1487109.

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Geophysics uses gravity to learn about the density variations of the Earth’s interior, whereas classical geodesy uses gravity to define the geoid. This difference in purpose has led to some confusion among geophysicists, and this tutorial attempts to clarify two points of the confusion. First, it is well known now that gravity anomalies after the “free‐air” correction are still located at their original positions. However, the “free‐air” reduction was thought historically to relocate gravity from its observation position to the geoid (mean sea level). Such an understanding is a geodetic fiction, invalid and unacceptable in geophysics. Second, in gravity corrections and gravity anomalies, the elevation has been used routinely. The main reason is that, before the emergence and widespread use of the Global Positioning System (GPS), height above the geoid was the only height measurement we could make accurately (i.e., by leveling). The GPS delivers a measurement of height above the ellipsoid. In principle, in the geophysical use of gravity, the ellipsoid height rather than the elevation should be used throughout because a combination of the latitude correction estimated by the International Gravity Formula and the height correction is designed to remove the gravity effects due to an ellipsoid of revolution. In practice, for minerals and petroleum exploration, use of the elevation rather than the ellipsoid height hardly introduces significant errors across the region of investigation because the geoid is very smooth. Furthermore, the gravity effects due to an ellipsoid actually can be calculated by a closed‐form expression. However, its approximation, by the International Gravity Formula and the height correction including the second‐order terms, is typically accurate enough worldwide.

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33

Portman, V., V. Shuster, Y. Rubenchik, and Y. Shneor. "Substitute Geometry of the Features of Size: Applications to Multidimensional Features." Journal of Computing and Information Science in Engineering 7, no. 1 (November 14, 2006): 52–65. http://dx.doi.org/10.1115/1.2410021.

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The problems of the substitute geometry for features of size are considered and an algorithm for synthesis of the substitute features (SF) is developed. Three and only three classes of surfaces are proved to have an incomplete set of position and orientation deviations within the SF equation: cylinders with any directrix, surfaces of revolution with any meridian, and helical surfaces with any profile. The form accuracy of multidimensional features relating to these classes is considered: ellipsoid of revolution, epitrochoidal cylinder, and Archimedean screw. The deterministic consideration is accompanied by evaluation of the uncertainty of the standard assessments of the geometric accuracy and capacity of the computing procedure.

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34

Angeles Alfonseca, M., and Jaegil Kim. "On the Local Convexity of Intersection Bodies of Revolution." Canadian Journal of Mathematics 67, no. 1 (February 1, 2015): 3–27. http://dx.doi.org/10.4153/cjm-2013-039-4.

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AbstractOne of the fundamental results in convex geometry is Busemann's theorem, which states that the intersection body of a symmetric convex body is convex. Thus, it is only natural to ask if there is a quantitative version of Busemann's theorem, i.e., if the intersection body operation actually improves convexity. In this paper we concentrate on the symmetric bodies of revolution to provide several results on the (strict) improvement of convexity under the intersection body operation. It is shown that the intersection body of a symmetric convex body of revolution has the same asymptotic behavior near the equator as the Euclidean ball. We apply this result to show that in sufficiently high dimension the double intersection body of a symmetric convex body of revolution is very close to an ellipsoid in the Banach–Mazur distance. We also prove results on the local convexity at the equator of intersection bodies in the class of star bodies of revolution.

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35

Akhsakhalyan, A. A., A. D. Akhsakhalyan, E. B. Klyuenkov, V. A. Murav’ev, N. N. Salashchenko, and A. I. Kharitonov. "A multilayer x-ray mirror in the form of an ellipsoid of revolution." Bulletin of the Russian Academy of Sciences: Physics 71, no. 1 (January 2007): 64–67. http://dx.doi.org/10.3103/s1062873807010169.

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36

Jha, R. M., D. J. Edwards, and R. Bhakthavathsalam. "Surface-ray tracing solution of ellipsoid of revolution for UTD mutual coupling applications." Electronics Letters 28, no. 4 (1992): 367. http://dx.doi.org/10.1049/el:19920230.

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37

Engels, J., and E. Grafarend. "The oblique Mercator projection of the ellipsoid of revolution IE a 2 ,b." Journal of Geodesy 70, no. 1-2 (November 1995): 38–50. http://dx.doi.org/10.1007/bf00863417.

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38

Devaney, Anthony J. "Generalized Projection-Slice Theorem for Fan Beam Diffraction Tomography." Ultrasonic Imaging 7, no. 3 (July 1985): 264–75. http://dx.doi.org/10.1177/016173468500700306.

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A generalized projection-slice theorem is derived for transmission fan beam diffraction tomography within the Born or Rytov approximations. The development is based on the use of the so-called paraxial approximation which requires that the object being probed subtend a small angle relative to the source point and to the measurement plane. Within this approximation it is shown that the transmitted field measured over a plane surface located on the opposite side of the object from the insonifying point source determines the three-dimensional spatial Fourier transform of the object profile over the surface of an ellipsoid of revolution in Fourier space. In the special case where the point source is in the far field of the object the semiaxes of the ellipsoid become equal and the surface degenerates to a sphere and the result reduces to the usual projection-slice theorem of plane beam diffraction tomography.

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39

Kerkovits, Krisztián, and Tünde Takáts. "Reference frame and map projection for irregular shaped celestial bodies." Abstracts of the ICA 2 (October 9, 2020): 1–2. http://dx.doi.org/10.5194/ica-abs-2-42-2020.

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Abstract. Recent advancements of technology resulted in greater knowledge of the Solar System and the need for mapping small celestial bodies significantly increased. However, creating a good map of such small objects is a big challenge for the cartographer: they are usually irregular shaped, the usual reference frames like the ellipsoid of revolution is inappropriate for their approximation.A method is presented to develop best-fitting irregular surfaces of revolution that can approximate any irregular celestial body. (Fig. 1.) Then a simple equal-area map projection is calculated to map this reference frame onto a plane. The shape of the resulting map in this projection resembles the shape of the original celestial body.The usefulness of the method is demonstrated on the example of the comet 67P/Churyumov-Gerasimenko. This comet has a highly irregular shape, which is hard to map. Previously used map projections for this comet include the simple cylindrical, which greatly distorts the surface and cannot depict the depressions of the object. Other maps used the combination of two triaxial ellipsoids as the reference frame, and the gained mapping had low distortion but at the expense of showing the tiny surface divided into 11 maps in different complicated map projections (Nyrtsov et. al., 2018). On the other hand, our mapping displays the comet in one single map with moderate distortion and the shape of the map frame suggests the original shape of the celestial body (Fig. 2. and 3.).

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40

Hansen, Steen. "Simultaneous estimation of the form factor and structure factor for globular particles in small-angle scattering." Journal of Applied Crystallography 41, no. 2 (March 8, 2008): 436–45. http://dx.doi.org/10.1107/s0021889808004937.

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Small-angle scattering data from non-dilute solutions of particles are often analysed by indirect Fourier transformation using a specific model structure factor to obtain an estimate of the distance distribution function that is free from concentration effects. A new approach is suggested here, whereby the concentration effects are expressed solely through real space functions without the use of an explicit structure factor. This is done by dividing the total distance distribution function for the scattering into three different contributions, as suggested by Kruglov [(2005).J. Appl. Cryst.38, 716–720]: (i) the single particle distribution which is due to intraparticle effects, (ii) the excluded volume distribution from excluded volume effects which is only dependent upon the geometry of the particles, and (iii) a structure distribution which is due to the remaining interaction between the particles. Only the single particle distribution and the structure distribution are allowed to vary freely (within the restrictions of a smoothness constraint). These two distributions may be separated mainly because they differ in their regions of support in real space. From the estimated distributions the structure factor can be calculated. For deviations of particles from spherical symmetry, the excluded volume distribution may be approximated by that of an ellipsoid of revolution. Excluded volume distributions have been calculated for ellipsoids of revolution of axial ratios between 0.1 and 10 and implemented in the programIFTc, which is described in the appendix. The validity of the approach is demonstrated for globular particles.

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41

Sagiv, A. "Inflation of an Axisymmetric Membrane: Stress Analysis." Journal of Applied Mechanics 57, no. 3 (September 1, 1990): 682–87. http://dx.doi.org/10.1115/1.2897077.

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An analysis of the stresses and deformations of an inflated axisymmetric membrane was obtained. Large deformations of Mooney material were assumed. The development of the governing differential equations is an extension of the Adkins-Rivlin equations. The extension is for a general form of undeformed profiles with symmetry of revolution. The equations obtained, when reduced identically to the special cases treated in the literature of flat, spherical, and half-ellipsoid undeformed shapes, are fully compatible. A family of axisymmetric ellipsoid curves were used as an example of undeformed shapes for numerical demonstration. A fourth-order Runge-Kutta method was applied to integrate the equations. The results show relations between the nondimensional parameters governing the deformed membrane, such as pressure, membrane height, undeformed profiles, stresses, and deformations. An analysis of a very large deformation was carried out. It was found that in this case the membrane surface approaches a spherical shape except near the support, regardless of its undeformed profile.

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42

Kholshevnikov, Konstantin V., Danila V. Milanov, and Vakhit Sh Shaidulin. "Stokes constants of an oblate ellipsoid of revolution with equidensites homothetic to its surface." Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy 4(62), no. 3 (2017): 516–24. http://dx.doi.org/10.21638/11701/spbu01.2017.313.

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43

Kholshevnikov, K. V., D. V. Milanov, and V. Sh Shaidulin. "Stokes constants of an oblate ellipsoid of revolution with equidensites homothetic to its surface." Vestnik St. Petersburg University, Mathematics 50, no. 3 (July 2017): 318–24. http://dx.doi.org/10.3103/s1063454117030098.

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44

Stolbov, Oleg, and Yuriy Raikher. "Large-Scale Shape Transformations of a Sphere Made of a Magnetoactive Elastomer." Polymers 12, no. 12 (December 8, 2020): 2933. http://dx.doi.org/10.3390/polym12122933.

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Magnetostriction effect, i.e., deformation under the action of a uniform applied field, is analyzed to detail for a spherical sample of a magnetoactive elastomer (MAE). A close analogy with the field-induced elongation of spherical ferrofluid droplets implies that similar characteristic effects viz. hysteresis stretching and transfiguration into a distinctively nonellipsoidal bodies, should be inherent to MAE objects as well. The absence until now of such studies seems to be due to very unfavorable conclusions which follow from the theoretical estimates, all of which are based on the assumption that a deformed sphere always retains the geometry of ellipsoid of revolution just changing its aspect ratio under field. Building up an adequate numerical modelling tool, we show that the ‘ellipsoidal’ approximation is misleading beginning right from the case of infinitesimal field strengths and strain increments. The results obtained show that the above-mentioned magnetodeformational effect should distinctively manifest itself in the objects made of quite ordinary MAEs, e.g., composites on the base of silicone cautchouc filled with micron-size carbonyl iron powder.

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45

Lewis, R. I. "An Inverse Method for the Design of Bodies of Revolution by Boundary Integral Modelling." Proceedings of the Institution of Mechanical Engineers, Part C: Mechanical Engineering Science 205, no. 2 (March 1991): 91–97. http://dx.doi.org/10.1243/pime_proc_1991_205_096_02.

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A surface vorticity boundary integral method is presented for the design of bodies of revolution in axisymmetric flow. The analysis finds the desired body shape to deliver a prescribed surface potential flow velocity or pressure distribution. To achieve this the body surface is simulated by a flexible vorticity sheet of prescribed strength. Starting from an arbitrary first guess for the body shape, normally an ellipsoid, the flexible vortex sheet is successively realigned with its own self-induced flow field during an iterative process which converges accurately onto the desired shape. A well-proven analysis method is also presented for back-checking the final design.

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46

Kawski, A. "On the Estimation of Excited-State Dipole Moments from Solvatochromic Shifts of Absorption and Fluorescence Spectra." Zeitschrift für Naturforschung A 57, no. 5 (May 1, 2002): 255–62. http://dx.doi.org/10.1515/zna-2002-0509.

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The effect of the solvent polarity parameters f (Ɛ, n) and ᵩ(Ɛ, n) appearing in solvatochromic theories, and the effect of the molecular shape of the Onsager cavity (sphere, ellipsoid of revolution) on the determined electric dipole moments μe in the singlet excited state are studied. It is found that the shape of the solute does not exhibit a significant effect on the determined values of μe, but only on the solvent parameters f (Ɛ, n) and ᵩ(Ɛ, n) as well as on the Onsager radius a. Passing from a sphere to an ellipsoid leads to such a change in the scale that induces a proportional change in the slope coefficients m1 and m2. Also the effect of α/a3 (a and a are the mean isotropic polarizability of the solute and the Onsager cavity radius in a homogeneous dielectric, respectively) on the determined values of m e has been studied, and it is found that the assumption α/a3 = 1/2 is valid in many cases.

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47

Munitsyna, M. A. "On Transients in the Dynamics of an Ellipsoid of Revolution on a Plane with Friction." Mechanics of Solids 54, no. 4 (July 2019): 545–50. http://dx.doi.org/10.3103/s0025654419040071.

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48

Crocetto, N. "POINT PROJECTION OF TOPOGRAPHIC SURFACE ONTO THE REFERENCE ELLIPSOID OF REVOLUTION IN GEOCENTRIC CARTESIAN COORDINATES." Survey Review 32, no. 250 (October 1993): 233–38. http://dx.doi.org/10.1179/sre.1993.32.250.233.

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49

Pereira, E. M. A., J. V. Silva, T. H. F. Andrade, S. R. de Farias Neto, and A. G. Barbosa de Lima. "Simulating Heat and Mass Transfer in Drying Process: Applications in Grains." Diffusion Foundations 3 (February 2015): 3–18. http://dx.doi.org/10.4028/www.scientific.net/df.3.3.

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The objective of this study was to investigate numerically heat and mass transport during drying of grains with particular reference to bean and rough rice. The proposed mathematical models based on the Fick’s and Fourier’s Laws consider constant physical properties and convective boundary condition at the surface of the solid. The solutions of the governing equations were performed using ANSYS CFX®software. The grains were regarded as an ellipsoid of revolution. Results of the drying and heating kinetics and moisture content and temperature distributions in the grains along the drying process are presented and analyzed.

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50

da Silva, José Vieira, Filipe Nascimento Silva, Tony Herbert Freire de Andrade, and Antônio Gilson Barbosa de Lima. "Drying of Rough Rice: A Numerical Investigation." Defect and Diffusion Forum 334-335 (February 2013): 131–36. http://dx.doi.org/10.4028/www.scientific.net/ddf.334-335.131.

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Drying can be defined as a simultaneous heat and mass transfer process between product and air drying. These phenomena have occurred due to thermal and moisture gradients and they are affected by different parameters. The inadequate management of this process can cause severe damage to the product such as cracking, loss of nutritional properties and even loss of the product quality. Thus, this paper proposes a numerical study of heat and mass transfer that occurs during the rough rice drying process, using the commercial software ANSYS CFX®. Herein the grain was considered as ellipsoid of revolution.

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