Journal articles: 'Ellipsoids of revolution'

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Rigby, Maurice. "Hard ellipsoids of revolution." Molecular Physics 66, no. 6 (April 20, 1989): 1261–68. http://dx.doi.org/10.1080/00268978900100851.

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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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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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Frenkel, Daan, Bela M. Mulder, and John P. Mctague. "Phase Diagram of Hard Ellipsoids of Revolution." Molecular Crystals and Liquid Crystals 123, no. 1 (February 1985): 119–28. http://dx.doi.org/10.1080/00268948508074770.

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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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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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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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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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Daghighi, Shahrzad, Mohammad Rouhi, Giovanni Zucco, and Paul M. Weaver. "Bend-free design of ellipsoids of revolution using variable stiffness composites." Composite Structures 233 (February 2020): 111630. http://dx.doi.org/10.1016/j.compstruct.2019.111630.

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Daghighi, Shahrzad, Giovanni Zucco, Mohammad Rouhi, and Paul M. Weaver. "Bend-free design of super ellipsoids of revolution composite pressure vessels." Composite Structures 245 (August 2020): 112283. http://dx.doi.org/10.1016/j.compstruct.2020.112283.

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Yi, Y. B., and A. M. Sastry. "Analytical approximation of the percolation threshold for overlapping ellipsoids of revolution." Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 460, no. 2048 (August 8, 2004): 2353–80. http://dx.doi.org/10.1098/rspa.2004.1279.

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Bautista-Carbajal, Gustavo, Arturo Moncho-Jordá, and Gerardo Odriozola. "Further details on the phase diagram of hard ellipsoids of revolution." Journal of Chemical Physics 138, no. 6 (February 14, 2013): 064501. http://dx.doi.org/10.1063/1.4789957.

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Ozerin, A. N., D. I. Svergun, V. V. Volkov, A. I. Kuklin, V. I. Gordelyi, A. Kh Islamov, L. A. Ozerina, and D. S. Zavorotnyuk. "The spatial structure of dendritic macromolecules." Journal of Applied Crystallography 38, no. 6 (November 12, 2005): 996–1003. http://dx.doi.org/10.1107/s0021889805032115.

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A low-resolutionab initioshape determination was performed from small-angle neutron and X-ray scattering (SANS and SAXS) curves from solutions of polycarbosilane dendrimers with the three-functional and the four-functional branching centre of the fourth, fifth, sixth, seventh and eighth generations. In all cases, anisometric dendrimer shapes were obtained. The overall shapes of the dendrimers with the three- and four-functional branching centres were oblate ellipsoids of revolution and triaxial ellipsoids, respectively. The restored bead models revealed a pronounced heterogeneity within the dendrimer structure. The density deficit was observed in the central part and close to the periphery of the dendrimers. The fraction of the overall volume of the dendrimers available for solvent penetration was about 0.2–0.3. These results may help in the design of new practical applications of dendrimer macromolecules.

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Kang, J. H., and A. W. Leissa. "Vibration analysis of solid ellipsoids and hollow ellipsoidal shells of revolution with variable thickness from a three-dimensional theory." Acta Mechanica 197, no. 1-2 (October 19, 2007): 97–117. http://dx.doi.org/10.1007/s00707-007-0491-3.

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Solomonov, A. I., O. M. Kushchenko, D. A. Yavsin, M. V. Rybin, and A. D. Sinelnik. "Active narrowband filter based on 2.5D metasurface from Ge2Sb2Te5." Journal of Physics: Conference Series 2015, no. 1 (November 1, 2021): 012147. http://dx.doi.org/10.1088/1742-6596/2015/1/012147.

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Abstract We propose a new concept of an active narrowband filter based on a 2.5D metasurface from Ge2Sb2Te5 (GST). In this paper, we present a numerical calculation of the transmission spectrum from a structure of ellipsoids of revolution. For this 2.5D metasurface, modulation of narrow peaks in the IR range for s- and p-polarization is shown. A manufacturing technique using two-photon lithography and laser electrodispersion is proposed.

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Sjöberg, Bo. "Small-angle scattering from collections of interacting hard ellipsoids of revolution studied by Monte Carlo simulations and other methods of statistical thermodynamics." Journal of Applied Crystallography 32, no. 5 (October 1, 1999): 917–23. http://dx.doi.org/10.1107/s0021889899006640.

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Computer simulations using Monte Carlo methods are used to investigate the effects of interparticle correlations on small-angle X-ray and neutron scattering from moderate or highly concentrated systems of ellipsoids of revolution. Both oblate and prolate ellipsoids, of varying eccentricities and concentrations, are considered. The advantage with Monte Carlo simulation is that completely general models, both regarding particle shapes and interaction potentials, can be considered. Equations are also given that relate the nonideal part of the chemical potential, βμni, with the scattering at zero angle,I(0), and the compressibility factor,z. The quantity βμnican be obtained during the Monte Carlo simulations by using Widom's test-particle method. For spherical particles, the simulations are compared with approximation formulas based on the Percus–Yevick equation. A method is also suggested for the calculation of both βμniandzfrom experimental values ofI(0) recorded as a function of concentration.

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SOULIÈRE, ANIK, and TADASHI TOKIEDA. "Periodic motions of vortices on surfaces with symmetry." Journal of Fluid Mechanics 460 (June 10, 2002): 83–92. http://dx.doi.org/10.1017/s0022112002008315.

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The theory of point vortices in a two-dimensional ideal fluid has a long history, but on surfaces other than the plane no method of finding periodic motions (except relative equilibria) of N vortices is known. We present one such method and find infinite families of periodic motions on surfaces possessing certain symmetries, including spheres, ellipsoids of revolution and cylinders. Our families exhibit bifurcations. N can be made arbitrarily large. Numerical plots of bifurcations are given.

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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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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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Greene, Daniel G., Daniel V. Ferraro, Abraham M. Lenhoff, and Norman J. Wagner. "A critical examination of the decoupling approximation for small-angle scattering from hard ellipsoids of revolution." Journal of Applied Crystallography 49, no. 5 (September 29, 2016): 1734–39. http://dx.doi.org/10.1107/s1600576716012929.

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The decoupling approximation, proposed by Kotlarchyk & Chen [J. Chem. Phys. (1983), 79, 2461–2469], is a first-order correction to the experimentally determined apparent structure factor that is necessary because of concentration effects in polydisperse and/or nonspherical systems. While the approximation is considered accurate for spheres with low polydispersity (<10%), the corresponding limitations for nonspherical particles are unknown. The validity of this approximation is studied for monodisperse dispersions of hard ellipsoids of revolution with aspect ratios ranging from 0.333 to 3 and a guide for its accuracy is provided.

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Theenhaus, Th, M. P. Allen, M. Letz, A. Latz, and R. Schilling. "Dynamical precursor of nematic order in a dense fluid of hard ellipsoids of revolution." European Physical Journal E 8, no. 3 (June 2002): 269–74. http://dx.doi.org/10.1140/epje/i2001-10093-7.

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Pujara, Nimish, and Evan A. Variano. "Rotations of small, inertialess triaxial ellipsoids in isotropic turbulence." Journal of Fluid Mechanics 821 (May 25, 2017): 517–38. http://dx.doi.org/10.1017/jfm.2017.256.

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The statistics of rotational motion of small, inertialess triaxial ellipsoids are computed along Lagrangian trajectories extracted from direct numerical simulations of homogeneous isotropic turbulence. The total particle angular velocity and its components along the three principal axes of the particle are considered, expanding on the results of Chevillard & Meneveau (J. Fluid Mech., vol. 737, 2013, pp. 571–596) who showed results of the rotation rate of the particle’s principal axes. The variance of the particle angular velocity, referred to as the particle enstrophy, is found to increase as particles become elongated, regardless of whether they are axisymmetric. This trend is explained by considering the contributions of vorticity and strain rate to particle rotation. It is found that the majority of particle enstrophy is due to fluid vorticity. Strain-rate-induced rotations, which are sensitive to shape, are mostly cancelled by strain–vorticity interactions. The remainder of the strain-rate-induced rotations are responsible for weak variations in particle enstrophy. For particles of all shapes, the majority of the enstrophy is in rotations about the longest axis, which is due to alignment between the longest axis and fluid vorticity. The integral time scale for particle angular velocities about different axes reveals that rotations are most persistent about the longest axis, but that a full revolution is rare.

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Plyushch, Artyom, Dmitry Lyakhov, Mantas Šimėnas, Dzmitry Bychanok, Jan Macutkevič, Dominik Michels, Jūras Banys, Patrizia Lamberti, and Polina Kuzhir. "Percolation and Transport Properties in The Mechanically Deformed Composites Filled with Carbon Nanotubes." Applied Sciences 10, no. 4 (February 15, 2020): 1315. http://dx.doi.org/10.3390/app10041315.

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The conductivity and percolation concentration of the composite material filled with randomly distributed carbon nanotubes were simulated as a function of the mechanical deformation. Nanotubes were modelled as the hard-core ellipsoids of revolution with high aspect ratio. The evident anisotropy was observed in the percolation threshold and conductivity. The minimal mean values of the percolation of 4.6 vol. % and maximal conductivity of 0.74 S/m were found for the isotropic composite. Being slightly aligned, the composite demonstrates lower percolation concentration and conductivity along the orientation of the nanotubes compared to the perpendicular arrangement.

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Pankratov, Alexander, Tatiana Romanova, Igor Litvinchev, and Jose Antonio Marmolejo-Saucedo. "An Optimized Covering Spheroids by Spheres." Applied Sciences 10, no. 5 (March 7, 2020): 1846. http://dx.doi.org/10.3390/app10051846.

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Covering spheroids (ellipsoids of revolution) by different spheres is studied. The research is motivated by packing non-spherical particles arising in natural sciences, e.g., in powder technologies. The concept of an ε -cover is introduced as an outer multi-spherical approximation of the spheroid with the proximity ε . A fast heuristic algorithm is proposed to construct an optimized ε -cover giving a reasonable balance between the value of the proximity parameter ε and the number of spheres used. Computational results are provided to demonstrate the efficiency of the approach.

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Wei, Gaoyuan, and B. E. Eichinger. "Shape distributions for Gaussian molecules: circular and linear chains as spheres and ellipsoids of revolution." Macromolecules 22, no. 8 (August 1989): 3429–35. http://dx.doi.org/10.1021/ma00198a039.

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Shim, Hyun-Ju, and Jae-Hoon Kang. "Free vibrations of solid and hollow hemi-ellipsoids of revolution from a three-dimensional theory." International Journal of Engineering Science 42, no. 17-18 (October 2004): 1793–815. http://dx.doi.org/10.1016/j.ijengsci.2004.04.008.

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Krivoshapko, Sergey N., and Vyacheslav N. Ivanov. "Simplified selection of optimal shell of revolution." Structural Mechanics of Engineering Constructions and Buildings 15, no. 6 (December 15, 2019): 438–48. http://dx.doi.org/10.22363/1815-5235-2019-15-6-438-448.

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Relevance. Architects and engineers, designing shells of revolution, use in their projects, as a rule, spherical shells, paraboloids, hyperboloids, and ellipsoids of revolution well proved themselves. But near hundreds of other surfaces of revolution, which can be applied with success in building and in machine-building, are known. Methods. Optimization problem of design of axisymmetric shell subjected to given external load is under consideration. As usual, the solution of this problem consists in the finding of shape of the meridian and in the distribution of the shell thickness along the meridian. In the paper, the narrower problem is considered. That is a selection of the shell shape from several known types, the middle surfaces of which can be given by parametrical equations. The results of static strength analyses of the domes of different Gaussian curvature with the same overall dimensions subjected to the uniformly distributed surface load are presented. Variational-difference energy method of analysis is used. Results. Comparison of results of strength analyses of six selected domes showed that a paraboloid of revolution and a dome with a middle surface in the form of the surface of rotation of the z = - a cosh( x/b ) curve around the Oz axis have the better indices of stress-strain state. These domes work almost in the momentless state and it is very well for thin-walled shell structures. New criterion of optimality can be called “minimum normal stresses in shells of revolution with the same overall dimensions, boundary conditions, and external load”.

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Hall, S. B., M. S. Bermel, Y. T. Ko, H. J. Palmer, G. Enhorning, and R. H. Notter. "Approximations in the measurement of surface tension on the oscillating bubble surfactometer." Journal of Applied Physiology 75, no. 1 (July 1, 1993): 468–77. http://dx.doi.org/10.1152/jappl.1993.75.1.468.

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This paper examines two factors, shape deformation and surface viscosity, that affect measurements of surface tension of lung surfactants with the oscillating bubble surfactometer. At lower surface tensions, the compressed bubble in this apparatus becomes deformed to an oblate ellipsoid that cannot be analyzed rigorously using the simplified (spherical) Laplace equation to calculate surface tension from interfacial pressure drop. However, for the small air bubbles present in this apparatus, analysis with more general equations for ellipsoids of revolution shows that deformation effects are limited to extremely low surface tensions, and the absolute error from the spherical approximation is minimal in practice. In contrast, this was not the case for the effects of surface dilational viscosity in oscillating bubble calculations. Direct measurements and values from the literature indicated that the surface dilational viscosities of lung surfactant, dipalmitoyl phosphatidylcholine, and palmitic acid were sufficient to give substantial errors if their effects on interfacial pressure drop were neglected during dynamic cycling. Surface tension calculations at maximum and minimum radii on the oscillating bubble apparatus remain accurate, because the time derivative of radius becomes zero and viscous effects vanish. However, surface tensions determined at points other than these extremes of bubble size should be interpreted with caution.

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Ram, Jokhan, and Yashwant Singh. "Density-functional theory of the nematic phase: Results for a system of hard ellipsoids of revolution." Physical Review A 44, no. 6 (September 1, 1991): 3718–31. http://dx.doi.org/10.1103/physreva.44.3718.

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Kovalchuk, Vasyl, and Ivaïlo M. Mladenov. "λ-Spheres as a New Reference Model for Geoid: Explicit Solutions of the Direct and Inverse Problems for Loxodromes (Rhumb Lines)." Mathematics 10, no. 18 (September 15, 2022): 3356. http://dx.doi.org/10.3390/math10183356.

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In this paper, we present a new reference model that approximates the actual shape of the Earth, based on the concept of the deformed spheres with the deformation parameter λ. These surfaces, which are called λ-spheres, were introduced in another setting by Faridi and Schucking as an alternative to the spheroids (i.e., ellipsoids of revolution). Using their explicit parametrizations that we have derived in our previous papers, here we have defined the corresponding isothermal (conformal) coordinates as well as obtained and solved the differential equation describing the loxodromes (or rhumb lines) on such surfaces. Next, the direct and inverse problems for loxodromes have been formulated and the explicit solutions for azimuths and arc lengths have been presented. Using these explicit solutions, we have assessed the value of the deformation parameter λ for our reference model on the basis of the values for the semi-major axis of the Earth a and the quarter-meridian mp (i.e., the distance between the Equator and the North or South Pole) for the current best ellipsoidal reference model for the geoid, i.e., WGS 84 (World Geodetic System 1984). The latter is designed for use as the reference system for the GPS (Global Positioning System). Finally, we have compared the results obtained with the use of the newly proposed reference model for the geoid with the corresponding results for the ellipsoidal (WGS 84) and spherical reference models used in the literature.

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Howells, Malcolm R., Joel Anspach, and John Bender. "An assessment of approximating aspheres with more easily manufactured surfaces." Journal of Synchrotron Radiation 5, no. 3 (May 1, 1998): 814–16. http://dx.doi.org/10.1107/s0909049597018633.

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In designing optical systems for synchrotron radiation, one is often led to conclude that optimal performance can be obtained from optical surfaces described by conic sections of revolution, usually paraboloids and ellipsoids. The resulting design can lead to prescriptions for three-dimensional optical surfaces that are difficult to fabricate accurately. Under some circumstances satisfactory system performance can be achieved through the use of more easily manufactured surfaces such as cylinders, cones, bent cones, toroids and elliptical cylinders. These surfaces often have the additional benefits of scalability to large aperture, lower surface roughness and improved surface figure accuracy. In this paper we explore some of the conditions under which these more easily manufactured surfaces can be utilized without sacrificing performance.

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Roig-Solvas, Biel, Dana Brooks, and Lee Makowski. "A direct approach to estimate the anisotropy of protein structures from small-angle X-ray scattering." Journal of Applied Crystallography 52, no. 2 (February 26, 2019): 274–83. http://dx.doi.org/10.1107/s1600576719000918.

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In the field of small-angle X-ray scattering (SAXS), the task of estimating the size of particles in solution is usually synonymous with the Guinier plot. The approximation behind this plot, developed by Guinier in 1939, provides a simple yet accurate characterization of the scattering behavior of particles at low scattering angle or momentum transfer q, together with a computationally efficient way of inferring their radii of gyration R G. Moreover, this approximation is valid beyond spherical scatterers, making its use ubiquitous in the SAXS world. However, when it is important to estimate further particle characteristics, such as the anisotropy of the scatterer's shape, no similar or extended approximations are available. Existing tools to characterize the shape of scatterers rely either on prior knowledge of the scatterers' geometry or on iterative procedures to infer the particle shape ab initio. In this work, a low-angle approximation of the scattering intensity I(q) for ellipsoids of revolution is developed and it is shown how the size and anisotropy information can be extracted from the parameters of that approximation. The goal of the approximation is not to estimate a particle's full structure in detail, and thus this approach will be less accurate than well known iterative and ab initio reconstruction tools available in the literature. However, it can be considered as an extension of the Guinier approximation and used to generate initial estimates for the aforementioned iterative techniques, which usually rely on R G and D max for initialization. This formulation also demonstrates that nonlinearity in the Guinier plot can arise from anisotropy in the scattering particles. Beyond ideal ellipsoids of revolution, it is shown that this approximation can be used to estimate the size and shape of molecules in solution, in both computational and experimental scenarios. The limits of the approach are discussed and the impact of a particle's anisotropy in the Guinier estimate of R G is assessed.

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Kuvshinnikova, Daria. "Nonlocal theory: modeling of thermomechanical processes in a composite with inclusions in the form of ellipsoids of revolution." IOP Conference Series: Materials Science and Engineering 1191, no. 1 (October 1, 2021): 012016. http://dx.doi.org/10.1088/1757-899x/1191/1/012016.

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Hansen, Steen. "Monte Carlo estimation of the structure factor for hard bodies in small-angle scattering." Journal of Applied Crystallography 45, no. 3 (April 25, 2012): 381–88. http://dx.doi.org/10.1107/s0021889812009557.

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The structure factor and the effect of deviation from spherical symmetry are studied for elongated particles with axial ratios between 0.1 and 10. This is done using Monte Carlo simulation of the excluded-volume distance distribution function for ellipsoids of revolution, for cylinders and for cylinders with hemispherical end caps. The method suggested is general and is applicable to particles of any shape. The results of the calculations are compared with the Percus–Yevick formula [Percus & Yevick (1958),Phys. Rev.110, 1–13] for hard spheres. The comparisons indicate that the Percus–Yevick formula should only be used for axial ratios close to 1. For larger deviations from spherical symmetry it is often better to use the excluded-volume distance distribution function for the particle, in combination with the single-particle distance distribution function, for the calculation of the structure factor.

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Zavizion, O. V. "On the density of self-graviting disks having external potentials which coincide with gravitational potentials of ellipsoids of revolution." Astronomical School’s Report 1, no. 1 (2000): 82–85. http://dx.doi.org/10.18372/2411-6602.01.1082.

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De Miguel, Enrique, and Michael P. Allen. "Hard ellipsoids of revolution with square wells: a comparison between computer simulation and theory in the liquid-vapour region." Molecular Physics 76, no. 6 (August 20, 1992): 1275–79. http://dx.doi.org/10.1080/00268979200102061.

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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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35

Perera, A., P. G. Kusalik, and G. N. Patey. "The solution of the hypernetted chain and Percus–Yevick approximations for fluids of hard nonspherical particles. Results for hard ellipsoids of revolution." Journal of Chemical Physics 87, no. 2 (July 15, 1987): 1295–306. http://dx.doi.org/10.1063/1.453313.

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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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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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37

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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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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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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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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Perera, A., P. G. Kusalik, and G. N. Patey. "Erratum: The solution of the hypernetted chain and Percus–Yevick approximations for fluids of hard nonspherical particles. Results for hard ellipsoids of revolution [J. Chem. Phys. 87, 1295 (1987)]." Journal of Chemical Physics 89, no. 9 (November 1988): 5969. http://dx.doi.org/10.1063/1.455752.

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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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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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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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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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42

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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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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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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Kholshevnikov, Konstantin V., Danila V. Milanov, and Vakhit Sh Shaidulin. "Laplace series of ellipsoidal figures of revolution." Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy 4(62), no. 4 (2017): 695–703. http://dx.doi.org/10.21638/11701/spbu01.2017.417.

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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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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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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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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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50

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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Journal articles on the topic 'Ellipsoids of revolution'

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