Spheroid

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Spheroids with vertical rotational axes
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oblateprolate

A spheroid, or ellipsoid of revolution, 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.

In mathematics, a quadric or quadric surface, is a generalization of conic sections. It is a hypersurface in a (D + 1)-dimensional space, and it is defined as the zero set of an irreducible polynomial of degree two in D + 1 variables. When the defining polynomial is not absolutely irreducible, the zero set is generally not considered a quadric, although it is often called a degenerate quadric or a reducible quadric.

Surface (mathematics) generalization of a plane which needs not be flat, that is, the curvature is not necessarily zero

In mathematics, a surface is a generalization of a plane which doesn't need to be flat – that is, the curvature is not necessarily zero. This is analogous to a curve generalizing a straight line. There are several more precise definitions, depending on the context and the mathematical tools that are used for the study.

Ellipse Plane curve: conic section

In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. As such, it generalizes a circle, which is the special type of ellipse in which the two focal points are the same. The elongation of an ellipse is measured by its eccentricity e, a number ranging from e = 0 to e = 1.

Contents

If the ellipse is rotated about its major axis, the result is a prolate (elongated) spheroid, shaped like an American football or rugby ball. If the ellipse is rotated about its minor axis, the result is an oblate (flattened) spheroid, shaped like a lentil. If the generating ellipse is a circle, the result is a sphere.

American football Team field sport

American football, referred to as football in the United States and Canada and also known as gridiron, is a team sport played by two teams of eleven players on a rectangular field with goalposts at each end. The offense, the team with possession of the oval-shaped football, attempts to advance down the field by running with the ball or passing it, while the defense, the team without possession of the ball, aims to stop the offense's advance and to take control of the ball for themselves. The offense must advance at least ten yards in four downs or plays; if they fail, they turn over the football to the defense, but if they succeed, they are given a new set of four downs to continue the drive. Points are scored primarily by advancing the ball into the opposing team's end zone for a touchdown or kicking the ball through the opponent's goalposts for a field goal. The team with the most points at the end of a game wins.

Rugby football refers to the team sports of rugby league and rugby union.

Lentil Species of plant

The lentil is an edible legume. It is a bushy annual plant known for its lens-shaped seeds. It is about 40 cm (16 in) tall, and the seeds grow in pods, usually with two seeds in each. As a food crop, the majority of world production comes from Canada, India, and Turkey.

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 the Earth is often approximated by an oblate spheroid 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.

Gravity Attractive force between objects with mass

Gravity, or gravitation, is a natural phenomenon by which all things with mass or energy—including planets, stars, galaxies, and even light—are brought toward one another. On Earth, gravity gives weight to physical objects, and the Moon's gravity causes the ocean tides. The gravitational attraction of the original gaseous matter present in the Universe caused it to begin coalescing, forming stars—and for the stars to group together into galaxies—so gravity is responsible for many of the large-scale structures in the Universe. Gravity has an infinite range, although its effects become increasingly weaker on farther objects.

Figure of the Earth Size and shape used to model the Earth for geodesy

Figure of the Earth is a term of art in geodesy that refers to the size and shape used to model Earth. The size and shape it refers to depend on context, including the precision needed for the model. The sphere is an approximation of the figure of the Earth that is satisfactory for many purposes. Several models with greater accuracy have been developed so that coordinate systems can serve the precise needs of navigation, surveying, cadastre, land use, and various other concerns.

Planet Class of astronomical body directly orbiting a star or stellar remnant

A planet is an astronomical body orbiting a star or stellar remnant that is massive enough to be rounded by its own gravity, is not massive enough to cause thermonuclear fusion, and has cleared its neighbouring region of planetesimals.

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

Geodesy The science of the geometric shape, orientation in space, and gravitational field of Earth

Geodesy is the Earth science of accurately measuring and understanding Earth's geometric shape, orientation in space and gravitational field. The field also incorporates studies of how these properties change over time and equivalent measurements for other planets. Geodynamical phenomena include crustal motion, tides and polar motion, which can be studied by designing global and national control networks, applying space and terrestrial techniques and relying on datums and coordinate systems.

Equation

The assignment of semi-axes on a spheroid. It is oblate if c < a (left) and prolate if c > a (right). Ellipsoid-rot-ax.svg
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:

The case of a = c reduces to a sphere.

Properties

Area

An oblate spheroid with c < a has surface area

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]

Eccentricity (mathematics) eccentricity of a conic section

In mathematics, the eccentricity of a conic section is a non-negative real number that uniquely characterizes its shape.

A prolate spheroid with c > a has surface area

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 Soblate can be used to calculate the surface area of a prolate spheroid and vice versa. However, e 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 . If is the equatorial diameter, and is the polar diameter, the volume is .

Curvature

If a spheroid is parameterized as

where β is the reduced or parametric latitude , λ is the longitude, and π/2 < β < +π/2 and −π < λ < +π, then its Gaussian curvature is

and its mean curvature is

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

Aspect ratio

Where c has been redefined without explanation in this section as b... The aspect ratio of an oblate spheroid/ellipse, b : 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.

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.

Oblate spheroids

The planet Jupiter is an oblate spheroid with flattening ratio 0.06487 Jupiter oblate spheroid.png
The planet Jupiter is an oblate spheroid with flattening ratio 0.06487

The oblate spheroid is the approximate shape of many planets and celestial bodies, including Saturn, Jupiter and the quickly-spinning star, Altair. 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. [5] [6] Earth's diverse cartographic and geodetic systems are based on reference ellipsoids, all of which are oblate.

A science-fiction example of an extremely oblate planet is Mesklin from Hal Clement's novel Mission of Gravity .

Prolate spheroids

An Australian rules football. Sherrin.png
An Australian rules football.

The prolate spheroid is the approximate shape of the ball in several sports, such as in rugby football.

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

The term is also used to describe the shape of some nebulae such as the Crab Nebula. [7] 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 elements are shaped like prolate spheroids.[ citation needed ] In anatomy, near-spheroid organs such as testis may be measured by their long and short axes. [8]

Many submarines have a shape which can be described as prolate spheroid. [9]

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 and 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: [10]

where M is the mass of the body defined as

See also

Related Research Articles

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In geography, latitude is a geographic coordinate that specifies the north–south position of a point on the Earth's surface. Latitude is an angle which ranges from 0° at the Equator to 90° at the poles. Lines of constant latitude, or parallels, run east–west as circles parallel to the equator. Latitude is used together with longitude to specify the precise location of features on the surface of the Earth. On its own, the term latitude should be taken to be the geodetic latitude as defined below. Briefly, geodetic latitude at a point is the angle formed by the vector perpendicular to the ellipsoidal surface from that point, and the equatorial plane. Also defined are six auxiliary latitudes which are used in special applications.

In electrodynamics, elliptical polarization is the polarization of electromagnetic radiation such that the tip of the electric field vector describes an ellipse in any fixed plane intersecting, and normal to, the direction of propagation. An elliptically polarized wave may be resolved into two linearly polarized waves in phase quadrature, with their polarization planes at right angles to each other. Since the electric field can rotate clockwise or counterclockwise as it propagates, elliptically polarized waves exhibit chirality.

Ellipsoid closed quadric surface that is a three dimensional analogue of an ellipse

An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation.

Earth radius mean distance from the Earths center to its surface

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Reference ellipsoid An ellipsoid that approximates the figure of the Earth

In geodesy, a reference ellipsoid is a mathematically defined surface that approximates the geoid, the truer figure of the Earth, or other planetary body. Because of their relative simplicity, reference ellipsoids are used as a preferred surface on which geodetic network computations are performed and point coordinates such as latitude, longitude, and elevation are defined.

Geodetic Reference System 1980 reference ellipsoid

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Flattening Measure of compression between circle to ellipse or sphere to an ellipsoid of revolution

Flattening is a measure of the compression of a circle or sphere along a diameter to form an ellipse or an ellipsoid of revolution (spheroid) respectively. Other terms used are ellipticity, or oblateness. The usual notation for flattening is f and its definition in terms of the semi-axes of the resulting ellipse or ellipsoid is

A Maclaurin spheroid is an oblate spheroid which arises when a self-gravitating fluid body of uniform density rotates with a constant angular velocity. This spheroid is named after the Scottish mathematician Colin Maclaurin, who formulated it for the shape of Earth in 1742. In fact the figure of the Earth is far less oblate than this, since the Earth is not homogeneous, but has a dense iron core. The Maclaurin spheroid is considered to be the simplest model of rotating ellipsoidal figures in equilibrium since it assumes uniform density.

Prolate spheroidal coordinates

Prolate spheroidal coordinates are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional elliptic coordinate system about the focal axis of the ellipse, i.e., the symmetry axis on which the foci are located. Rotation about the other axis produces oblate spheroidal coordinates. Prolate spheroidal coordinates can also be considered as a limiting case of ellipsoidal coordinates in which the two smallest principal axes are equal in length.

In hydrodynamics, the Perrin friction factors are multiplicative adjustments to the translational and rotational friction of a rigid spheroid, relative to the corresponding frictions in spheres of the same volume. These friction factors were first calculated by Jean-Baptiste Perrin.

Angular eccentricity

Angular eccentricity is one of many parameters which arise in the study of the ellipse or ellipsoid. It is denoted here by α (alpha). It may be defined in terms of the eccentricity, e, or the aspect ratio, b/a :

Great ellipse

A great ellipse is an ellipse passing through two points on a spheroid and having the same center as that of the spheroid. Equivalently, it is an ellipse on the surface of a spheroid and centered on the origin, or the curve formed by intersecting the spheroid by a plane through its center. For points that are separated by less than about a quarter of the circumference of the earth, about , the length of the great ellipse connecting the points is close to the geodesic distance.

In geodesy, a meridian arc measurement is the distance between two points with the same longitude, i.e., a segment of a meridian curve or its length. Two or more such determinations at different locations then specify the shape of the reference ellipsoid which best approximates the shape of the geoid. This process is called the determination of the figure of the Earth. The earliest determinations of the size of a spherical Earth required a single arc. The latest determinations use astro-geodetic measurements and the methods of satellite geodesy to determine the reference ellipsoids.

Geographical distance Distance measured along the surface of the earth

Geographical distance is the distance measured along the surface of the earth. The formulae in this article calculate distances between points which are defined by geographical coordinates in terms of latitude and longitude. This distance is an element in solving the second (inverse) geodetic problem.

Semi-major and semi-minor axes longest radius of an ellipse

In geometry, the major axis of an ellipse is its longest diameter: a line segment that runs through the center and both foci, with ends at the widest points of the perimeter.

The article Transverse Mercator projection restricts itself to general features of the projection. This article describes in detail one of the (two) implementations developed by Louis Krüger in 1912; that expressed as a power series in the longitude difference from the central meridian. These series were recalculated by Lee in 1946, by Redfearn in 1948, and by Thomas in 1952. They are often referred to as the Redfearn series, or the Thomas series. This implementation is of great importance since it is widely used in the U.S. State Plane Coordinate System, in national and also international mapping systems, including the Universal Transverse Mercator coordinate system (UTM). They are also incorporated into the Geotrans coordinate converter made available by the United States National Geospatial-Intelligence Agency. When paired with a suitable geodetic datum, the series deliver high accuracy in zones less than a few degrees in east-west extent.

Jacobi ellipsoid oblate ellipsoid under equilibrium

A Jacobi ellipsoid is a triaxial ellipsoid under equilibrium which arises when a self-gravitating fluid body of uniform density rotates with a constant angular velocity. It is named after the German mathematician Carl Gustav Jacob Jacobi.

Geodesics on an ellipsoid application of geodesics on ellipsoid geometries

The study of geodesics on an ellipsoid arose in connection with geodesy specifically with the solution of triangulation networks. The figure of the Earth is well approximated by an oblate ellipsoid, a slightly flattened sphere. A geodesic is the shortest path between two points on a curved surface, analogous to a straight line on a plane surface. The solution of a triangulation network on an ellipsoid is therefore a set of exercises in spheroidal trigonometry.

References

  1. Torge, Wolfgang (2001). Geodesy (3rd ed.). Walter de Gruyter. p. 104.
  2. A derivation of this result may be found at "Oblate Spheroid - from Wolfram MathWorld". Mathworld.wolfram.com. Retrieved 24 June 2014.
  3. A derivation of this result may be found at "Prolate Spheroid - from Wolfram MathWorld". Mathworld.wolfram.com. 7 October 2003. Retrieved 24 June 2014.
  4. Brial P., Shaalan C.(2009), Introduction à la Géodésie et au geopositionnement par satellites, p.8
  5. Greenburg, John L. (1995). "Isaac Newton and the Problem of the Earth's Shape". History of Exact Sciences. 49 (4). Springer. pp. 371–391. JSTOR   https://www.jstor.org/stable/41134011.
  6. Durant, Will; Durant, Ariel (28 July 1997). The Story of Civilization: The Age of Louis XIV. MJF Books. ISBN   1567310192.
  7. Trimble, Virginia Louise (October 1973), "The Distance to the Crab Nebula and NP 0532", Publications of the Astronomical Society of the Pacific, 85 (507): 579, Bibcode:1973PASP...85..579T, doi:10.1086/129507
  8. Page 559 in: John Pellerito, Joseph F Polak (2012). Introduction to Vascular Ultrasonography (6 ed.). Elsevier Health Sciences. ISBN   9781455737666.
  9. "What Do a Submarine, a Rocket and a Football Have in Common?". Scientific American . 8 November 2010. Retrieved 13 June 2015.
  10. Weisstein, Eric W. ""Spheroid."". MathWorld--A Wolfram Web Resource. Retrieved 16 May 2018.