World Geodetic System
The World Geodetic System is a standard for use in cartography and satellite navigation including GPS. This standard includes the definition of the coordinate system's fundamental and derived constants, the ellipsoidal Earth Gravitational Model, a description of the associated World Magnetic Model, a current list of local datum transformations; the latest revision is WGS 84, established in 1984 and last revised in 2004. Earlier schemes included WGS 72, WGS 66, WGS 60. WGS 84 is the reference coordinate system used by the Global Positioning System; the coordinate origin of WGS 84 is meant to be located at the Earth's center of mass. The WGS 84 meridian of zero longitude is the IERS Reference Meridian, 5.3 arc seconds or 102 metres east of the Greenwich meridian at the latitude of the Royal Observatory. The WGS 84 datum surface is an oblate spheroid with equatorial radius a = 6378137 m at the equator and flattening f = 1/298.257223563. The polar semi-minor axis b equals a × = 6356752.3142 m. WGS 84 uses the Earth Gravitational Model 2008.
This geoid defines the nominal sea level surface by means of a spherical harmonics series of degree 360. The deviations of the EGM96 geoid from the WGS 84 reference ellipsoid range from about −105 m to about +85 m. EGM96 differs from the original WGS 84 geoid, referred to as EGM84. WGS 84 uses the World Magnetic Model 2015v2; the new version of WMM 2015 became necessary due to extraordinarily large and erratic movements of the north magnetic pole. The next regular update will occur in late 2019. Efforts to supplement the various national surveying systems began in the 19th century with F. R. Helmert's famous book Mathematische und Physikalische Theorien der Physikalischen Geodäsie. Austria and Germany founded the Zentralbüro für die Internationale Erdmessung, a series of global ellipsoids of the Earth were derived. A unified geodetic system for the whole world became essential in the 1950s for several reasons: International space science and the beginning of astronautics; the lack of inter-continental geodetic information.
The inability of the large geodetic systems, such as European Datum, North American Datum, Tokyo Datum, to provide a worldwide geo-data basis Need for global maps for navigation and geography. Western Cold War preparedness necessitated a standardised, NATO-wide geospatial reference system, in accordance with the NATO Standardisation AgreementIn the late 1950s, the United States Department of Defense, together with scientists of other institutions and countries, began to develop the needed world system to which geodetic data could be referred and compatibility established between the coordinates of separated sites of interest. Efforts of the U. S. Army and Air Force were combined leading to the DoD World Geodetic System 1960; the term datum as used here refers to a smooth surface somewhat arbitrarily defined as zero elevation, consistent with a set of surveyor's measures of distances between various stations, differences in elevation, all reduced to a grid of latitudes and elevations. Heritage surveying methods found elevation differences from a local horizontal determined by the spirit level, plumb line, or an equivalent device that depends on the local gravity field.
As a result, the elevations in the data are referenced to the geoid, a surface, not found using satellite geodesy. The latter observational method is more suitable for global mapping. Therefore, a motivation, a substantial problem in the WGS and similar work is to patch together data that were not only made separately, for different regions, but to re-reference the elevations to an ellipsoid model rather than to the geoid. In accomplishing WGS 60, a combination of available surface gravity data, astro-geodetic data and results from HIRAN and Canadian SHORAN surveys were used to define a best-fitting ellipsoid and an earth-centered orientation for each of selected datum; the sole contribution of satellite data to the development of WGS 60 was a value for the ellipsoid flattening, obtained from the nodal motion of a satellite. Prior to WGS 60, the U. S. Army and U. S. Air Force had each developed a world system by using different approaches to the gravimetric datum orientation method. To determine their gravimetric orientation parameters, the Air Force used the mean of the differences between the gravimetric and astro-geodetic deflections and geoid heights at selected stations in the areas of the major datums.
The Army performed an adjustment to minimize the difference between astro-geodetic and gravimetric geoids. By matching the relative astro-geodetic geoids of the selected datums with an earth-centered gravimetric geoid, the selected datums were reduced to an earth-centered orientation. Since the Army and Air Force systems agreed remarkably well for the NAD, ED and TD areas, they were consolidated and became WGS 60. Improvements to the global system included the Astrogeoid of Irene Fischer and the astronautic Mercury datum. In January 1966, a World Geodetic System Committee composed of representatives from the United States Army and Air Force was charged with developing an improved WGS, needed to satisfy mapping and geodetic requirements. Additional surface gravity observa
Geodesy, is the Earth science of measuring and understanding Earth's geometric shape, orientation in space, gravitational field. The field incorporates studies of how these properties change over time and equivalent measurements for other planets. Geodynamical phenomena include crustal motion and polar motion, which can be studied by designing global and national control networks, applying space and terrestrial techniques, relying on datums and coordinate systems; the word "geodesy" comes from the Ancient Greek word γεωδαισία geodaisia. It is concerned with positioning within the temporally varying gravity field. Geodesy in the German-speaking world is divided into "higher geodesy", concerned with measuring Earth on the global scale, "practical geodesy" or "engineering geodesy", concerned with measuring specific parts or regions of Earth, which includes surveying; such geodetic operations are applied to other astronomical bodies in the solar system. It is the science of measuring and understanding Earth's geometric shape, orientation in space, gravity field.
To a large extent, the shape of Earth is the result of rotation, which causes its equatorial bulge, the competition of geological processes such as the collision of plates and of volcanism, resisted by Earth's gravity field. This applies to the liquid surface and Earth's atmosphere. For this reason, the study of Earth's gravity field is called physical geodesy; the geoid is the figure of Earth abstracted from its topographical features. It is an idealized equilibrium surface of sea water, the mean sea level surface in the absence of currents and air pressure variations, continued under the continental masses; the geoid, unlike the reference ellipsoid, is irregular and too complicated to serve as the computational surface on which to solve geometrical problems like point positioning. The geometrical separation between the geoid and the reference ellipsoid is called the geoidal undulation, it varies globally between ± 110 m. A reference ellipsoid, customarily chosen to be the same size as the geoid, is described by its semi-major axis a and flattening f.
The quantity f = a − b/a, where b is the semi-minor axis, is a purely geometrical one. The mechanical ellipticity of Earth can be determined to high precision by observation of satellite orbit perturbations, its relationship with the geometrical flattening is indirect. The relationship depends on the internal density distribution, or, in simplest terms, the degree of central concentration of mass; the 1980 Geodetic Reference System posited a 1:298.257 flattening. This system was adopted at the XVII General Assembly of the International Union of Geodesy and Geophysics, it is the basis for geodetic positioning by the Global Positioning System and is thus in widespread use outside the geodetic community. The numerous systems that countries have used to create maps and charts are becoming obsolete as countries move to global, geocentric reference systems using the GRS 80 reference ellipsoid; the geoid is "realizable", meaning it can be located on Earth by suitable simple measurements from physical objects like a tide gauge.
The geoid can, therefore, be considered a real surface. The reference ellipsoid, has many possible instantiations and is not realizable, therefore it is an abstract surface; the third primary surface of geodetic interest—the topographic surface of Earth—is a realizable surface. The locations of points in three-dimensional space are most conveniently described by three cartesian or rectangular coordinates, X, Y and Z. Since the advent of satellite positioning, such coordinate systems are geocentric: the Z-axis is aligned with Earth's rotation axis. Prior to the era of satellite geodesy, the coordinate systems associated with a geodetic datum attempted to be geocentric, but their origins differed from the geocenter by hundreds of meters, due to regional deviations in the direction of the plumbline; these regional geodetic data, such as ED 50 or NAD 27 have ellipsoids associated with them that are regional "best fits" to the geoids within their areas of validity, minimizing the deflections of the vertical over these areas.
It is only because GPS satellites orbit about the geocenter, that this point becomes the origin of a coordinate system defined by satellite geodetic means, as the satellite positions in space are themselves computed in such a system. Geocentric coordinate systems used in geodesy can be divided into two classes: Inertial reference systems, where the coordinate axes retain their orientation relative to the fixed stars, or equivalently, to the rotation axes of ideal gyroscopes; the X-axis lies within the Greenwich observatory's meridian plane. The coordinate transformation between these two systems is described to good approximation by sidereal time, which takes into account variations in Earth's axial rotation. A more accurate description takes polar motion into account, a phenomenon monitored by geodesists. In surveying and mapping, important fields of application of geodesy, two general types of coordinate systems are used in the plane
Earth radius is the distance from the center of Earth to a point on its surface. Its value ranges from 6,378 kilometres at the equator to 6,357 kilometres at a pole. Earth's radius can be defined in different ways; the surface to which a radius extends is chosen to be on an ellipsoid representing the shape of Earth. Like the surface, what point gets used for the center of Earth is a matter of definition and therefore contributes to the diverse ways of defining Earth's radius; when only one radius is stated, the International Astronomical Union prefers that it be the equatorial radius. The International Union of Geodesy and Geophysics gives three global average radii: the arithmetic mean of the radii of the ellipsoid. All three of those radii are about 6,371 kilometres. Many other ways to define Earth radius have been described; some appear below. A few definitions yield values outside the range between polar radius and equatorial radius because they include local or geoidal topology or because they depend on abstract geometrical considerations.
Earth's rotation, internal density variations, external tidal forces cause its shape to deviate systematically from a perfect sphere. Local topography increases the variance. Our descriptions of Earth's surface must be simpler than reality. Hence, we create models to approximate characteristics of Earth's surface relying on the simplest model that suits the need; each of the models in common use involve some notion of the geometric radius. Speaking, spheres are the only solids to have radii, but broader uses of the term radius are common in many fields, including those dealing with models of Earth; the following is a partial list of models of Earth's surface, ordered from exact to more approximate: The actual surface of Earth The geoid, defined by mean sea level at each point on the real surface A spheroid called an ellipsoid of revolution, geocentric to model the entire Earth, or else geodetic for regional work A sphereIn the case of the geoid and ellipsoids, the fixed distance from any point on the model to the specified center is called "a radius of the Earth" or "the radius of the Earth at that point".
It is common to refer to any mean radius of a spherical model as "the radius of the earth". When considering the Earth's real surface, on the other hand, it is uncommon to refer to a "radius", since there is no practical need. Rather, elevation above or below sea level is useful. Regardless of the model, any radius falls between the polar minimum of about 6,357 km and the equatorial maximum of about 6,378 km. Hence, the Earth deviates from a perfect sphere by only a third of a percent, which supports the spherical model in many contexts and justifies the term "radius of the Earth". While specific values differ, the concepts in this article generalize to any major planet. Rotation of a planet causes it to approximate an oblate ellipsoid/spheroid with a bulge at the equator and flattening at the North and South Poles, so that the equatorial radius a is larger than the polar radius b by aq; the oblateness constant q is given by q = a 3 ω 2 G M, where ω is the angular frequency, G is the gravitational constant, M is the mass of the planet.
For the Earth 1/q ≈ 289, close to the measured inverse flattening 1/f ≈ 298.257. Additionally, the bulge at the equator shows slow variations; the bulge had been decreasing, but since 1998 the bulge has increased due to redistribution of ocean mass via currents. The variation in density and crustal thickness causes gravity to vary across the surface and in time, so that the mean sea level differs from the ellipsoid; this difference is the geoid height, positive above or outside the ellipsoid, negative below or inside. The geoid height variation is under 110 m on Earth; the geoid height can change abruptly due to earthquakes or reduction in ice masses. Not all deformations originate within the Earth; the gravity of the Moon and Sun cause the Earth's surface at a given point to undulate by tenths of meters over a nearly 12-hour period. Given local and transient influences on surface height, the values defined below are based on a "general purpose" model, refined as globally as possible within 5 m of reference ellipsoid height, to within 100 m of mean sea level.
Additionally, the radius can be estimated from the curvature of the Earth at a point. Like a torus, the curvature at a point will be greatest in one direction and smallest perpendicularly; the corresponding radius of curvature depends on the location and direction of measurement from that point. A consequence is that a distance to the true horizon at the equator is shorter in the north/south direction than in the east-west direction. In summary, local variations in terrain prevent defining a single "precise" radius. One can only adopt an idealized model. Since the estimate by Eratosthenes, many models have been created; these models were based on regional topography, giving the best reference ellipsoid for the area under survey. As satellite remote sensing and the Global Positioning System ga
Quasi-Zenith Satellite System
The Quasi-Zenith Satellite System is a project of the Japanese government for the development of a four-satellite regional time transfer system and a satellite-based augmentation system for the United States operated Global Positioning System to be receivable in the Asia-Oceania regions, with a focus on Japan. The goal of QZSS is to provide precise and stable positioning services in the Asia-Oceania region, compatible with GPS. Four-satellite QZSS services are available on a trial basis as of January 12, 2018, started on November 1, 2018. In 2002, the Japanese government authorized the development of QZSS, as a three-satellite regional time transfer system and a satellite-based augmentation system for the United States operated Global Positioning System to be receivable within Japan. A contract was awarded to Advanced Space Business Corporation, that began concept development work, Mitsubishi Electric, GNSS Technologies Inc. However, ASBC collapsed in 2007, the work was taken over by the Satellite Positioning Research and Application Center, owned by four Japanese government departments: the Ministry of Education, Sports and Technology, the Ministry of Internal Affairs and Communications, the Ministry of Economy and Industry, the Ministry of Land, Infrastructure and Tourism.
The first satellite "Michibiki" was launched on 11 September 2010. Full operational status was expected by 2013. In March 2013, Japan's Cabinet Office announced the expansion of QZSS from three satellites to four; the $526 million contract with Mitsubishi Electric for the construction of three satellites was scheduled for launch before the end of 2017. The third satellite was launched into orbit on 19 August 2017, the fourth was launched on 10 October 2017; the basic four-satellite system was announced as operational on November 1, 2018. QZSS uses three satellites, in inclined elliptical, geosynchronous orbits; each orbit is 120° apart from the other two. Because of this inclination, they are not geostationary. Instead, their ground traces are asymmetrical figure-8 patterns, designed to ensure that one is directly overhead over Japan at all times; the nominal orbital elements are: The primary purpose of QZSS is to increase the availability of GPS in Japan's numerous urban canyons, where only satellites at high elevation can be seen.
A secondary function is performance enhancement, increasing the accuracy and reliability of GPS derived navigation solutions. The Quasi-Zenith Satellites transmit signals compatible with the GPS L1C/A signal, as well as the modernized GPS L1C, L2C signal and L5 signals; this minimizes changes to existing GPS receivers. Compared to standalone GPS, the combined system GPS plus QZSS delivers improved positioning performance via ranging correction data provided through the transmission of submeter-class performance enhancement signals L1-SAIF and LEX from QZSS, it improves reliability by means of failure monitoring and system health data notifications. QZSS provides other support data to users to improve GPS satellite acquisition. According to its original plan, QZSS was to carry two types of space-borne atomic clocks; the development of a passive hydrogen maser for QZSS was abandoned in 2006. The positioning signal will be generated by a Rb clock and an architecture similar to the GPS timekeeping system will be employed.
QZSS will be able to use a Two-Way Satellite Time and Frequency Transfer scheme, which will be employed to gain some fundamental knowledge of satellite atomic standard behavior in space as well as for other research purposes. Although the first generation QZSS timekeeping system will be based on the Rb clock, the first QZSS satellites will carry a basic prototype of an experimental crystal clock synchronization system. During the first half of the two year in-orbit test phase, preliminary tests will investigate the feasibility of the atomic clock-less technology which might be employed in the second generation QZSS; the mentioned QZSS TKS technology is a novel satellite timekeeping system which does not require on-board atomic clocks as used by existing navigation satellite systems such as GPS, GLONASS, NAVIC or Galileo system. This concept is differentiated by the employment of a synchronization framework combined with lightweight steerable on-board clocks which act as transponders re-broadcasting the precise time remotely provided by the time synchronization network located on the ground.
This allows the system to operate optimally when satellites are in direct contact with the ground station, making it suitable for a system like the Japanese QZSS. Low satellite mass and low satellite manufacturing and launch cost are significant advantages of this system. An outline of this concept as well as two possible implementations of the time synchronization network for QZSS were studied and published in Remote Synchronization Method for the Quasi-Zenith Satellite System and Remote Synchronization Method for the Quasi-Zenith Satellite System: study of a novel satellite timekeeping system which does not require on-board atomic clocks. Global Navigation Satellite System Multi-functional Satellite Augmentation System Inclined orbit Government Of Japan QZSS site JAXA QZSS site JAXA MICHIBIKI data site JAXA MICHIBIKI data site, English subsite JAXA Quasi-Zenith Satellite-1 "MICHIBIKI" JAXA MICHIBIKI Special Site JAXA QZSS Twitter account ESA Navipedia QZSS article
Carl Friedrich Gauss
Johann Carl Friedrich Gauss (. Sometimes referred to as the Princeps mathematicorum and "the greatest mathematician since antiquity", Gauss had an exceptional influence in many fields of mathematics and science, is ranked among history's most influential mathematicians. Johann Carl Friedrich Gauss was born on 30 April 1777 in Brunswick, in the Duchy of Brunswick-Wolfenbüttel, to poor, working-class parents, his mother was illiterate and never recorded the date of his birth, remembering only that he had been born on a Wednesday, eight days before the Feast of the Ascension. Gauss solved this puzzle about his birthdate in the context of finding the date of Easter, deriving methods to compute the date in both past and future years, he was christened and confirmed in a church near the school he attended as a child. Gauss was a child prodigy. In his memorial on Gauss, Wolfgang Sartorius von Waltershausen says that when Gauss was three years old he corrected a math error his father made. Many versions of this story have been retold since that time with various details regarding what the series was – the most frequent being the classical problem of adding all the integers from 1 to 100.
There are many other anecdotes about his precocity while a toddler, he made his first groundbreaking mathematical discoveries while still a teenager. He completed his magnum opus, Disquisitiones Arithmeticae, in 1798, at the age of 21—though it was not published until 1801; this work was fundamental in consolidating number theory as a discipline and has shaped the field to the present day. Gauss's intellectual abilities attracted the attention of the Duke of Brunswick, who sent him to the Collegium Carolinum, which he attended from 1792 to 1795, to the University of Göttingen from 1795 to 1798. While at university, Gauss independently rediscovered several important theorems, his breakthrough occurred in 1796 when he showed that a regular polygon can be constructed by compass and straightedge if the number of its sides is the product of distinct Fermat primes and a power of 2. This was a major discovery in an important field of mathematics. Gauss was so pleased with this result that he requested that a regular heptadecagon be inscribed on his tombstone.
The stonemason declined, stating that the difficult construction would look like a circle. The year 1796 was productive for both Gauss and number theory, he discovered a construction of the heptadecagon on 30 March. He further advanced modular arithmetic simplifying manipulations in number theory. On 8 April he became the first to prove the quadratic reciprocity law; this remarkably general law allows mathematicians to determine the solvability of any quadratic equation in modular arithmetic. The prime number theorem, conjectured on 31 May, gives a good understanding of how the prime numbers are distributed among the integers. Gauss discovered that every positive integer is representable as a sum of at most three triangular numbers on 10 July and jotted down in his diary the note: "ΕΥΡΗΚΑ! num = Δ + Δ' + Δ". On 1 October he published a result on the number of solutions of polynomials with coefficients in finite fields, which 150 years led to the Weil conjectures. Gauss remained mentally active into his old age while suffering from gout and general unhappiness.
For example, at the age of 62, he taught himself Russian. In 1840, Gauss published his influential Dioptrische Untersuchungen, in which he gave the first systematic analysis on the formation of images under a paraxial approximation. Among his results, Gauss showed that under a paraxial approximation an optical system can be characterized by its cardinal points and he derived the Gaussian lens formula. In 1845, he became an associated member of the Royal Institute of the Netherlands. In 1854, Gauss selected the topic for Bernhard Riemann's inaugural lecture "Über die Hypothesen, welche der Geometrie zu Grunde liegen". On the way home from Riemann's lecture, Weber reported that Gauss was full of excitement. On 23 February 1855, Gauss died of a heart attack in Göttingen. Two people gave eulogies at his funeral: Gauss's son-in-law Heinrich Ewald, Wolfgang Sartorius von Waltershausen, Gauss's close friend and biographer. Gauss's brain was preserved and was studied by Rudolf Wagner, who found its mass to be above average, at 1,492 grams, the cerebral area equal to 219,588 square millimeters.
Developed convolutions were found, which in the early 20th century were suggested as the explanation of his genius. Gauss was a Lutheran Protestant, a member of the St. Albans Evangelical Lutheran church in Göttingen. Potential evidence that Gauss believed in God comes from his response after solving a problem that had defeated him: "Finally, two days ago, I succeeded—not on account of my hard efforts, but by th
Satellite geodesy is geodesy by means of artificial satellites — the measurement of the form and dimensions of Earth, the location of objects on its surface and the figure of the Earth's gravity field by means of artificial satellite techniques. It belongs to the broader field of space geodesy. Traditional astronomical geodesy is not considered a part of satellite geodesy, although there is considerable overlap between the techniques; the main goals of satellite geodesy are: Determination of the figure of the Earth and navigation Determination of geoid, Earth's gravity field and its temporal variations Measurement of geodynamical phenomena, such as crustal dynamics and polar motionSatellite geodetic data and methods can be applied to diverse fields such as navigation, hydrography and geophysics. Satellite geodesy relies on orbital mechanics. Satellite geodesy began shortly after the launch of Sputnik in 1957. Observations of Explorer 1 and Sputnik 2 in 1958 allowed for an accurate determination of Earth's flattening.
The 1960s saw the launch of the Doppler satellite Transit-1B and the balloon satellites Echo 1, Echo 2, PAGEOS. The first dedicated geodetic satellite was ANNA-1B, a collaborative effort between NASA, the DoD, other civilian agencies. ANNA-1B carried the first of the US Army's SECOR instruments; these missions led to the accurate determination of the leading spherical harmonic coefficients of the geopotential, the general shape of the geoid, linked the world's geodetic datums. Soviet military satellites undertook geodesic missions to assist in ICBM targeting in the late 1960s and early 1970s; the Transit satellite system was used extensively for Doppler surveying and positioning. Observations of satellites in the 1970s by worldwide triangulation networks allowed for the establishment of the World Geodetic System; the development of GPS by the United States in the 1980s allowed for precise navigation and positioning and soon became a standard tool in surveying. In the 1980s and 1990s satellite geodesy began to be used for monitoring of geodynamic phenomena, such as crustal motion, Earth rotation, polar motion.
The 1990s were focused on the development of reference frames. Dedicated satellites were launched to measure Earth's gravity field in the 2000s, such as CHAMP, GRACE, GOCE. Techniques of satellite geodesy may be classified by instrument platform: A satellite may be observed with ground-based instruments, carry an instrument or sensor as part of its payload to observe the Earth, or use its instruments to track or be tracked by another satellite. Global navigation satellite systems are dedicated radio positioning services, which can locate a receiver to within a few meters; the most prominent system, GPS, consists of a constellation of 31 satellites in high, 12-hour circular orbits, distributed in six planes with 55° inclinations. The principle of location is based on trilateration; each satellite transmits a precise ephemeris with information on its own position and a message containing the exact time of transmission. The receiver compares this time of transmission with its own clock at the time of reception and multiplies the difference by the speed of light to obtain a "pseudorange."
Four pseudoranges are needed to obtain the precise time and the receiver's position within a few meters. More sophisticated methods, such as real-time kinematic can yield positions to within a few millimeters. In geodesy, GNSS is used as an economical tool for time transfer, it is used for monitoring Earth's rotation, polar motion, crustal dynamics. The presence of the GPS signal in space makes it suitable for orbit determination and satellite-to-satellite tracking. Examples: GPS, GLONASS, Galileo In satellite laser ranging a global network of observation stations measure the round trip time of flight of ultrashort pulses of light to satellites equipped with retroreflectors; this provides instantaneous range measurements of millimeter level precision which can be accumulated to provide accurate orbit parameters, gravity field parameters, Earth rotation parameters, tidal Earth's deformations and velocities of SLR stations, other substantial geodetic data. Satellite laser ranging is a proven geodetic technique with significant potential for important contributions to scientific studies of the Earth/Atmosphere/Oceans system.
It is the most accurate technique available to determine the geocentric position of an Earth satellite, allowing for the precise calibration of radar altimeters and separation of long-term instrumentation drift from secular changes in ocean surface topography. Satellite laser ranging contributes to the definition of the international terrestrial reference frames by providing the information about the scale and the origin of the reference frame, the so-called geocenter coordinates. Example: LAGEOS Doppler positioning involves recording the Doppler shift of a radio signal of stable frequency emitted from a satellite as the satellite approaches and recedes from the observer; the observed frequency depends on the radial velocity of the satellite relative to the observer, constrained by orbital mechanics. If the observer knows the orbit of the satellite the recording the Doppler profile determines the observer's position. Conversely, if the observer's position is known the orbit of the satellite can be determined and used to study the Earth's gravity.
In DORIS, the ground station emits the satellite receives. Examples: Tr
An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more of an affine transformation. An ellipsoid is a quadric surface. Among quadric surfaces, an ellipsoid is characterized by either of the two following properties; every planar cross section is empty, or is reduced to a single point. It is bounded. An ellipsoid has three pairwise perpendicular axes of symmetry which intersect at a center of symmetry, called the center of the ellipsoid; the line segments that are delimited on the axes of symmetry by the ellipsoid are called the principal axes, or axes of the ellipsoid. If the three axes have different lengths, the ellipsoid is said to be tri-axial or scalene, the axes are uniquely defined. If two of the axes have the same length the ellipsoid is an "ellipsoid of revolution" called a spheroid. In this case, the ellipsoid is invariant under a rotation around the third axis, there are thus infinitely many ways of choosing the two perpendicular axes of the same length.
If the third axis is shorter, the ellipsoid is an oblate spheroid. If the three axes have the same length, the ellipsoid is a sphere. Using a Cartesian coordinate system in which the origin is the center of the ellipsoid and the coordinate axes are axes of the ellipsoid, the implicit equation of the ellipsoid has the standard form x 2 a 2 + y 2 b 2 + z 2 c 2 = 1, where a, b, c are positive real numbers; the points, lie on the surface. The line segments from the origin to these points are called the principal semi-axes of the ellipsoid, because a, b, c are half the length of the principal axes, they correspond to semi-minor axis of an ellipse. If a = b > c, one has an oblate spheroid. The ellipsoid may be parameterized in several ways, which are simpler to express when the ellipsoid axes coincide with coordinate axes. A common choice is x = a cos cos , y = b cos sin , z = c sin , where − π 2 ≤ θ ≤ π 2, − π ≤ φ ≤ π; these parameters may be interpreted as spherical coordinates, where π / 2 − θ is the polar angle, φ is the azimuth angle of the point of the ellipsoid.
The volume bounded by the ellipsoid is V = 4 3 π a b c. Alternatively expressed, where A, B and C are the lengths of the principal semi-axes: V = π 6 A B C ≈ 0.523 A B C. Note that this equation reduces to that of the volume of a sphere when all three elliptic radii are equal, to that of an oblate or prolate spheroid when two of them are equal; the volume of an ellipsoid is 2 3 the volume of a circumscribed elliptic cylinder, π 6 the volume of the circumscribed box. The volumes of the inscribed and circumscribed boxes are respectively: V inscribed = 8 3 3 a b c, V circumscribed = 8 a b c; the surface area of a general ellipsoid is S = 2 π c 2 + 2 π a b sin ( E ( φ