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The prolate spheroid is generated by rotation about the z-axis of an ellipse with semi-major axis c and semi-minor axis a; therefore, e may again be identified as the eccentricity. (See ellipse.) [3] These formulas are identical in the sense that the formula for S oblate can
This equation reduces to that of the volume of a sphere when all three elliptic radii are equal, and to that of an oblate or prolate spheroid when two of them are equal. The volume of an ellipsoid is 2 / 3 the volume of a circumscribed elliptic cylinder, and π / 6 the volume of the circumscribed box.
For a Maclaurin spheroid of eccentricity greater than 0.812670, [3] a Jacobi ellipsoid of the same angular momentum has lower total energy. If such a spheroid is composed of a viscous fluid (or in the presence of gravitational radiation reaction), and if it suffers a perturbation which breaks its rotational symmetry, then it will gradually elongate into the Jacobi ellipsoidal form, while ...
An example of a spherical cap in blue (and another in red) In geometry, a spherical cap or spherical dome is a portion of a sphere or of a ball cut off by a plane.It is also a spherical segment of one base, i.e., bounded by a single plane.
Electrodynamics of Continuous Media (Volume 8 of the Course of Theoretical Physics) (2nd ed.). New York: Pergamon Press. pp. 19– 29. ISBN 978-0-7506-2634-7. Treats the prolate spheroidal coordinates as a limiting case of the general ellipsoidal coordinates. Uses (ξ, η, ζ) coordinates that have the units of distance squared.
The red oblate spheroid (flattened sphere) corresponds to μ = 1, whereas the blue half-hyperboloid corresponds to ν = 45°. The azimuth φ = −60° measures the dihedral angle between the green xz half-plane and the yellow half-plane that includes the point P. The Cartesian coordinates of P are roughly (1.09, −1.89, 1.66).
If the radius of the sphere is denoted by r and the height of the cap by h, the volume of the spherical sector is =. This may also be written as V = 2 π r 3 3 ( 1 − cos φ ) , {\displaystyle V={\frac {2\pi r^{3}}{3}}(1-\cos \varphi )\,,} where φ is half the cone aperture angle, i.e., φ is the angle between the rim of the cap and the ...
An ellipsoidal model describes only the ellipsoid's geometry and a normal gravity field formula to go with it. Commonly an ellipsoidal model is part of a more encompassing geodetic datum. For example, the older ED-50 (European Datum 1950) is based on the Hayford or International Ellipsoid. WGS-84 is peculiar in that the same name is used for ...