Hubbry Logo
search
logo
Spheroid avatar
Spheroid
Spheroid
current hub
2299991

Spheroid

logo
Community Hub0 Subscribers
Read side by side
Spheroid
View on Wikipedia
from Wikipedia

Spheroids with vertical rotational axes
oblate prolate

A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters. A spheroid has circular symmetry.

If the ellipse is rotated about its major axis, the result is a prolate spheroid, elongated like a rugby ball. The American football is similar but has a pointier end than a spheroid could. If the ellipse is rotated about its minor axis, the result is an oblate spheroid, flattened like a lentil or a plain M&M. If the generating ellipse is a circle, the result is a sphere.

Due to the combined effects of gravity and rotation, the figure of the Earth (and of all planets) is not quite a sphere, but instead is slightly flattened in the direction of its axis of rotation. For that reason, in cartography and geodesy the Earth is often approximated by an oblate spheroid, known as the reference ellipsoid, instead of a sphere. The current World Geodetic System model uses a spheroid whose radius is 6,378.137 km (3,963.191 mi) at the Equator and 6,356.752 km (3,949.903 mi) at the poles.

The word spheroid originally meant "an approximately spherical body", admitting irregularities even beyond the bi- or tri-axial ellipsoidal shape; that is how the term is used in some older papers on geodesy (for example, referring to truncated spherical harmonic expansions of the Earth's gravity geopotential model).[1]

Equation

[edit]
The assignment of semi-axes on a spheroid. It is oblate if c < a (left) and prolate if c > a (right).

The equation of a tri-axial ellipsoid centred at the origin with semi-axes a, b and c aligned along the coordinate axes is

The equation of a spheroid with z as the symmetry axis is given by setting a = b:

The semi-axis a is the equatorial radius of the spheroid, and c is the distance from centre to pole along the symmetry axis. There are two possible cases:

  • c < a: oblate spheroid
  • c > a: prolate spheroid

The case of a = c reduces to a sphere.

Properties

[edit]

Circumference

[edit]

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

The meridional or polar circumference of a spheroid is measured through its poles and is given as: The volumetric circumference of a spheroid is the circumference of a sphere of equal volume as the spheroid and is given as:

Area

[edit]

An oblate spheroid with c < a has surface area

where . A prolate spheroid with c > a has surface area

where In both cases, eo and ep may be identified as the eccentricity (see ellipse).[2][3]

These formulas are identical in the sense that the formula for So can be used to calculate the surface area of a prolate spheroid and vice versa. However, eo 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

[edit]

The volume inside a spheroid (of any kind) is

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

Curvature

[edit]
See also: Radius of the Earth § Radii of curvature

Let a spheroid be parameterized as

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:

and its mean curvature is

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

[edit]

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:

The first eccentricity (usually simply eccentricity, as above) is often used instead of flattening.[4] It is defined by:

The relations between eccentricity and flattening are:

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

[edit]

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

[edit]
The planet Jupiter is a slight oblate spheroid with a flattening of 0.06487

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.[5] 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.[6][7][8] Earth's diverse cartographic and geodetic systems are based on reference ellipsoids, all of which are oblate.

Prolate spheroids

[edit]
A rugby ball

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.[9] 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.[10] In anatomy, near-spheroid organs such as testis may be measured by their long and short axes.[11]

Many submarines have a shape which can be described as prolate spheroid.[12]

Dynamical properties

[edit]
See also: Ellipsoid § 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:[13]

where M is the mass of the body defined as

See also

[edit]

References

[edit]
[edit]
Revisions and contributorsEdit on WikipediaRead on Wikipedia

Spheroid

View on Grokipedia
from Grokipedia
A spheroid is a quadric surface formed by rotating an ellipse about one of its principal axes, resulting in a shape with two equal semi-axes and one distinct semi-axis, distinguishing it from a general ellipsoid with three unequal axes.[1] This surface of revolution approximates a sphere but exhibits elongation or flattening depending on the axis of rotation.[2] Spheroids are classified into two primary types: oblate and prolate. An oblate spheroid arises from rotating an ellipse about its minor axis, producing a flattened shape at the poles with an equatorial radius greater than the polar radius.[3] In contrast, a prolate spheroid results from rotation about the major axis, yielding an elongated form where the polar radius exceeds the equatorial radius.[4] In geodesy and planetary science, the oblate spheroid serves as a fundamental model for Earth's shape, accounting for the planet's equatorial bulge due to rotational forces, with the equatorial radius approximately 21 kilometers larger than the polar radius.[5] This approximation underpins reference ellipsoids used in global positioning systems (GPS) and cartographic projections, enabling precise mapping and navigation.[6] Prolate spheroids, while less common in natural contexts, appear in models of certain asteroids and in theoretical physics for symmetric potentials.[7]

Fundamentals

Definition

A spheroid is a quadric surface formed by rotating an ellipse about one of its principal axes, generating a surface of revolution that extends the two-dimensional ellipse into three dimensions.[8] This rotation preserves the elliptical cross-sections while creating a symmetric shape around the axis of rotation, distinguishing it from a general ellipsoid, which has three unequal axes, as a spheroid specifically has two equal semi-axes.[1] Visually, a spheroid resembles a sphere that has been stretched or compressed along one direction, resulting in a more flattened or elongated profile depending on the axis of rotation. For instance, rotating an ellipse about its minor axis produces a shape widened at the equator and narrowed at the poles, akin to spinning a flattened circle to form a disc-like solid.[1] The key dimensions are the equatorial radius aa, which measures the distance from the center to the equator along the plane perpendicular to the rotation axis, and the polar radius cc, which measures along the rotation axis itself.[9] Spheroids are classified into two types based on the relative sizes of these radii: an oblate spheroid occurs when the equatorial radius exceeds the polar radius (a>ca > c), creating a flattened appearance at the poles, while a prolate spheroid has a longer polar radius than equatorial (a<ca < c), resulting in an elongated, rugby-ball-like form.[9] A sphere represents the special case of a spheroid where the equatorial and polar radii are equal (a=ca = c).[1]

Historical Context

The concept of the Earth's shape as a sphere was recognized by ancient Greek philosophers as early as the 5th century BCE, with Aristotle providing empirical evidence around 330 BCE through observations of lunar eclipses and the varying positions of stars, establishing a qualitative understanding of a rounded planet.[10] However, the notion of a spheroid—an ellipsoid of revolution deviated from a perfect sphere—emerged much later, with formal mathematical modeling beginning in the 17th century amid advances in mechanics and astronomy. In his Philosophiæ Naturalis Principia Mathematica published in 1687, Isaac Newton theorized that the Earth's rotation would cause centrifugal forces to flatten it at the poles and bulge it at the equator, predicting an oblate spheroid shape for a rotating fluid body in equilibrium.[11] This marked a pivotal shift from qualitative descriptions to quantitative predictions based on universal gravitation, influencing subsequent geodetic inquiries. During the 18th century, mathematicians refined Newton's model for practical geodetic applications. Colin Maclaurin provided a rigorous proof in 1740 for the equilibrium figure of a homogeneous rotating fluid, deriving the oblate spheroid as the stable form and enabling calculations of gravitational variations.[12] Independently, Alexis-Claude Clairaut developed a more general theory in his 1743 work Théorie de la figure de la Terre, accounting for density variations and confirming the oblate shape through differential equations that linked ellipticity to rotational effects, which supported expeditions measuring meridional arcs.[13] Advancements in the 19th and 20th centuries focused on precise ellipsoid approximations for global mapping and surveying, transitioning from theoretical models to standardized reference surfaces. Efforts culminated in the development of reference ellipsoids, such as the World Geodetic System 1984 (WGS84), adopted for international consistency in positioning and adopted by organizations like the U.S. Department of Defense.[14] This evolution in astronomy and geodesy progressed from Newton's qualitative insights to quantitative frameworks essential for accurate planetary modeling.

Mathematical Formulation

Cartesian Equation

The Cartesian equation of a spheroid, which describes its surface in three-dimensional Cartesian coordinates, is
x2+y2a2+z2c2=1, \frac{x^2 + y^2}{a^2} + \frac{z^2}{c^2} = 1,
where aa denotes the semi-axis length in the equatorial plane (spanned by the xx- and yy-axes) and cc denotes the semi-axis length along the polar zz-axis.[1] This equation represents the canonical form for a spheroid aligned with the coordinate axes, distinguishing it as a special case of a quadric surface defined by a second-degree polynomial equation in xx, yy, and zz./04%3A_Coordinate_Geometry_in_Three_Dimensions/4.03%3A_The_Ellipsoid) This form arises from the rotation of a two-dimensional ellipse about its symmetry axis. Consider the ellipse equation in the xzx z-plane given by x2a2+z2c2=1\frac{x^2}{a^2} + \frac{z^2}{c^2} = 1; rotating this curve about the zz-axis generates the spheroid, replacing x2x^2 with x2+y2x^2 + y^2 to account for the circular symmetry in the equatorial plane./04%3A_Coordinate_Geometry_in_Three_Dimensions/4.03%3A_The_Ellipsoid)[15] The resulting surface is oblate if a>ca > c (flattened at the poles, like Earth) or prolate if a<ca < c (elongated along the polar axis, like a rugby ball).[1] In this convention, aa is the equatorial semi-axis, reflecting the radius of the circular cross-section in the xyxy-plane at z=0z = 0, while cc is the polar semi-axis, corresponding to the extent along the axis of rotation.[16] This alignment ensures the equation captures the spheroid's rotational symmetry without loss of generality for axisymmetric cases.[15]

Parametric Equations

The parametric equations provide an explicit representation of points on the surface of a spheroid, facilitating computations and visualizations in three-dimensional space. For a spheroid aligned with the z-axis, where aa is the equatorial semi-axis and cc is the polar semi-axis, the coordinates are given by
x=asin⁡θcos⁡ϕ,y=asin⁡θsin⁡ϕ,z=ccos⁡θ, \begin{align*} x &= a \sin \theta \cos \phi, \\ y &= a \sin \theta \sin \phi, \\ z &= c \cos \theta, \end{align*}
with the polar angle θ\theta ranging from 0 to π\pi and the azimuthal angle ϕ\phi ranging from 0 to 2π2\pi.[1] These parameters θ\theta and ϕ\phi parameterize the surface in a manner analogous to spherical coordinates for a sphere, but adjusted for the spheroid's eccentricity: θ\theta measures the colatitude from the positive z-pole, while ϕ\phi describes the longitude around the axis of rotation, ensuring full coverage of the surface without overlap except at the poles.[1] The equations derive from rotating a parametric ellipse in the xz-plane around the z-axis. The ellipse (x/a)2+(z/c)2=1(x/a)^2 + (z/c)^2 = 1 is parameterized as x=asin⁡θx = a \sin \theta, z=ccos⁡θz = c \cos \theta for θ∈[0,π]\theta \in [0, \pi], and rotation by angle ϕ\phi yields the x and y components via the cylindrical transformation x′=xcos⁡ϕx' = x \cos \phi, y′=xsin⁡ϕy' = x \sin \phi, with z unchanged.[1] This parameterization offers advantages in numerical methods, such as generating surface plots by evaluating at discrete θ\theta and ϕ\phi grids, or performing surface integrals by leveraging the metric tensor derived from partial derivatives with respect to θ\theta and ϕ\phi.[1]

Geometric Properties

Surface Area

The surface area of a spheroid is obtained by evaluating the surface integral over its parametric representation. The parametric equations are given by
r(θ,ϕ)=(asin⁡θcos⁡ϕ,asin⁡θsin⁡ϕ,ccos⁡θ), \mathbf{r}(\theta, \phi) = (a \sin \theta \cos \phi, a \sin \theta \sin \phi, c \cos \theta),
where aa is the equatorial semi-axis, cc is the polar semi-axis, 0≤θ≤π0 \leq \theta \leq \pi, and 0≤ϕ<2π0 \leq \phi < 2\pi. The partial derivatives are rθ=(acos⁡θcos⁡ϕ,acos⁡θsin⁡ϕ,−csin⁡θ)\mathbf{r}_\theta = (a \cos \theta \cos \phi, a \cos \theta \sin \phi, -c \sin \theta) and rϕ=(−asin⁡θsin⁡ϕ,asin⁡θcos⁡ϕ,0)\mathbf{r}_\phi = (-a \sin \theta \sin \phi, a \sin \theta \cos \phi, 0). The magnitude of their cross product is ∣∣rθ×rϕ∣∣=asin⁡θa2cos⁡2θ+c2sin⁡2θ||\mathbf{r}_\theta \times \mathbf{r}_\phi|| = a \sin \theta \sqrt{a^2 \cos^2 \theta + c^2 \sin^2 \theta}. Thus, the surface area SS is
S=∫02π∫0πasin⁡θa2cos⁡2θ+c2sin⁡2θ dθ dϕ=2πa∫0πsin⁡θa2cos⁡2θ+c2sin⁡2θ dθ. S = \int_0^{2\pi} \int_0^\pi a \sin \theta \sqrt{a^2 \cos^2 \theta + c^2 \sin^2 \theta} \, d\theta \, d\phi = 2\pi a \int_0^\pi \sin \theta \sqrt{a^2 \cos^2 \theta + c^2 \sin^2 \theta} \, d\theta.
This integral evaluates to distinct closed-form expressions for oblate and prolate spheroids, derived by substitution u=cos⁡θu = \cos \theta and recognizing the resulting form as solvable via hyperbolic or trigonometric functions, equivalent to certain elliptic integrals that simplify for the axisymmetric case.[17][18] For an oblate spheroid (a>ca > c), the eccentricity is e=1−(c/a)2e = \sqrt{1 - (c/a)^2}. The exact surface area is
S=2πa2[1+1−e2etanh⁡−1e]=2πa2+πc2eln⁡(1+e1−e), S = 2\pi a^2 \left[ 1 + \frac{1 - e^2}{e} \tanh^{-1} e \right] = 2\pi a^2 + \frac{\pi c^2}{e} \ln \left( \frac{1 + e}{1 - e} \right),
where tanh⁡−1e=12ln⁡(1+e1−e)\tanh^{-1} e = \frac{1}{2} \ln \left( \frac{1 + e}{1 - e} \right). This can also be expressed using the complete elliptic integral of the second kind E(e)=∫0π/21−e2sin⁡2ψ dψE(e) = \int_0^{\pi/2} \sqrt{1 - e^2 \sin^2 \psi} \, d\psi, though the elementary form is preferred for computation. An alternative closed-form expression using the Gauss hypergeometric function is
S = 2\pi \left( a^2 + c^2 \, _2F_1\left(\frac{1}{2}, 1; \frac{3}{2}; 1 - \frac{c^2}{a^2}\right) \right).
[16] For a prolate spheroid (c>ac > a), the eccentricity is e=1−(a/c)2e = \sqrt{1 - (a/c)^2}. The exact surface area is
S=2πa2[1+1e1−e2sin⁡−1e]=2πa2+2πacesin⁡−1e. S = 2\pi a^2 \left[ 1 + \frac{1}{e} \sqrt{1 - e^2} \sin^{-1} e \right] = 2\pi a^2 + \frac{2\pi a c}{e} \sin^{-1} e.
As with the oblate case, this arises from evaluating the parametric integral, reducing the elliptic form to elementary functions.[4] For small eccentricity (e≪1e \ll 1), applicable to near-spherical spheroids like Earth's oblate shape, the surface area approximates that of a sphere of radius aa with corrections:
S≈4πa2(1−13e2−130e4). S \approx 4\pi a^2 \left( 1 - \frac{1}{3} e^2 - \frac{1}{30} e^4 \right).
Higher-order terms include −1840e6-\frac{1}{840} e^6, but the e2e^2 term provides the primary deviation from sphericity. A similar expansion holds for prolate spheroids.[19]

Volume

The volume $ V $ of a spheroid, defined by the equation x2+y2a2+z2c2=1\frac{x^2 + y^2}{a^2} + \frac{z^2}{c^2} = 1 where $ a $ is the equatorial semi-axis and $ c $ is the polar semi-axis, is given by the formula
V=43πa2c. V = \frac{4}{3} \pi a^2 c.
This expression arises as a special case of the general ellipsoid volume $ V = \frac{4}{3} \pi a b c $ when the intermediate axis equals the equatorial axis ($ b = a $).[20][1] A direct derivation can be obtained using the disk method, integrating the cross-sectional areas perpendicular to the polar (z) axis. At a fixed height $ z $ between $ -c $ and $ c $, the cross-section is a disk with radius $ r(z) = a \sqrt{1 - \frac{z^2}{c^2}} $, so the area is $ \pi [r(z)]^2 = \pi a^2 \left(1 - \frac{z^2}{c^2}\right) $. The volume is then the integral
V=∫−ccπa2(1−z2c2) dz=2πa2∫0c(1−z2c2) dz=2πa2[z−z33c2]0c=2πa2(c−c3)=43πa2c. V = \int_{-c}^{c} \pi a^2 \left(1 - \frac{z^2}{c^2}\right) \, dz = 2 \pi a^2 \int_{0}^{c} \left(1 - \frac{z^2}{c^2}\right) \, dz = 2 \pi a^2 \left[ z - \frac{z^3}{3 c^2} \right]_{0}^{c} = 2 \pi a^2 \left( c - \frac{c}{3} \right) = \frac{4}{3} \pi a^2 c.
This approach works identically for both oblate ($ c < a )andprolate() and prolate ( c > a $) spheroids. An alternative derivation employs triple integration in cylindrical coordinates or a change of variables scaling from the unit ball, yielding the same closed-form result.[21][22] When $ a = c = r $, the spheroid degenerates to a sphere, and the volume simplifies to the familiar $ V = \frac{4}{3} \pi r^3 $. For a non-spherical spheroid, flattening (deviation of $ c $ from $ a $) scales the volume relative to a sphere of equivalent "average" radius by the factor $ \frac{a^2 c}{r^3} $, reducing it for oblate forms and increasing it for prolate ones compared to a sphere of radius $ a $ or $ c $. Unlike the surface area, which requires elliptic integrals, the volume formula is elementary and does not involve special functions.[23]

Circumference

The equatorial circumference of a spheroid, which lies in the plane perpendicular to the axis of rotation, forms a great circle of radius aa, the semi-major axis. Thus, its length is given by
Ce=2πa. C_e = 2\pi a.
This expression follows directly from the geometry of a circle and applies to both oblate and prolate spheroids, where aa is the equatorial radius.[24] The meridional circumference traces a closed meridian ellipse in a plane containing the axis of rotation, with semi-axes aa (equatorial) and cc (polar). Its total length is the perimeter of this ellipse. For an oblate spheroid (a>ca > c), it is expressed as
Cm=4aE(e), C_m = 4a E(e),
where e=1−(c/a)2e = \sqrt{1 - (c/a)^2} is the eccentricity and E(e)E(e) denotes the complete elliptic integral of the second kind, defined by
E(e)=∫0π/21−e2sin⁡2θ dθ. E(e) = \int_0^{\pi/2} \sqrt{1 - e^2 \sin^2 \theta} \, d\theta.
For a prolate spheroid (c>ac > a),
Cm=4cE(e′), C_m = 4c E(e'),
where e′=1−(a/c)2e' = \sqrt{1 - (a/c)^2}.
This integral arises in geodesy for computing distances on ellipsoidal models of Earth. To derive the meridional circumference for the oblate case, consider the parametric equations of the meridian ellipse in the xx-zz plane: x(θ)=acos⁡θx(\theta) = a \cos \theta, z(θ)=csin⁡θz(\theta) = c \sin \theta, where θ\theta is the parametric angle ranging from 0 to 2π2\pi. The arc length element is ds=(dx/dθ)2+(dz/dθ)2 dθ=a2sin⁡2θ+c2cos⁡2θ dθds = \sqrt{(dx/d\theta)^2 + (dz/d\theta)^2} \, d\theta = \sqrt{a^2 \sin^2 \theta + c^2 \cos^2 \theta} \, d\theta. The quarter arc from equator to pole ( θ=0\theta = 0 to π/2\pi/2 ) is aE(e)a E(e), so the full circumference is four times this value. This formulation holds for the reference ellipsoid in coordinate systems like WGS84.[24] Circumferences of parallels at other latitudes ϕ\phi (measured from the equator) are circles parallel to the equator, with radius ν(ϕ)cos⁡ϕ\nu(\phi) \cos \phi, where ν(ϕ)=a/1−e2sin⁡2ϕ\nu(\phi) = a / \sqrt{1 - e^2 \sin^2 \phi} is the prime vertical radius of curvature for oblate spheroids. The length is thus
C(ϕ)=2πacos⁡ϕ1−e2sin⁡2ϕ. C(\phi) = 2\pi \frac{a \cos \phi}{\sqrt{1 - e^2 \sin^2 \phi}}.
For prolate spheroids, the formula requires adjustment using prolate coordinates. Unlike the meridian, this does not involve elliptic integrals, as each parallel is a true circle. For an oblate spheroid like Earth's, C(ϕ)C(\phi) decreases from the equator toward the poles.[24]

Curvature

The curvature of a spheroid surface deviates from the uniform positive Gaussian curvature of a sphere, varying with position due to the eccentricity induced by the differing semi-axes. For an oblate spheroid with equatorial semi-axis aa and polar semi-axis c<ac < a, and eccentricity e=1−(c/a)2e = \sqrt{1 - (c/a)^2}, the Gaussian curvature KK is computed using the first and second fundamental forms of the parametrized surface r(ϕ,θ)=(acos⁡ϕcos⁡θ,acos⁡ϕsin⁡θ,csin⁡ϕ)\mathbf{r}(\phi, \theta) = (a \cos \phi \cos \theta, a \cos \phi \sin \theta, c \sin \phi), where ϕ∈[−π/2,π/2]\phi \in [-\pi/2, \pi/2] is the latitude parameter and θ∈[0,2π)\theta \in [0, 2\pi) is the azimuthal angle. The first fundamental form coefficients are E=a2(1−e2cos⁡2ϕ)E = a^2 (1 - e^2 \cos^2 \phi), F=0F = 0, and G=a2cos⁡2ϕG = a^2 \cos^2 \phi. The second fundamental form yields the principal curvatures, leading to K=1−e2a2(1−e2cos⁡2ϕ)2K = \frac{1 - e^2}{a^2 (1 - e^2 \cos^2 \phi)^2}.[1] This expression highlights the positional variation: at the equator (ϕ=0\phi = 0), K=1/c2>1/a2K = 1/c^2 > 1/a^2, exceeding the spherical value, while at the poles (ϕ=±π/2\phi = \pm \pi/2), K=c2/a4=(1−e2)/a2<1/a2K = c^2 / a^4 = (1 - e^2)/a^2 < 1/a^2, showing a negative deviation from spherical uniformity. For a prolate spheroid (c>ac > a), the formula adjusts with e=1−(a/c)2e = \sqrt{1 - (a/c)^2} and interchanged roles, yielding higher curvature at the poles and lower at the equator. The Gaussian curvature remains positive everywhere, confirming the spheroid as an elliptic point surface, but the deviation scales with ee, which depends on the aspect ratio c/ac/a.[1][25] The mean curvature HH, defined as the average of the principal curvatures, is H=c(a2+E)2aE3/2H = \frac{c (a^2 + E)}{2 a E^{3/2}}, where E=a2(1−e2cos⁡2ϕ)E = a^2 (1 - e^2 \cos^2 \phi). At the poles, H=c/a2H = c / a^2, smaller than the spherical 1/a1/a for oblate cases. At the equator, H=(a2+c2)/(2ac2)H = (a^2 + c^2)/(2 a c^2), which is larger than 1/a1/a. These values reflect the surface's embedding in space, with oblate spheroids exhibiting relatively flatter poles (lower HH) and sharper equatorial bending compared to a sphere of radius aa. The derivation via fundamental forms ensures intrinsic consistency, as Gaussian curvature is independent of embedding, while mean curvature captures extrinsic bending.[1][25]

Structural Characteristics

Aspect Ratio

In the context of spheroids, the aspect ratio serves as a key parameter to describe the shape's deviation from perfect sphericity, primarily through the flattening factor $ f $, which measures the relative difference between the equatorial semi-axis $ a $ and the polar semi-axis $ c $. For an oblate spheroid, where $ a > c $, the flattening is defined as $ f = \frac{a - c}{a} $, with $ 0 < f < 1 $ indicating the degree of polar compression.[26] This formulation arises from the geometry of the ellipse rotated about its minor axis, where $ f = 0 $ corresponds to a sphere and increasing $ f $ reflects greater oblateness.[27] For a prolate spheroid, where $ c > a $, the aspect ratio is often expressed as the reciprocal form $ \frac{c}{a} > 1 $, quantifying the elongation along the polar axis instead of compression.[9] This reciprocal approach maintains consistency in describing the ratio of the distinct axes, adapting the flattening concept to highlight extension rather than contraction. The flattening $ f $ is closely related to the spheroid's eccentricity $ e $, a measure of how much the generating ellipse deviates from a circle. For an oblate spheroid, $ e = \sqrt{1 - \left( \frac{c}{a} \right)^2 } = \sqrt{1 - (1 - f)^2 } $, linking the two parameters such that small $ f $ yields small $ e $, both approaching zero for near-spherical shapes.[16] This relationship underscores how $ f $ and $ e $ interchangeably characterize the same geometric distortion, with $ e $ emphasizing the angular spread and $ f $ the linear compression.[28] The parameter $ f $ (or its reciprocal for prolate cases) plays a crucial role in classifying spheroids by their elongation or oblateness relative to a sphere. For instance, Earth's oblate spheroid has $ f \approx \frac{1}{298.257} $, a very small value indicating minimal deviation from sphericity due to rotational forces.[14] In contrast, values of $ f $ closer to 1 denote highly oblate forms approaching a disk-like shape, while reciprocals substantially greater than 1 identify highly elongated prolate structures. This classification aids in distinguishing nearly spherical bodies from those with pronounced asymmetry, informing applications in geometry and physics where shape influences properties like volume and surface area.[27]

Axes and Radii

In a spheroid, the principal axes are defined by two distinct semi-axes lengths: the equatorial semi-axis aa, which measures the radius in the plane perpendicular to the axis of symmetry (or rotation), and the polar semi-axis cc, which extends along the axis of symmetry from the center to the pole.[1] These axes represent the semi-major and semi-minor dimensions of the ellipse that serves as the generating curve for the spheroid, formed by revolving this ellipse about one of its principal axes.[1] The equatorial axis aa thus governs the width across the equator, while the polar axis cc sets the height along the rotational symmetry.[1] The relative measurements of these axes distinguish the two primary types of spheroids and dictate their geometric form. In an oblate spheroid, the polar semi-axis cc is shorter than the equatorial semi-axis aa (c<ac < a), yielding a compressed, disk-like shape analogous to a squashed sphere.[16] This configuration arises from rotating an ellipse about its minor axis, emphasizing the equatorial bulge.[16] By contrast, a prolate spheroid features a longer polar semi-axis cc than equatorial semi-axis aa (c>ac > a), creating an extended, rugby-ball-like profile generated by rotation about the ellipse's major axis.[4] Collectively, the values of aa and cc establish the spheroid's overall dimensions and morphological characteristics, with equal lengths (a=ca = c) reducing it to a sphere and increasing disparities amplifying the ellipsoidal deviation.[1] The aspect ratio, simply the proportion c/ac/a, provides a dimensionless measure of this elongation or flattening.[1]

Types

Oblate Spheroid

An oblate spheroid is a quadric surface formed by rotating an ellipse about its minor axis, producing a shape that is compressed along the polar axis and expanded at the equator.[16] This configuration arises in rotating fluid bodies, where centrifugal forces cause material to migrate outward along the equatorial plane, resulting in an equatorial radius aa greater than the polar radius cc.[29] The resulting form is characteristic of self-gravitating, rotating masses in hydrostatic equilibrium, such as planets and stars with sufficient spin rates.[30] The mathematical distinction of an oblate spheroid from a sphere or prolate form lies in its eccentricity, defined as e=1−(c/a)2e = \sqrt{1 - (c/a)^2}, where e>0e > 0 indicates the deviation from sphericity due to a>ca > c.[16] This parameter quantifies the degree of polar flattening, with the flattening ratio f=(a−c)/af = (a - c)/a providing a related measure of oblateness. Key properties include a larger equatorial cross-section and rotational symmetry about the polar axis, making oblate spheroids suitable models for bodies where rotation dominates over other deformational forces.[29] Prominent examples include Earth, which exhibits a modest oblateness with f≈1/298.257f \approx 1/298.257, attributable to its diurnal rotation over 24 hours.[31] Jupiter displays a more pronounced equatorial bulge, with f≈0.065f \approx 0.065, driven by its rapid rotation period of approximately 10 hours, which amplifies centrifugal effects relative to its gravitational binding.[29] These cases illustrate how rotational dynamics shape oblate spheroids in natural systems.

Prolate Spheroid

A prolate spheroid is formed by rotating an ellipse about its major axis, where the semi-major axis length cc exceeds the semi-minor axis length aa, leading to an elongated shape along the polar direction.[4] This rotation generates a surface of revolution characterized by polar extension, with the equatorial radius equal to aa and the polar radius equal to c>ac > a.[32] The resulting geometry resembles a rugby ball, featuring tapered ends and a reduced equatorial cross-section compared to a sphere of equivalent volume.[4] Mathematically, it is distinguished from an oblate spheroid by its positive eccentricity squared, defined as e2=1−(a/c)2>0e^2 = 1 - (a/c)^2 > 0, which quantifies the degree of elongation.[4] Prolate spheroids occur in various natural contexts, such as rotating liquid drops of lower density within a denser medium, where centrifugal forces stabilize the elongated form.[33] Certain asteroids, like (1620) Geographos, are approximated as prolate spheroids due to their irregular, elongated profiles derived from rotational dynamics and collisional evolution.[34] In biology, columnar epithelial cells lining the stomach and intestines adopt a prolate spheroidal shape under structural constraints, facilitating directional functions like absorption.[35]

Applications and Occurrences

In Geodesy and Earth Modeling

In geodesy, spheroids serve as mathematical approximations of Earth's oblate shape, enabling precise calculations for mapping, surveying, and navigation by defining reference surfaces for latitude and longitude coordinates. Unlike spherical models, which assume uniform radius and introduce errors up to about 0.3% in polar-equatorial distances, oblate spheroids account for the planet's equatorial bulge, providing a closer fit to the irregular geoid—the true equipotential surface of gravity—thus improving accuracy in geodetic computations by reducing distortions in positional data. This approximation is essential for transforming three-dimensional Earth coordinates into usable two-dimensional representations, with errors minimized to sub-meter levels in modern systems. The historical evolution in geodesy shifted from simplistic spherical Earth models, prevalent in early 19th-century surveys for their computational ease, to oblate spheroid references as measurements confirmed Earth's flattening. A pivotal advancement was the Clarke 1866 ellipsoid, developed for North American Datum 1927 (NAD27), with a semi-major axis a=6,378,206.4a = 6,378,206.4 m and inverse flattening 1/f=294.9786982141/f = 294.978698214, which better matched regional gravity observations but was locally oriented rather than geocentric. Subsequent global refinements included the Geodetic Reference System 1980 (GRS80), adopted by the International Union of Geodesy and Geophysics, featuring a=6,378,137a = 6,378,137 m and 1/f=298.2572221011/f = 298.257222101, designed for worldwide consistency based on satellite and gravimetric data. The World Geodetic System 1984 (WGS84), nearly identical to GRS80 with a=6,378,137a = 6,378,137 m and 1/f=298.2572235631/f = 298.257223563, became the standard for international applications, reflecting a transition to geocentric models aligned with Earth's center of mass. Spheroids underpin map projections and global navigation systems by serving as the baseline for coordinate transformations. In projections like the Mercator, which preserves angles for navigation, the spheroid's parameters determine scale factors and distortions, ensuring rhumb lines project as straight lines with minimal angular error on nautical charts. For GPS and satellite-based positioning, WGS84 defines the ellipsoidal coordinates, allowing receivers to compute positions relative to this reference surface, with latitude and longitude calculated via geodetic formulas that incorporate flattening to yield accuracies better than 1 meter under ideal conditions. This integration facilitates seamless interoperability in aviation, maritime, and terrestrial surveying, where spherical approximations would inflate errors in high-latitude regions by up to 21 km in radius mismatch.

In Astronomy and Physics

In astronomy, spheroids play a key role in modeling the shapes of rotating celestial bodies, where rotational forces distort gravitational equilibrium from sphericity. Oblate spheroids are prevalent among planets, particularly gas giants, due to centrifugal effects counteracting self-gravity. Saturn exemplifies this, with an equatorial radius of 60,268 km and a polar radius of 54,364 km, yielding a flattening factor $ f \approx 0.0986 $, or roughly 1/10. This pronounced oblateness arises from Saturn's rapid rotation period of approximately 10.7 hours combined with its low mean density of 0.687 g/cm³, which amplifies the equatorial bulge relative to polar compression. Jupiter shows a milder case with $ f \approx 0.0649 $, reflecting its slower rotation (9.9 hours) and higher density (1.326 g/cm³). These oblate forms influence planetary atmospheres, ring systems, and magnetic fields, as observed by missions like Cassini. Prolate spheroids, elongated along the rotation axis, occur in specific stellar and binary contexts where magnetic or tidal forces dominate over uniform rotation. In rotating stars, strong internal magnetic fields, such as toroidal configurations, can counteract centrifugal flattening and induce prolate shapes. Theoretical models confirm that prolate spheroids form in uniformly dense stars with non-uniform density profiles or magnetic support, affecting epicyclic frequencies around such bodies and resembling prograde orbits in Kerr metrics without a marginally stable orbit.[36] In binary star systems, tidal deformations often result in prolate configurations; close binaries are modeled as pairs of prolate ellipsoids in synchronous rotation, where mutual gravitational pull elongates each component along the line connecting their centers, influencing orbital dynamics and light curves.[37] Examples include systems like Algol, where Roche lobe overflow and tidal locking produce such elongated shapes.[37] Spheroids extend to general relativity for describing non-spherical gravitational fields, particularly for rotating masses where the metric deviates from Schwarzschild symmetry. The Quevedo-Mashhoon formalism generates exact solutions for spinning oblate or prolate spheroids by generalizing the Zipoy-Voorhees metric, enabling computation of the gravitational potential and frame-dragging effects outside such bodies. These metrics incorporate multipole expansions that account for mass and current moments, crucial for modeling the exterior fields of rapidly rotating neutron stars or black hole companions with asphericity.[38] For instance, in binary systems involving compact objects, prolate distortions alter the gravitational waveform, as seen in post-Newtonian approximations where higher-order spheroidal harmonics contribute to the binding energy. In fluid dynamics, spheroidal models describe the equilibrium and evolution of self-gravitating bodies, balancing hydrostatic pressure, gravity, and rotation. The Maclaurin spheroid provides a foundational solution for a uniformly dense, rotating incompressible fluid, forming an oblate shape where the angular velocity $ \omega $ relates to the flattening via $ \omega^2 / (2\pi G \rho) = (3-2e^2) \sqrt{1-e^2} / e^3 - 3(1-e^2)/e^2 \arcsin e $, with $ e $ the eccentricity and $ \rho $ the density; this sequence bifurcates to Jacobi ellipsoids at higher rotations. Advanced applications use nonspherical Boussinesq approximations for stably stratified, self-gravitating fluids, constructing oblate models that capture thermal convection and baroclinic instabilities in planetary interiors or stellar cores without assuming spherical symmetry.[39] These frameworks, often solved via perturbation theory, predict the onset of thermal instabilities in rapidly rotating spheroids, relevant to the cores of Jupiter-like planets.

Dynamical Properties

Spheroids exhibit distinct dynamical properties when subjected to rotational forces, particularly in the context of self-gravitating fluids. For a uniformly rotating, constant-density body in Newtonian gravity, the equilibrium configuration is an oblate spheroid known as the Maclaurin spheroid, where the flattening arises to balance gravitational and centrifugal forces.[40] This oblate form represents a stable equilibrium for angular velocities below a critical value, beyond which the spheroid becomes unstable and may bifurcate into more complex shapes like Jacobi ellipsoids.[41] The rotational stability of these spheroids is governed by the interplay between the body's self-gravity and the centrifugal potential, ensuring that perturbations do not grow uncontrollably for sufficiently slow rotations. The moments of inertia for a uniform-density oblate spheroid differ from those of a sphere due to the asymmetry along the polar axis. For an oblate spheroid with equatorial radius aa and polar radius c<ac < a, the principal moments are Ixx=Iyy=15M(a2+c2)I_{xx} = I_{yy} = \frac{1}{5} M (a^2 + c^2) about the equatorial axes and Izz=25Ma2I_{zz} = \frac{2}{5} M a^2 about the polar axis, where MM is the mass.[42] In contrast, a sphere has all moments equal to 25Ma2\frac{2}{5} M a^2. This difference in IzzI_{zz} and the equatorial moments leads to torque-free motion characterized by precession and nutation when the spheroid rotates about a non-principal axis. For Earth, modeled as an oblate spheroid, this manifests as the Chandler wobble, a free nutation with a period of approximately 433 days, resulting from the slight misalignment between the rotation axis and the principal axis of maximum inertia.[43][44] The shape of a rotating spheroid evolves to achieve hydrostatic equilibrium under the combined influence of gravity and centrifugal forces. The governing equation is the hydrostatic equilibrium condition in the rotating frame: 1ρ∇P=−∇Φ+Ω2s\frac{1}{\rho} \nabla P = -\nabla \Phi + \Omega^2 \mathbf{s}, where ρ\rho is density, PP is pressure, Φ\Phi is the gravitational potential, Ω\Omega is the angular velocity, and s\mathbf{s} is the cylindrical radius vector perpendicular to the rotation axis representing the centrifugal acceleration.[45] For small rotations, this leads to an oblate shape where the flattening ϵ=(a−c)/a\epsilon = (a - c)/a is proportional to Ω2\Omega^2, as derived in the Maclaurin sequence. In radial coordinates, the pressure gradient balances the effective gravity: ∂P∂r=−ρ(g−Ω2rsin⁡2θ)\frac{\partial P}{\partial r} = -\rho \left( g - \Omega^2 r \sin^2 \theta \right), with gg the gravitational acceleration and θ\theta the colatitude, ensuring the surfaces of constant pressure align with the spheroidal equipotentials.[46]

References

  1. Spheroid -- from Wolfram MathWorld
  2. Spheroid: Definition, Properties, Shapes & Uses in Maths - Vedantu
  3. Oblate Spheroid - an overview | ScienceDirect Topics
  4. Prolate Spheroid -- from Wolfram MathWorld
  5. The Geopotential Surface - National Geodetic Survey - NOAA
  6. Spheroids and spheres - ArcMap Resources for ArcGIS Desktop
  7. User Manual 3.3 Spheroids - Patrius - CNES
  8. Spheroids - Geometry - TechnologyUK
  9. The History of Geodes: Global Positioning Tutorial
  10. How Newton Derived Shape of Earth | American Physical Society
  11. Chapter 5. Gravity surveying and the 'Figure of the Earth' from ...
  12. Alexis Clairaut (1713 - Biography - MacTutor History of Mathematics
  13. World Geodetic System 1984 (WGS 84) - NGA - Office of Geomatics
  14. [PDF] 1 CHAPTER 3 Ellipsoids and Hyperboloids (Central Quadrics) 3.1 ...
  15. Oblate Spheroid -- from Wolfram MathWorld
  16. [PDF] Surface Area of an Ellipsoid - MIT OpenCourseWare
  17. [PDF] Integrals and Series Related to the Surface Area of Arbitrary Ellipsoids
  18. Surface Area of an Ellipsoid - Scalene Ellipsoid - Numericana
  19. Ellipsoid -- from Wolfram MathWorld
  20. 17.2 Volume of an Ellipsoid: the Disk Method
  21. volume of ellipsoid - PlanetMath
  22. 6.2 Determining Volumes by Slicing - Calculus Volume 1 | OpenStax
  23. [PDF] Map projections--a working manual - USGS Publications Warehouse
  24. [1104.5145] Surface Area and Curvature of the general Ellipsoid
  25. Flattening -- from Wolfram MathWorld
  26. Convert between ellipse eccentricity, flattening, aspect ratio
  27. oblateSpheroid - Oblate ellipsoid of revolution - MATLAB - MathWorks
  28. Rotational flattening
  29. Oblateness - an overview | ScienceDirect Topics
  30. axial_precession_physics.html - UNLV Physics
  31. 4.3: The Ellipsoid - Physics LibreTexts
  32. (PDF) On the Stability of Rotating Drops - ResearchGate
  33. Minor planets and related objects. XV. Asteroid (1620) Geographos
  34. Application of Mie theory to determine the structure of spheroidal ...
  35. Shape of a slowly rotating star measured by asteroseismology
  36. Epicyclic frequencies of spheroidal stars with non-uniform density
  37. Close Binary Stars Modeled by Two Prolate Ellipsoids ... - IOP Science
  38. On relativistic multipole moments of stationary space–times - Journals
  39. None
  40. Orbital and epicyclic frequencies of Maclaurin spheroids
  41. On the Stability of a Maclaurin Spheroid of Small Viscosity.
  42. [PDF] Moment of Inertia and Strain January 5, 2003 1
  43. Wobble and nutation of the Earth - Oxford Academic
  44. First Detection of a Built-In Wobble on Another Planet - Eos.org
  45. 2-dimensional models of rapidly rotating stars - II. Hydrostatic and ...
  46. Maclaurin Spheroids - Richard Fitzpatrick
User Avatar
No comments yet.