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The three coordinate surfaces of prolate spheroidal coordinates. The red prolate spheroid (stretched sphere) corresponds to μ = 1, and the blue two-sheet hyperboloid corresponds to ν = 45°. The yellow half-plane corresponds to φ = −60°, which is measured relative to the x-axis (highlighted in green).
The oblate spheroid is generated by rotation about the z-axis of an ellipse with semi-major axis a and semi-minor axis c, therefore e may be identified as the eccentricity. (See ellipse.) [2] A prolate spheroid with c > a has surface area
Pioneering work in this area was performed by Slepian and Pollak, [4] Landau and Pollak, [5] [6] and Slepian. [7] [8] Prolate spheroidal wave functions whose domain is a (portion of) the surface of the unit sphere are more generally called "Slepian functions". [9] These are of great utility in disciplines such as geodesy, [10] cosmology, [11 ...
An ellipsoid is a surface that can be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation. An ellipsoid is a quadric surface; that is, a surface that may be defined as the zero set of a polynomial of degree two in three variables. Among quadric surfaces, an ellipsoid is ...
The unusual cosmic abundance of alpha nuclides has inspired geometric arrangements of alpha particles as a solution to nuclear shapes, although the atomic nucleus generally assumes a prolate spheroid shape. Nuclides can also be discus-shaped (oblate deformation), triaxial (a combination of oblate and prolate deformation) or pear-shaped. [7] [8]
In prolate spheroids, ... boundary conditions, i.e., it is assumed that the velocity of the fluid is zero at the surface of the spheroid. Perrin S factor ...
The motion is simplified in the case of an axisymmetric body, in which the moment of inertia is the same about two of the principal axes. These cases include rotation of a prolate spheroid (the shape of an American football), or rotation of an oblate spheroid (the shape of a flattened sphere). In this case, the angular velocity describes a cone ...
In the well-known case of calcite, for example, the index ellipsoid is an oblate spheroid, so that one sheet of the index surface is a sphere touching that oblate spheroid at the equator, while the other sheet of the index surface is a prolate spheroid touching the sphere at the poles, with an equatorial radius (extraordinary index) equal to ...