Equations of an ellipsoid

Symmetry 50. Intercepts 50. Asymptotes 50. Equations of Slope 51. Tangents 51. Equations of a Straight Line 52. Equations of a Circle 53. Equations of a Parabola 53. Equations of an Ellipse of Eccentricity e 54. Equations of a Hyperbola 55. Equations of Three-Dimensional Coordinate Systems 56. Equations of a Plane 56. Equations of a Line 57. Equations of Angles 57. Equation of a Sphere 57. Equation of an Ellipsoid 57. Equations of Hyperboloids and Paraboloids 58. Equation of an Elliptic Cone 59. Equation of an Elliptic Cylinder 59. 

This is an equation of an ellipsoid arbitrary oriented with respect to any Cartesian frame [1]. The frame may be chosen in such a way that the eUipsoid will be oriented with its principal axes along the co-ordinate axes, fri the new frame x, y, z, the tensor is diagonal that is all the off-diagonal terms vanish ... 

Further, if the plane x — z cuts through the ellipsoid y = y, where y < b, then the equation of an ellipsoid becomes ... 

In Section 2.4 we have studied the behavior of the gravitational field of the spheroid outside of masses. Now let us focus our attention on the field of attraction inside masses. It may be proper to notice that the determination of the field caused by masses in the spheroid and, in general, by an ellipsoid, was a subject of classical works performed by Maclaurin, Lagrange, Laplace, Poisson, and others. As is well known, the equation of the ellipsoid, when the major axes are directed along coordinate lines is... 

Exercise. The rotation of an ellipsoidal particle suspended in a fluid obeys the macroscopic equation of motion... 

Atoms in crystals seldom have isotropic environments, and a better approximation (but still an approximation) is to describe the atomic motion in terms of an ellipsoid, with larger amplitudes of vibration in some directions than in others. Six parameters, the anisotropic vibration or displacement parameters, are introduced for each atom. Three of these parameters per atom give the orientations of the principal axes of the ellipsoid with respect to the unit cell axes. One of these principal axes is the direction of maximum displacement and the other two are perpendicular to this and also to each other. The other three parameters per atom represent the amounts of displacement along these three ellipsoidal axes. Some equations used to express anisotropic displacement parameters, which may be reported as 71, Uij, or jdjj, axe listed in Table 13.1. Most crystal structure determinations of all but the largest molecules include anisotropic temperature parameters for all atoms, except hydrogen, in the least-squares refinement. Usually, for brevity, the equivalent isotropic displacement factor Ueq, is published. This is expressed as ... 

The polarizability tensor is defined by an array of nine components but, in most cases, the tensor is symmetric so that a y = oty, Uy = a y, and ot = a x. The resulting six components of the polarizability tensor can be better visualized by considering an equation for an ellipsoid in Cartesian coordinates ... 

Fig. 4. Orientation of an ellipsoidal molecule in a flowing liquid of constant velocity gradient. The positive Z axis points perpendicularly upward from the plane of the paper. The projection of the axis of revolution of the ellipsoid on the XV plane is denoted by aa. The movement of the Liquid is parallel to the X axis, and is described by the equation GY, where is the velocity and G is the velocity gradient. The significance of the angle

Notably, the anisotropic thermal displacement factors form the elements of a 3x3 symmetric matrix. The physically meaningful form of this matrix when it is positive-definite is that of an ellipsoidal probability surface centered at the equilibrium atom position. An alternative form for Equation (22) frequently used in crystallography ... 

D. Wachner, M. Simeonova and J. Gimsa, Estimating the subcelluar absorption of electric field energy equations for an ellipsoidal single shell model, Bioelectroch., 56, 211-213 (2002). 

He also assumed that the surface charge density undergoes a tangential as well as a vertical variation when an electric field is applied. By solving the continuity equation with appropriate boundary conditions, he obtained an equation for an ellipsoid which has the same form as Fricke s Equation 17 except for the magnitude of the excess conductivity. 

The equation can also be solved for spherical particles diffusing into a hole which has the shape of an ellipsoid of rotation but for most other shapes no exact solution is available. The differential equation for spheres of radius R diffusing into a hole of any given shape is the same as for the electrical capacity C of a closed surface enclosing the hole at a distance R. 

Simha concerning the dissymetry of protein molecules. The axis ratio/ of an ellipsoid of revolution, which represents the particle, is calculated from diffusion measurements in the ultracentrifuge (column 2) and compared with values obtained from viscosity measurements using equation (103) on page 285. The agreement between the two methods is rather satisfactory. 

The similar equation for calculation of wall thickness of an ellipsoidal end cap (head Box 2.4) is Equation 2.4, which uses the same variables and units as in Equation 2.3. 

Assuming an ellipsoidal geometry for the LV, the myocardial stiffness or incremental modulus (E ) at the equator of the ellipsoid may be approximated by the expression (Mirsky, 1984)... 

The rotational diffusion coefficient, 0, obtainable from a variety of experimental methods, as indicated in Table I, may be treated in a similar manner. From 0 it is possible to compute a rotational frictional coefficient, f,of an ellipsoidal particle by means of the equation... 

This equation represents an ellipsoid with axes along the principal directions. The lengths of its semi-axes are A,i, A,n, Am along I, II and III principal directions, respectively, as shown in Figure 3.3. 

A second class of models directly relates flow to blend structure without the assumption of an ellipsoidal droplet shape. This description was initiated by Doi and Ohta for an equiviscous blend with equal compositions of both components [34], Coupling this method with a constraint of constant volume of the inclusions, leads again to equations for microstructural dynamics in blends with a droplet-matrix morphology [35], An alternative way to develop these microstructural theories is the use of nonequilibrium thermodynamics. This way, Grmela et al. showed that the phenomenological Maffettone-Minale model can be retrieved for a specific choice of the free energy [36], An in-depth review of the different available models for droplet dynamics can be found in the work of Minale [20]. 

Jeffery made the first calculations on ellipsoidal particles neglecting BROWNian motion. He solved the hydrodynamic problem, determined how the orientation of an ellipsoid changes with time and from that calculated the surplus dissipation of energy. The viscosity calculated in this way is, however, not constant and depends rather sensitively upon the initial orientations. Jeffery therefore did not give an equation for the viscosity of a suspension of ellipsoids. 

Tandon and Weng have derived a complete set of explicit relationships for moduli of a composite model in which randomly distributed ellipsoidal particles (i.e., short-fiber like) are unidirectionally aligned. The Mori-Tanaka s average strain concept is also used as the transformation tensor described by Eshelby when he solved the problem of an ellipsoidal inclusion in an elastic field. The Eshelby s transformation tensor is a 4 order tensor whose components depend only the aspect ratio of the inclusion and the elastic moduli of the matrix. Some of the equations obtained by Tandon and Weng must however be solved iteratively and handling them implies calculating at first quite a impressive number of "constants," in fact various... 

The weight of nozzles and internals can be estimated at 5 to 10% of the sum of the shell and head weights. The weight of a skirt can be estimated as the same weight per foot as the shell with a length given by Equation 12-8 for an ellipsoidal head and Equation 12-9 for a conical head. 

An ellipsoidal nucleus with two spherons in the inner core has major radius greater than the minor radii by the radius of a spheron. about 1.5 f, which is about 25 percent of the mean radius. The amount of deformation given by this model is accordingly in rough agreement with that observed (18). In a detailed treatment it would be necessary to take into account the effect of electrostatic repulsion in causing the helions to tend to occupy the poles of the prolate mantle, with tritons tending to the equator. 

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