Spheroid

Properties

Circumference

The equatorial circumference of a spheroid is measured around its equator and is given as:

${\displaystyle C_{\text{e}}=2\pi a}$

The meridional or polar circumference of a spheroid is measured through its poles and is given as: $${\displaystyle C_{\text{p}}=4a{\int }_0^{\pi /2}\sqrt{1-e^2{\sin }^2\theta }d\theta }$$ The volumetric circumference of a spheroid is the circumference of a sphere of equal volume as the spheroid and is given as:

${\displaystyle C_{\text{v}}=2\sqrt[3]{a^{2}c}}$

Area

An oblate spheroid with c < a has surface area

${\displaystyle S_{\text{o}}=2\pi a^2\left(1+\frac{1-e_o^2}{e_o}\mathrm{arctanh}{e}_o\right)=2\pi a^2+\pi \frac{c^2}{e_o}\ln \left(\frac{1+e_o}{1-e_o}\right)}$

where ${\displaystyle e_o^2=1-\frac{c^2}{a^2}}$. A prolate spheroid with c > a has surface area

${\displaystyle S_{\text{p}}=2\pi a^2\left(1+\frac{c}{a{e}_p}\arcsin e_p\right)}$

where ${\displaystyle e_p^2=1-\frac{a^2}{c^2}.}$ In both cases, e_o and e_p may be identified as the eccentricity (see ellipse).

These formulas are identical in the sense that the formula for S_o can be used to calculate the surface area of a prolate spheroid and vice versa. However, e_o then becomes imaginary and can no longer directly be identified with the eccentricity. Both of these results may be cast into many other forms using standard mathematical identities and relations between parameters of the ellipse.

Volume

The volume inside a spheroid (of any kind) is

${\displaystyle V=\frac{4}{3}\pi a^{2}c\approx 4.19a^{2}c.}$

If A = 2a is the equatorial diameter, and C = 2c is the polar diameter, the volume is

${\displaystyle V=\frac{\pi }{6}{A}^{2}C\approx 0.523A^{2}C.}$

Curvature

Let a spheroid be parameterized as

${\displaystyle \mathit{\sigma }(\beta ,\lambda )=(a\cos \beta \cos \lambda ,a\cos \beta \sin \lambda ,c\sin \beta ),}$

where β is the reduced latitude or parametric latitude and λ is the longitude, with domain −⁠π/2⁠ < β < +⁠π/2⁠ and −π < λ < +π, respectively. Then, the spheroid's Gaussian curvature is:

${\displaystyle K(\beta )=\frac{c^2}{{\left(a^2+\left(c^2-a^2\right){\cos }^2\beta \right)}^2},}$

and its mean curvature is

${\displaystyle H(\beta )=\frac{c\left(2a^2+\left(c^2-a^2\right){\cos }^2\beta \right)}{2a{\left(a^2+\left(c^2-a^2\right){\cos }^2\beta \right)}^{\frac{3}{2}}}.}$

Both of these curvatures are a function of latitude only and are always positive, so that every point on a spheroid is elliptic.

Aspect ratio

The aspect ratio of an oblate spheroid/ellipse, c : a, is the ratio of the polar to equatorial lengths, while the flattening (also called oblateness) f, is the ratio of the equatorial-polar length difference to the equatorial length:

${\displaystyle f=\frac{a-c}{a}=1-\frac{c}{a}.}$

The first eccentricity (usually simply eccentricity, as above) is often used instead of flattening. It is defined by:

${\displaystyle e=\sqrt{1-\frac{c^2}{a^2}}}$

The relations between eccentricity and flattening are:

${\displaystyle \begin{array}{cc}e& =\sqrt{2f-f^2}\\ f& =1-\sqrt{1-e^2}\end{array}}$

All modern geodetic ellipsoids are defined by the semi-major axis plus either the semi-minor axis (giving the aspect ratio), the flattening, or the first eccentricity. While these definitions are mathematically interchangeable, real-world calculations must lose some precision. To avoid confusion, an ellipsoidal definition considers its own values to be exact in the form it gives.

Occurrence and applications

The most common shapes for the density distribution of protons and neutrons in an atomic nucleus are spherical, prolate, and oblate spheroidal, where the polar axis is assumed to be the spin axis (or direction of the spin angular momentum vector). Deformed nuclear shapes occur as a result of the competition between electromagnetic repulsion between protons, surface tension and quantum shell effects.

Spheroids are common in 3D cell cultures. Rotating equilibrium spheroids include the Maclaurin spheroid and the Jacobi ellipsoid. Spheroid is also a shape of archaeological artifacts.

Oblate spheroids

The oblate spheroid is the approximate shape of rotating planets and other celestial bodies, including Earth, Saturn, Jupiter, and the quickly spinning star Altair. Saturn is the most oblate planet in the Solar System, with a flattening of 0.09796. See planetary flattening and equatorial bulge for details.

Enlightenment scientist Isaac Newton, working from Jean Richer's pendulum experiments and Christiaan Huygens's theories for their interpretation, reasoned that Jupiter and Earth are oblate spheroids owing to their centrifugal force. Earth's diverse cartographic and geodetic systems are based on reference ellipsoids, all of which are oblate.

Prolate spheroids

The prolate spheroid is the approximate shape of the ball used in American football and in rugby.

Several moons of the Solar System approximate prolate spheroids in shape, though they are closer to triaxial ellipsoids. Examples are Saturn's satellites Mimas, Enceladus, and Tethys and Uranus's satellite Miranda.

In contrast to being distorted into oblate spheroids via rapid rotation, celestial objects distort slightly into prolate spheroids via tidal forces when they orbit a massive body in a close orbit. The most extreme example is Jupiter's moon Io, which becomes slightly more or less prolate in its orbit due to a slight eccentricity, causing intense volcanism. The major axis of the prolate spheroid does not run through the satellite's poles in this case, but through the two points on its equator directly facing toward and away from the primary. This combines with the smaller oblate distortion from the synchronous rotation to cause the body to become triaxial.

The term is also used to describe the shape of some nebulae such as the Crab Nebula. Fresnel zones, used to analyze wave propagation and interference in space, are a series of concentric prolate spheroids with principal axes aligned along the direct line-of-sight between a transmitter and a receiver.

The atomic nuclei of the actinide and lanthanide elements are shaped like prolate spheroids. In anatomy, near-spheroid organs such as testis may be measured by their long and short axes.

Many submarines have a shape which can be described as prolate spheroid.

Dynamical properties

For a spheroid having uniform density, the moment of inertia is that of an ellipsoid with an additional axis of symmetry. Given a description of a spheroid as having a major axis c, and minor axes a = b, the moments of inertia along these principal axes are C, A, and B. However, in a spheroid the minor axes are symmetrical. Therefore, our inertial terms along the major axes are:

${\displaystyle \begin{array}{cc}A=B& =\frac{1}{5}M\left(a^2+c^2\right),\\ C& =\frac{1}{5}M\left(a^2+b^2\right)=\frac{2}{5}M\left(a^2\right),\end{array}}$

where M is the mass of the body defined as

${\displaystyle M=\frac{4}{3}\pi a^{2}c\rho .}$