{"history":[],"article":{"id":"781bf959-6a4c-4cbb-83c7-fffc4233d07b","title":"Spheroid","slug":"spheroid","content":":::summary\n:::\n\n# \nA **spheroid**, also known as an **[ellipsoid](https://bliptext.com/articles/ellipsoid) of revolution** or **rotational [ellipsoid](https://bliptext.com/articles/ellipsoid)**, is a [quadric](https://bliptext.com/articles/quadric) [surface](https://bliptext.com/articles/surface-mathematics) obtained by [rotating](https://bliptext.com/articles/surface-of-revolution) an [ellipse](https://bliptext.com/articles/ellipse) about one of its principal axes; in other words, an [ellipsoid](https://bliptext.com/articles/ellipsoid) with two equal [semi-diameters](https://bliptext.com/articles/semi-diameter).\nA spheroid has [circular symmetry](https://bliptext.com/articles/circular-symmetry).\nIf the ellipse is rotated about its [major axis](https://bliptext.com/articles/major-axis), the result is a **prolate spheroid**, elongated like a [rugby ball](https://bliptext.com/articles/rugby-ball).\nThe [American football](https://bliptext.com/articles/ball-gridiron-football) is similar but has a pointier end than a spheroid could.\nIf the ellipse is rotated about its [minor axis](https://bliptext.com/articles/minor-axis), the result is an **oblate spheroid**, flattened like a [lentil](https://bliptext.com/articles/lentil) or a plain [M&M](https://bliptext.com/articles/m-m-s).\nIf the generating ellipse is a circle, the result is a [sphere](https://bliptext.com/articles/sphere).\nDue to the combined effects of [gravity](https://bliptext.com/articles/gravity) and [rotation](https://bliptext.com/articles/rotation-of-the-earth), the [figure of the Earth](https://bliptext.com/articles/figure-of-the-earth) (and of all [planets](https://bliptext.com/articles/planet)) is not quite a sphere, but instead is slightly [flattened](https://bliptext.com/articles/flattening) in the direction of its axis of [rotation](https://bliptext.com/articles/rotation-of-the-earth).\nFor that reason, in [cartography](https://bliptext.com/articles/cartography) and [geodesy](https://bliptext.com/articles/geodesy) the Earth is often approximated by an oblate spheroid, known as the [reference ellipsoid](https://bliptext.com/articles/reference-ellipsoid), instead of a sphere.\nThe current [World Geodetic System](https://bliptext.com/articles/world-geodetic-system) model uses a spheroid whose radius is 6,378.137 km at the [Equator](https://bliptext.com/articles/equator) and 6,356.752 km at the [poles](https://bliptext.com/articles/geographical-pole).\nThe 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](https://bliptext.com/articles/earth-s-gravity) [geopotential model](https://bliptext.com/articles/geopotential-model)).\n\n# Equation\n[[File:ellipsoid-rot-ax.svg|thumb|350px| The assignment of semi-axes on a spheroid. It is oblate if\nc < a\n(left) and prolate if\nc > a\n(right).]]\nThe equation of a tri-axial ellipsoid centred at the origin with semi-axes a, b and c aligned along the coordinate axes is\nThe equation of a spheroid with z as the [symmetry axis](https://bliptext.com/articles/symmetry-axis) is given by setting\nThe semi-axis a is the equatorial radius of the spheroid, and c is the distance from centre to pole along the symmetry axis.\nThere are two possible cases:\nc < a\nc > a\nThe case of\nreduces to a sphere.\n\n# Properties\n\n## Area\nAn oblate spheroid with\nc < a\nhas [surface area](https://bliptext.com/articles/surface-area)\nThe 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](https://bliptext.com/articles/eccentricity-mathematics).\n(See [ellipse](https://bliptext.com/articles/ellipse).)\nA prolate spheroid with\nc > a\nhas surface area\nThe 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](https://bliptext.com/articles/eccentricity-mathematics).\n(See [ellipse](https://bliptext.com/articles/ellipse).)\nThese formulas are identical in the sense that the formula for\nSoblate\ncan be used to calculate the surface area of a prolate spheroid and vice versa.\nHowever, e then becomes [imaginary](https://bliptext.com/articles/imaginary-number) and can no longer directly be identified with the eccentricity.\nBoth of these results may be cast into many other forms using standard mathematical identities and relations between parameters of the ellipse.\n\n## Volume\nThe volume inside a spheroid (of any kind) is If\nis the equatorial diameter, and\nis the polar diameter, the volume is\n\n## Curvature\nLet a spheroid be parameterized as where β is the reduced latitude or [parametric latitude](https://bliptext.com/articles/parametric-latitude), λ is the [longitude](https://bliptext.com/articles/longitude), and\n−π⁄2 < β < +π⁄2\nand\n−π < λ < +π\n. Then, the spheroid's [Gaussian curvature](https://bliptext.com/articles/gaussian-curvature) is and its [mean curvature](https://bliptext.com/articles/mean-curvature) is\nBoth of these curvatures are always positive, so that every point on a spheroid is elliptic.\n\n## Aspect ratio\nThe [aspect ratio](https://bliptext.com/articles/aspect-ratio) of an oblate spheroid/ellipse,\nc : a\n, is the ratio of the polar to equatorial lengths, while the [flattening](https://bliptext.com/articles/flattening) (also called oblateness) f, is the ratio of the equatorial-polar length difference to the equatorial length:\nThe first eccentricity (usually simply eccentricity, as above) is often used instead of flattening.\nIt is defined by:\nThe relations between eccentricity and flattening are:\nAll 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.\nWhile these definitions are mathematically interchangeable, real-world calculations must lose some precision.\nTo avoid confusion, an ellipsoidal definition considers its own values to be exact in the form it gives.\n\n# Occurrence and applications\nThe most common shapes for the density distribution of protons and neutrons in an [atomic nucleus](https://bliptext.com/articles/atomic-nucleus) are [spherical](https://bliptext.com/articles/spherical), prolate, and oblate spheroidal, where the polar axis is assumed to be the spin axis (or direction of the spin [angular momentum](https://bliptext.com/articles/angular-momentum) vector).\nDeformed nuclear shapes occur as a result of the competition between [electromagnetic](https://bliptext.com/articles/electromagnetic-force) repulsion between protons, [surface tension](https://bliptext.com/articles/surface-tension) and [quantum](https://bliptext.com/articles/quantum-mechanics) [shell effects](https://bliptext.com/articles/nuclear-shell-model).\nSpheroids are common in [3D cell cultures](https://bliptext.com/articles/3d-cell-culture).\nRotating equilibrium spheroids include the [Maclaurin spheroid](https://bliptext.com/articles/maclaurin-spheroid) and the [Jacobi ellipsoid](https://bliptext.com/articles/jacobi-ellipsoid).\n[Spheroid](https://bliptext.com/articles/spheroid-lithic) is also a shape of archaeological artifacts.\n\n## Oblate spheroids\nThe oblate spheroid is the approximate shape of rotating [planets](https://bliptext.com/articles/planet) and other [celestial bodies](https://bliptext.com/articles/astronomical-object), including Earth, [Saturn](https://bliptext.com/articles/saturn), [Jupiter](https://bliptext.com/articles/jupiter), and the quickly spinning star [Altair](https://bliptext.com/articles/altair).\nSaturn is the most oblate planet in the [Solar System](https://bliptext.com/articles/solar-system), with a [flattening](https://bliptext.com/articles/flattening) of 0.09796.\nSee [planetary flattening](https://bliptext.com/articles/planetary-flattening) and [equatorial bulge](https://bliptext.com/articles/equatorial-bulge) for details.\n[Enlightenment](https://bliptext.com/articles/age-of-enlightenment) scientist [Isaac Newton](https://bliptext.com/articles/isaac-newton), working from [Jean Richer](https://bliptext.com/articles/jean-richer)'s pendulum experiments and [Christiaan Huygens](https://bliptext.com/articles/christiaan-huygens)'s theories for their interpretation, reasoned that Jupiter and [Earth](https://bliptext.com/articles/earth) are oblate spheroids owing to their [centrifugal force](https://bliptext.com/articles/centrifugal-force).\nEarth's diverse cartographic and geodetic systems are based on [reference ellipsoids](https://bliptext.com/articles/reference-ellipsoid), all of which are oblate.\n# [The planet Jupiter is a slight oblate spheroid with a flattening of 0.06487 | upload.wikimedia.org/wikipedia/commons/e/e5/Jupiter///oblate///spheroid.png]\n\n## Prolate spheroids\nThe prolate spheroid is the approximate shape of the ball in several sports, such as in the [rugby ball](https://bliptext.com/articles/rugby-ball).\nSeveral [moons](https://bliptext.com/articles/moons) of the Solar System approximate prolate spheroids in shape, though they are actually [triaxial ellipsoids](https://bliptext.com/articles/triaxial-ellipsoid).\nExamples are [Saturn](https://bliptext.com/articles/saturn)'s satellites [Mimas](https://bliptext.com/articles/mimas-moon), [Enceladus](https://bliptext.com/articles/enceladus-moon), and [Tethys](https://bliptext.com/articles/tethys-moon) and [Uranus](https://bliptext.com/articles/uranus)' satellite [Miranda](https://bliptext.com/articles/miranda-moon).\nIn contrast to being distorted into oblate spheroids via rapid rotation, celestial objects distort slightly into prolate spheroids via [tidal forces](https://bliptext.com/articles/tide) when they orbit a massive body in a close orbit.\nThe most extreme example is Jupiter's moon [Io](https://bliptext.com/articles/io-moon), which becomes slightly more or less prolate in its orbit due to a slight eccentricity, causing intense [volcanism](https://bliptext.com/articles/volcanism).\nThe 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.\nThis combines with the smaller oblate distortion from the synchronous rotation to cause the body to become triaxial.\nThe term is also used to describe the shape of some [nebulae](https://bliptext.com/articles/nebula) such as the [Crab Nebula](https://bliptext.com/articles/crab-nebula).\n[Fresnel zones](https://bliptext.com/articles/fresnel-zone), 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.\nThe [atomic nuclei](https://bliptext.com/articles/atomic-nucleus) of the [actinide](https://bliptext.com/articles/actinide) and [lanthanide](https://bliptext.com/articles/lanthanide) elements are shaped like prolate spheroids.\nIn anatomy, near-spheroid organs such as [testis](https://bliptext.com/articles/testicle) may be measured by their [long and short axes](https://bliptext.com/articles/anatomical-terms-of-location).\nMany submarines have a shape which can be described as prolate spheroid.\n# [A rugby ball. | upload.wikimedia.org/wikipedia/commons/1/1a/Gilbert///rugby///ball///on///grass.jpg]\n\n## Dynamical properties\nFor a spheroid having uniform density, the [moment of inertia](https://bliptext.com/articles/moment-of-inertia) is that of an ellipsoid with an additional axis of symmetry.\nGiven a description of a spheroid as having a [major axis](https://bliptext.com/articles/major-axis) c, and minor axes, the moments of inertia along these principal axes are C, A, and B. However, in a spheroid the minor axes are symmetrical.\nTherefore, our inertial terms along the major axes are:\nwhere M is the mass of the body defined as","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T17:48:42.196Z","updated_at":"2025-01-25T17:48:42.196Z","search_vector":"'spheroid':1","search_content":" \n \n\n# \nA **spheroid**, also known as an **[ellipsoid](https://bliptext.com/articles/ellipsoid) of revolution** or **rotational [ellipsoid](https://bliptext.com/articles/ellipsoid)**, is a [quadric](https://bliptext.com/articles/quadric) [surface](https://bliptext.com/articles/surface-mathematics) obtained by [rotating](https://bliptext.com/articles/surface-of-revolution) an [ellipse](https://bliptext.com/articles/ellipse) about one of its principal axes; in other words, an [ellipsoid](https://bliptext.com/articles/ellipsoid) with two equal [semi-diameters](https://bliptext.com/articles/semi-diameter).\nA spheroid has [circular symmetry](https://bliptext.com/articles/circular-symmetry).\nIf the ellipse is rotated about its [major axis](https://bliptext.com/articles/major-axis), the result is a **prolate spheroid**, elongated like a [rugby ball](https://bliptext.com/articles/rugby-ball).\nThe [American football](https://bliptext.com/articles/ball-gridiron-football) is similar but has a pointier end than a spheroid could.\nIf the ellipse is rotated about its [minor axis](https://bliptext.com/articles/minor-axis), the result is an **oblate spheroid**, flattened like a [lentil](https://bliptext.com/articles/lentil) or a plain [M&M](https://bliptext.com/articles/m-m-s).\nIf the generating ellipse is a circle, the result is a [sphere](https://bliptext.com/articles/sphere).\nDue to the combined effects of [gravity](https://bliptext.com/articles/gravity) and [rotation](https://bliptext.com/articles/rotation-of-the-earth), the [figure of the Earth](https://bliptext.com/articles/figure-of-the-earth) (and of all [planets](https://bliptext.com/articles/planet)) is not quite a sphere, but instead is slightly [flattened](https://bliptext.com/articles/flattening) in the direction of its axis of [rotation](https://bliptext.com/articles/rotation-of-the-earth).\nFor that reason, in [cartography](https://bliptext.com/articles/cartography) and [geodesy](https://bliptext.com/articles/geodesy) the Earth is often approximated by an oblate spheroid, known as the [reference ellipsoid](https://bliptext.com/articles/reference-ellipsoid), instead of a sphere.\nThe current [World Geodetic System](https://bliptext.com/articles/world-geodetic-system) model uses a spheroid whose radius is 6,378.137 km at the [Equator](https://bliptext.com/articles/equator) and 6,356.752 km at the [poles](https://bliptext.com/articles/geographical-pole).\nThe 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](https://bliptext.com/articles/earth-s-gravity) [geopotential model](https://bliptext.com/articles/geopotential-model)).\n\n# Equation\n[[File:ellipsoid-rot-ax.svg|thumb|350px| The assignment of semi-axes on a spheroid. It is oblate if\nc a\n(right).]]\nThe equation of a tri-axial ellipsoid centred at the origin with semi-axes a, b and c aligned along the coordinate axes is\nThe equation of a spheroid with z as the [symmetry axis](https://bliptext.com/articles/symmetry-axis) is given by setting\nThe semi-axis a is the equatorial radius of the spheroid, and c is the distance from centre to pole along the symmetry axis.\nThere are two possible cases:\nc a\nThe case of\nreduces to a sphere.\n\n# Properties\n\n## Area\nAn oblate spheroid with\nc a\nhas surface area\nThe 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](https://bliptext.com/articles/eccentricity-mathematics).\n(See [ellipse](https://bliptext.com/articles/ellipse).)\nThese formulas are identical in the sense that the formula for\nSoblate\ncan be used to calculate the surface area of a prolate spheroid and vice versa.\nHowever, e then becomes [imaginary](https://bliptext.com/articles/imaginary-number) and can no longer directly be identified with the eccentricity.\nBoth of these results may be cast into many other forms using standard mathematical identities and relations between parameters of the ellipse.\n\n## Volume\nThe volume inside a spheroid (of any kind) is If\nis the equatorial diameter, and\nis the polar diameter, the volume is\n\n## Curvature\nLet a spheroid be parameterized as where β is the reduced latitude or [parametric latitude](https://bliptext.com/articles/parametric-latitude), λ is the [longitude](https://bliptext.com/articles/longitude), and\n−π⁄2 < β < +π⁄2\nand\n−π < λ < +π\n. Then, the spheroid's [Gaussian curvature](https://bliptext.com/articles/gaussian-curvature) is and its [mean curvature](https://bliptext.com/articles/mean-curvature) is\nBoth of these curvatures are always positive, so that every point on a spheroid is elliptic.\n\n## Aspect ratio\nThe [aspect ratio](https://bliptext.com/articles/aspect-ratio) of an oblate spheroid/ellipse,\nc : a\n, is the ratio of the polar to equatorial lengths, while the [flattening](https://bliptext.com/articles/flattening) (also called oblateness) f, is the ratio of the equatorial-polar length difference to the equatorial length:\nThe first eccentricity (usually simply eccentricity, as above) is often used instead of flattening.\nIt is defined by:\nThe relations between eccentricity and flattening are:\nAll 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.\nWhile these definitions are mathematically interchangeable, real-world calculations must lose some precision.\nTo avoid confusion, an ellipsoidal definition considers its own values to be exact in the form it gives.\n\n# Occurrence and applications\nThe most common shapes for the density distribution of protons and neutrons in an [atomic nucleus](https://bliptext.com/articles/atomic-nucleus) are [spherical](https://bliptext.com/articles/spherical), prolate, and oblate spheroidal, where the polar axis is assumed to be the spin axis (or direction of the spin [angular momentum](https://bliptext.com/articles/angular-momentum) vector).\nDeformed nuclear shapes occur as a result of the competition between [electromagnetic](https://bliptext.com/articles/electromagnetic-force) repulsion between protons, [surface tension](https://bliptext.com/articles/surface-tension) and [quantum](https://bliptext.com/articles/quantum-mechanics) [shell effects](https://bliptext.com/articles/nuclear-shell-model).\nSpheroids are common in [3D cell cultures](https://bliptext.com/articles/3d-cell-culture).\nRotating equilibrium spheroids include the [Maclaurin spheroid](https://bliptext.com/articles/maclaurin-spheroid) and the [Jacobi ellipsoid](https://bliptext.com/articles/jacobi-ellipsoid).\n[Spheroid](https://bliptext.com/articles/spheroid-lithic) is also a shape of archaeological artifacts.\n\n## Oblate spheroids\nThe oblate spheroid is the approximate shape of rotating [planets](https://bliptext.com/articles/planet) and other [celestial bodies](https://bliptext.com/articles/astronomical-object), including Earth, [Saturn](https://bliptext.com/articles/saturn), [Jupiter](https://bliptext.com/articles/jupiter), and the quickly spinning star [Altair](https://bliptext.com/articles/altair).\nSaturn is the most oblate planet in the [Solar System](https://bliptext.com/articles/solar-system), with a [flattening](https://bliptext.com/articles/flattening) of 0.09796.\nSee [planetary flattening](https://bliptext.com/articles/planetary-flattening) and [equatorial bulge](https://bliptext.com/articles/equatorial-bulge) for details.\n[Enlightenment](https://bliptext.com/articles/age-of-enlightenment) scientist [Isaac Newton](https://bliptext.com/articles/isaac-newton), working from [Jean Richer](https://bliptext.com/articles/jean-richer)'s pendulum experiments and [Christiaan Huygens](https://bliptext.com/articles/christiaan-huygens)'s theories for their interpretation, reasoned that Jupiter and [Earth](https://bliptext.com/articles/earth) are oblate spheroids owing to their [centrifugal force](https://bliptext.com/articles/centrifugal-force).\nEarth's diverse cartographic and geodetic systems are based on [reference ellipsoids](https://bliptext.com/articles/reference-ellipsoid), all of which are oblate.\n# [The planet Jupiter is a slight oblate spheroid with a flattening of 0.06487 | upload.wikimedia.org/wikipedia/commons/e/e5/Jupiter///oblate///spheroid.png]\n\n## Prolate spheroids\nThe prolate spheroid is the approximate shape of the ball in several sports, such as in the [rugby ball](https://bliptext.com/articles/rugby-ball).\nSeveral [moons](https://bliptext.com/articles/moons) of the Solar System approximate prolate spheroids in shape, though they are actually [triaxial ellipsoids](https://bliptext.com/articles/triaxial-ellipsoid).\nExamples are [Saturn](https://bliptext.com/articles/saturn)'s satellites [Mimas](https://bliptext.com/articles/mimas-moon), [Enceladus](https://bliptext.com/articles/enceladus-moon), and [Tethys](https://bliptext.com/articles/tethys-moon) and [Uranus](https://bliptext.com/articles/uranus)' satellite [Miranda](https://bliptext.com/articles/miranda-moon).\nIn contrast to being distorted into oblate spheroids via rapid rotation, celestial objects distort slightly into prolate spheroids via [tidal forces](https://bliptext.com/articles/tide) when they orbit a massive body in a close orbit.\nThe most extreme example is Jupiter's moon [Io](https://bliptext.com/articles/io-moon), which becomes slightly more or less prolate in its orbit due to a slight eccentricity, causing intense [volcanism](https://bliptext.com/articles/volcanism).\nThe 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.\nThis combines with the smaller oblate distortion from the synchronous rotation to cause the body to become triaxial.\nThe term is also used to describe the shape of some [nebulae](https://bliptext.com/articles/nebula) such as the [Crab Nebula](https://bliptext.com/articles/crab-nebula).\n[Fresnel zones](https://bliptext.com/articles/fresnel-zone), 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.\nThe [atomic nuclei](https://bliptext.com/articles/atomic-nucleus) of the [actinide](https://bliptext.com/articles/actinide) and [lanthanide](https://bliptext.com/articles/lanthanide) elements are shaped like prolate spheroids.\nIn anatomy, near-spheroid organs such as [testis](https://bliptext.com/articles/testicle) may be measured by their [long and short axes](https://bliptext.com/articles/anatomical-terms-of-location).\nMany submarines have a shape which can be described as prolate spheroid.\n# [A rugby ball. | upload.wikimedia.org/wikipedia/commons/1/1a/Gilbert///rugby///ball///on///grass.jpg]\n\n## Dynamical properties\nFor a spheroid having uniform density, the [moment of inertia](https://bliptext.com/articles/moment-of-inertia) is that of an ellipsoid with an additional axis of symmetry.\nGiven a description of a spheroid as having a [major axis](https://bliptext.com/articles/major-axis) c, and minor axes, the moments of inertia along these principal axes are C, A, and B. However, in a spheroid the minor axes are symmetrical.\nTherefore, our inertial terms along the major axes are:\nwhere M is the mass of the body defined as","revisionCount":0},"hasMore":false,"nextCursor":null}

History