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91-488: In spatial (3D) geometry , as applied in astronomy , the equator of a rotating spheroid (such as a planet ) is the parallel (circle of latitude) at which latitude is defined to be 0°. It is an imaginary line on the spheroid, equidistant from its poles , dividing it into northern and southern hemispheres. In other words, it is the intersection of the spheroid with the plane perpendicular to its axis of rotation and midway between its geographical poles . On and near

182-414: A = 0 {\displaystyle a=0} . While not explicitly studied by Hamilton, this indirectly introduced notions of basis, here given by the quaternion elements i , j , k {\displaystyle i,j,k} , as well as the dot product and cross product , which correspond to (the negative of) the scalar part and the vector part of the product of two vector quaternions. It

273-434: A n -dimensional Euclidean space and a Cartesian coordinate system . When n = 3 , this space is called the three-dimensional Euclidean space (or simply "Euclidean space" when the context is clear). In classical physics , it serves as a model of the physical universe , in which all known matter exists. When relativity theory is considered, it can be considered a local subspace of space-time . While this space remains

364-487: A parallelogram , and hence are coplanar. A sphere in 3-space (also called a 2-sphere because it is a 2-dimensional object) consists of the set of all points in 3-space at a fixed distance r from a central point P . The solid enclosed by the sphere is called a ball (or, more precisely a 3-ball ). The volume of the ball is given by V = 4 3 π r 3 , {\displaystyle V={\frac {4}{3}}\pi r^{3},} and

455-554: A sphere flattened 0.336% along its axis. This makes the Equator 0.16% longer than a meridian (a great circle passing through the two poles). The IUGG standard meridian is, to the nearest millimetre, 40,007.862917 kilometres (24,859.733480 mi), one arc-minute of which is 1,852.216 metres (6,076.82 ft), explaining the SI standardization of the nautical mile as 1,852 metres (6,076 ft), more than 3 metres (9.8 ft) less than

546-517: A three-dimensional space ( 3D space , 3-space or, rarely, tri-dimensional space ) is a mathematical space in which three values ( coordinates ) are required to determine the position of a point . Most commonly, it is the three-dimensional Euclidean space , that is, the Euclidean space of dimension three, which models physical space . More general three-dimensional spaces are called 3-manifolds . The term may also refer colloquially to

637-502: A 12-hour day and 12-hour night. The name is derived from medieval Latin word aequator , in the phrase circulus aequator diei et noctis , meaning 'circle equalizing day and night', from the Latin word aequare 'make equal'. The latitude of the Earth's equator is, by definition, 0° (zero degrees ) of arc. The equator is one of the five notable circles of latitude on Earth;

728-522: A choice of basis, corresponding to a set of axes. But in rotational symmetry, there is no reason why one set of axes is preferred to say, the same set of axes which has been rotated arbitrarily. Stated another way, a preferred choice of axes breaks the rotational symmetry of physical space. Computationally, it is necessary to work with the more concrete description R 3 {\displaystyle \mathbb {R} ^{3}} in order to do concrete computations. A more abstract description still

819-541: A combination of atmospheric and oceanic processes, with the dominant excitation mechanism being ocean‐bottom pressure fluctuations. Current and historic polar motion data is available from the International Earth Rotation and Reference Systems Service 's Earth orientation parameters . Note in using this data that the convention is to define p x to be positive along 0° longitude and p y to be positive along 90°E longitude. There

910-528: A field , which is not commutative nor associative , but is a Lie algebra with the cross product being the Lie bracket. Specifically, the space together with the product, ( R 3 , × ) {\displaystyle (\mathbb {R} ^{3},\times )} is isomorphic to the Lie algebra of three-dimensional rotations, denoted s o ( 3 ) {\displaystyle {\mathfrak {so}}(3)} . In order to satisfy

1001-412: A given plane, intersect that plane in a unique point, or be parallel to the plane. In the last case, there will be lines in the plane that are parallel to the given line. A hyperplane is a subspace of one dimension less than the dimension of the full space. The hyperplanes of a three-dimensional space are the two-dimensional subspaces, that is, the planes. In terms of Cartesian coordinates, the points of

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1092-400: A hyperplane satisfy a single linear equation , so planes in this 3-space are described by linear equations. A line can be described by a pair of independent linear equations—each representing a plane having this line as a common intersection. Varignon's theorem states that the midpoints of any quadrilateral in R 3 {\displaystyle \mathbb {R} ^{3}} form

1183-561: A linear displacement of either geographical pole amounting to several meters on the surface of the Earth: 100 mas subtends an arc length of 3.082 m, when converted to radians and multiplied by the Earth's polar radius (6,356,752.3 m). Using the geometric axis as the primary axis of a new body-fixed coordinate system, one arrives at the Euler equation of a gyroscope describing

1274-419: A plane curve about a fixed line in its plane as an axis is called a surface of revolution . The plane curve is called the generatrix of the surface. A section of the surface, made by intersecting the surface with a plane that is perpendicular (orthogonal) to the axis, is a circle. Simple examples occur when the generatrix is a line. If the generatrix line intersects the axis line, the surface of revolution

1365-443: A pressure amplitude, Θ −3 a Hough function describing the latitude distribution of the atmospheric pressure on the ground, θ the geographic co-latitude, t the time of year, t 0 a time delay, ν A = 1.003 the normalized frequency of one solar year, λ the longitude, and λ 0 the longitude of maximum pressure. The Hough function in a first approximation is proportional to sin θ cos θ. Such standing wave represents

1456-465: A subset of space, a three-dimensional region (or 3D domain ), a solid figure . Technically, a tuple of n numbers can be understood as the Cartesian coordinates of a location in a n -dimensional Euclidean space. The set of these n -tuples is commonly denoted R n , {\displaystyle \mathbb {R} ^{n},} and can be identified to the pair formed by

1547-442: A subtle way. By definition, there exists a basis B = { e 1 , e 2 , e 3 } {\displaystyle {\mathcal {B}}=\{e_{1},e_{2},e_{3}\}} for V {\displaystyle V} . This corresponds to an isomorphism between V {\displaystyle V} and R 3 {\displaystyle \mathbb {R} ^{3}} :

1638-445: A unique plane, so skew lines are lines that do not meet and do not lie in a common plane. Two distinct planes can either meet in a common line or are parallel (i.e., do not meet). Three distinct planes, no pair of which are parallel, can either meet in a common line, meet in a unique common point, or have no point in common. In the last case, the three lines of intersection of each pair of planes are mutually parallel. A line can lie in

1729-421: A vector A is denoted by || A || . The dot product of a vector A = [ A 1 , A 2 , A 3 ] with itself is which gives the formula for the Euclidean length of the vector. Without reference to the components of the vectors, the dot product of two non-zero Euclidean vectors A and B is given by where θ is the angle between A and B . The cross product or vector product

1820-627: Is a binary operation on two vectors in three-dimensional space and is denoted by the symbol ×. The cross product A × B of the vectors A and B is a vector that is perpendicular to both and therefore normal to the plane containing them. It has many applications in mathematics, physics , and engineering . In function language, the cross product is a function × : R 3 × R 3 → R 3 {\displaystyle \times :\mathbb {R} ^{3}\times \mathbb {R} ^{3}\rightarrow \mathbb {R} ^{3}} . The components of

1911-758: Is a right circular cone with vertex (apex) the point of intersection. However, if the generatrix and axis are parallel, then the surface of revolution is a circular cylinder . In analogy with the conic sections , the set of points whose Cartesian coordinates satisfy the general equation of the second degree, namely, A x 2 + B y 2 + C z 2 + F x y + G y z + H x z + J x + K y + L z + M = 0 , {\displaystyle Ax^{2}+By^{2}+Cz^{2}+Fxy+Gyz+Hxz+Jx+Ky+Lz+M=0,} where A , B , C , F , G , H , J , K , L and M are real numbers and not all of A , B , C , F , G and H are zero,

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2002-466: Is called a quadric surface . There are six types of non-degenerate quadric surfaces: The degenerate quadric surfaces are the empty set, a single point, a single line, a single plane, a pair of planes or a quadratic cylinder (a surface consisting of a non-degenerate conic section in a plane π and all the lines of R through that conic that are normal to π ). Elliptic cones are sometimes considered to be degenerate quadric surfaces as well. Both

2093-421: Is defined as one arc-minute of the Equator, so it has different values depending on which radius is assumed. For example, by WSG-84, the distance is 1,855.3248 metres (6,087.024 ft), while by IAU-2000, it is 1,855.3257 metres (6,087.027 ft). This is a difference of less than one millimetre (0.039 in) over the total distance (approximately 1.86 kilometres or 1.16 miles). Earth is commonly modeled as

2184-465: Is difficult to estimate the effect of the ocean, which may slightly increase the value of maximum ground pressure necessary to generate the annual wobble. This ocean effect has been estimated to be of the order of 5–10%. It is improbable that the internal parameters of the Earth responsible for the Chandler wobble would be time dependent on such short time intervals. Moreover, the observed stability of

2275-411: Is directed toward a fixed point in space. If the earth were perfectly symmetrical and rigid, M would remain aligned with its axis of symmetry, which would also be its axis of rotation . In the case of the Earth, it is almost identical with its axis of rotation, with the discrepancy due to shifts of mass on the planet's surface. The vector of the figure axis F of the system (or maximum principal axis,

2366-406: Is found in linear algebra , where the idea of independence is crucial. Space has three dimensions because the length of a box is independent of its width or breadth. In the technical language of linear algebra, space is three-dimensional because every point in space can be described by a linear combination of three independent vectors . A vector can be pictured as an arrow. The vector's magnitude

2457-425: Is its length, and its direction is the direction the arrow points. A vector in R 3 {\displaystyle \mathbb {R} ^{3}} can be represented by an ordered triple of real numbers. These numbers are called the components of the vector. The dot product of two vectors A = [ A 1 , A 2 , A 3 ] and B = [ B 1 , B 2 , B 3 ] is defined as: The magnitude of

2548-404: Is now general agreement that the annual component of polar motion is a forced motion excited predominantly by atmospheric dynamics. There exist two external forces to excite polar motion: atmospheric winds, and pressure loading. The main component is pressure forcing, which is a standing wave of the form: (3)   p = p 0 Θ −3 (θ) cos[2πν A (t − t 0 )] cos(λ − λ 0 ) with p 0

2639-669: Is the Kronecker delta . Written out in full, the standard basis is E 1 = ( 1 0 0 ) , E 2 = ( 0 1 0 ) , E 3 = ( 0 0 1 ) . {\displaystyle E_{1}={\begin{pmatrix}1\\0\\0\end{pmatrix}},E_{2}={\begin{pmatrix}0\\1\\0\end{pmatrix}},E_{3}={\begin{pmatrix}0\\0\\1\end{pmatrix}}.} Therefore R 3 {\displaystyle \mathbb {R} ^{3}} can be viewed as

2730-534: Is the Levi-Civita symbol . It has the property that A × B = − B × A {\displaystyle \mathbf {A} \times \mathbf {B} =-\mathbf {B} \times \mathbf {A} } . Its magnitude is related to the angle θ {\displaystyle \theta } between A {\displaystyle \mathbf {A} } and B {\displaystyle \mathbf {B} } by

2821-400: Is the normalized Euler frequency (in units of reciprocal years), C = 8.04 × 10  kg m is the polar moment of inertia of the Earth, A is its mean equatorial moment of inertia, and C − A = 2.61 × 10  kg m . The observed angle between the figure axis of the Earth F and its angular momentum M is a few hundred milliarcseconds (mas). This rotation can be interpreted as

Equator - Misplaced Pages Continue

2912-429: Is tilted towards or away from the sun, resulting in either summer or winter in both hemispheres. This also results in a corresponding movement of the equator away from the subsolar point, which is then situated over or near the relevant tropic circle . Nevertheless, temperatures are high year-round due to the Earth's axial tilt of 23.5° not being enough to create a low minimum midday declination to sufficiently weaken

3003-418: Is to model physical space as a three-dimensional affine space E ( 3 ) {\displaystyle E(3)} over the real numbers. This is unique up to affine isomorphism. It is sometimes referred to as three-dimensional Euclidean space. Just as the vector space description came from 'forgetting the preferred basis' of R 3 {\displaystyle \mathbb {R} ^{3}} ,

3094-512: The Apollo Moon landings . The precise location of the Equator is not truly fixed; the true equatorial plane is perpendicular to the Earth's rotation axis , which drifts about 9 metres (30 ft) during a year. Geological samples show that the Equator significantly changed positions between 48 and 12 million years ago, as sediment deposited by ocean thermal currents at the Equator shifted. The deposits by thermal currents are determined by

3185-518: The celestial equator . In the cycle of Earth's seasons , the equatorial plane runs through the Sun twice a year : on the equinoxes in March and September . To a person on Earth, the Sun appears to travel along the equator (or along the celestial equator) at these times. Locations on the equator experience the shortest sunrises and sunsets because the Sun's daily path is nearly perpendicular to

3276-537: The geographical mile . The sea-level surface of Earth (the geoid ) is irregular, so the actual length of the Equator is not so easy to determine. Aviation Week and Space Technology on 9 October 1961 reported that measurements using the Transit IV-A satellite had shown the equatorial diameter from longitude 11° West to 169° East to be 1,000 feet (305 m) greater than its diameter ninety degrees away. Download coordinates as: The Equator passes through

3367-424: The horizon for most of the year. The length of daylight (sunrise to sunset) is almost constant throughout the year; it is about 14 minutes longer than nighttime due to atmospheric refraction and the fact that sunrise begins (or sunset ends) as the upper limb, not the center, of the Sun's disk contacts the horizon. Earth bulges slightly at the Equator; its average diameter is 12,742 km (7,918 mi), but

3458-513: The 19th century, developments of the geometry of three-dimensional space came with William Rowan Hamilton 's development of the quaternions . In fact, it was Hamilton who coined the terms scalar and vector , and they were first defined within his geometric framework for quaternions . Three dimensional space could then be described by quaternions q = a + u i + v j + w k {\displaystyle q=a+ui+vj+wk} which had vanishing scalar component, that is,

3549-399: The Chandler wobble, however, varies by a factor of three, and its frequency by up to 7%. Its maximum amplitude during the last 100 years never exceeded 230 mas. The Chandler wobble is usually considered a resonance phenomenon, a free nutation that is excited by a source and then dies away with a time constant τ D of the order of 100 years. It is a measure of the elastic reaction of

3640-480: The Earth is the motion of the Earth's rotational axis relative to its crust . This is measured with respect to a reference frame in which the solid Earth is fixed (a so-called Earth-centered, Earth-fixed or ECEF reference frame). This variation is a few meters on the surface of the Earth. Polar motion is defined relative to a conventionally defined reference axis, the CIO ( Conventional International Origin ), being

3731-471: The Earth. It is also the explanation for the deviation of the Chandler period from the Euler period. However, rather than dying away, the Chandler wobble, continuously observed for more than 100 years, varies in amplitude and shows a sometimes rapid frequency shift within a few years. This reciprocal behavior between amplitude and frequency has been described by the empirical formula: (2)   m = 3.7/(ν − 0.816)   (for 0.83 < ν < 0.9) with m

Equator - Misplaced Pages Continue

3822-444: The Equator (on Earth), noontime sunlight appears almost directly overhead (no more than about 23° from the zenith ) every day, year-round. Consequently, the Equator has a rather stable daytime temperature throughout the year. On the equinoxes (approximately March 20 and September 23) the subsolar point crosses Earth's equator at a shallow angle, sunlight shines perpendicular to Earth's axis of rotation, and all latitudes have nearly

3913-565: The Equator. However, its island of Annobón is 155 km (96 mi) south of the Equator, and the rest of the country lies to the north. France , Norway ( Bouvet Island ), and the United Kingdom are the other three Northern Hemisphere -based countries which have territories in the Southern Hemisphere . Seasons result from the tilt of Earth's axis away from a line perpendicular to the plane of its revolution around

4004-487: The Euler equation with pressure loading as in eq.(3), however now with a slowly changing frequency ν, and replaces the frequency ν by a complex frequency ν + iν D , where ν D simulates dissipation due to the elastic reaction of the Earth's interior. As in Figure 2, the result is the sum of a prograde and a retrograde circular polarized wave. For frequencies ν < 0.9 the retrograde wave can be neglected, and there remains

4095-518: The IUGG at its Canberra, Australia meeting of 1979 has an equatorial radius of 6,378.137 km (3,963.191 mi). The WGS 84 (World Geodetic System 1984) which is a standard for use in cartography, geodesy, and satellite navigation including GPS , also has an equatorial radius of 6,378.137 km (3,963.191 mi). For both GRS 80 and WGS 84, this results in a length for the Equator of 40,075.0167 km (24,901.4609 mi). The geographical mile

4186-689: The International Astronomical Union (IAU) use an equatorial radius of 6,378.1366 km (3,963.1903 mi) (codified as the IAU 2009 value). This equatorial radius is also in the 2003 and 2010 IERS Conventions. It is also the equatorial radius used for the IERS 2003 ellipsoid. If it were really circular, the length of the equator would then be exactly 2π times the radius, namely 40,075.0142 km (24,901.4594 mi). The GRS 80 (Geodetic Reference System 1980) as approved and adopted by

4277-513: The Sun's rays even during the solstices. High year-round temperatures extend to about 25° north or south of the equator, although the moderate seasonal temperature difference is defined by the opposing solstices (as it is at higher latitudes) near the poleward limits of this range. Near the equator, there is little temperature change throughout the year, though there may be dramatic differences in rainfall and humidity. The terms summer, autumn, winter and spring do not generally apply. Lowlands around

4368-545: The Sun. Throughout the year, the Northern and Southern hemispheres are alternately turned either toward or away from the Sun, depending on Earth's position in its orbit. The hemisphere turned toward the Sun receives more sunlight and is in summer, while the other hemisphere receives less sun and is in winter (see solstice ). At the equinoxes , Earth's axis is perpendicular to the Sun rather than tilted toward or away, meaning that day and night are both about 12 hours long across

4459-399: The above-mentioned systems. Two distinct points always determine a (straight) line . Three distinct points are either collinear or determine a unique plane . On the other hand, four distinct points can either be collinear, coplanar , or determine the entire space. Two distinct lines can either intersect, be parallel or be skew . Two parallel lines, or two intersecting lines , lie in

4550-495: The abstract vector space, together with the additional structure of a choice of basis. Conversely, V {\displaystyle V} can be obtained by starting with R 3 {\displaystyle \mathbb {R} ^{3}} and 'forgetting' the Cartesian product structure, or equivalently the standard choice of basis. As opposed to a general vector space V {\displaystyle V} ,

4641-399: The affine space description comes from 'forgetting the origin' of the vector space. Euclidean spaces are sometimes called Euclidean affine spaces for distinguishing them from Euclidean vector spaces. This is physically appealing as it makes the translation invariance of physical space manifest. A preferred origin breaks the translational invariance. Polar motion Polar motion of

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4732-477: The afternoon and 23 °C (73 °F) around sunrise. Rainfall is very high away from cold ocean current upwelling zones, from 2,500 to 3,500 mm (100 to 140 in) per year. There are about 200 rainy days per year and average annual sunshine hours are around 2,000. Despite high year-round sea level temperatures, some higher altitudes such as the Andes and Mount Kilimanjaro have glaciers. The highest point on

4823-529: The annual component argues against any hypothesis of a variable Chandler resonance frequency. One possible explanation for the observed frequency-amplitude behavior would be a forced, but slowly changing quasi-periodic excitation by interannually varying atmospheric dynamics. Indeed, a quasi-14 month period has been found in coupled ocean-atmosphere general circulation models, and a regional 14-month signal in regional sea surface temperature has been observed. To describe such behavior theoretically, one starts with

4914-486: The apparent motion of the rotation axis about the geometric axis of the Earth. This is the so-called polar motion. Observations show that the figure axis exhibits an annual wobble forced by surface mass displacement via atmospheric and/or ocean dynamics, while the free nutation is much larger than the Euler period and of the order of 435 to 445 sidereal days. This observed free nutation is called Chandler wobble . There exist, in addition, polar motions with smaller periods of

5005-775: The axioms of a Lie algebra, instead of associativity the cross product satisfies the Jacobi identity . For any three vectors A , B {\displaystyle \mathbf {A} ,\mathbf {B} } and C {\displaystyle \mathbf {C} } A × ( B × C ) + B × ( C × A ) + C × ( A × B ) = 0 {\displaystyle \mathbf {A} \times (\mathbf {B} \times \mathbf {C} )+\mathbf {B} \times (\mathbf {C} \times \mathbf {A} )+\mathbf {C} \times (\mathbf {A} \times \mathbf {B} )=0} One can in n dimensions take

5096-614: The axis of Earth, which determines solar coverage of Earth's surface . Changes in Earth's axis can also be observed in the geographical layout of volcanic island chains, which are created by shifting hot spots under Earth's crust as the axis and crust move. This is consistent with the Indian tectonic plate colliding with the Eurasian tectonic plate , which is causing the Himalayan uplift. The International Association of Geodesy (IAG) and

5187-514: The axis which yields the largest value of moment of inertia) wobbles around M . This motion is called Euler 's free nutation . For a rigid Earth which is an oblate spheroid to a good approximation, the figure axis F would be its geometric axis defined by the geographic north and south pole, and identical with the axis of its polar moment of inertia. The Euler period of free nutation is (1)   τ E = 1/ν E = A/(C − A) sidereal days ≈ 307 sidereal days ≈ 0.84 sidereal years ν E = 1.19

5278-509: The circular propagating prograde wave where the vector of polar motion moves on a circle in anti-clockwise direction. The magnitude of m becomes: (6)   m = 14.5 p 0 ν C /[(ν − ν C ) + ν D ]   (for ν < 0.9) It is a resonance curve which can be approximated at its flanks by (7)   m ≈ 14.5 p 0 ν C /|ν − ν C |   (for (ν − ν C ) ≫ ν D ) The maximum amplitude of m at ν = ν C becomes (8)   m max = 14.5 p 0 ν C /ν D In

5369-630: The construction for the isomorphism is found here . However, there is no 'preferred' or 'canonical basis' for V {\displaystyle V} . On the other hand, there is a preferred basis for R 3 {\displaystyle \mathbb {R} ^{3}} , which is due to its description as a Cartesian product of copies of R {\displaystyle \mathbb {R} } , that is, R 3 = R × R × R {\displaystyle \mathbb {R} ^{3}=\mathbb {R} \times \mathbb {R} \times \mathbb {R} } . This allows

5460-491: The construction of the five regular Platonic solids in a sphere. In the 17th century, three-dimensional space was described with Cartesian coordinates , with the advent of analytic geometry developed by René Descartes in his work La Géométrie and Pierre de Fermat in the manuscript Ad locos planos et solidos isagoge (Introduction to Plane and Solid Loci), which was unpublished during Fermat's lifetime. However, only Fermat's work dealt with three-dimensional space. In

5551-880: The cross product are A × B = [ A 2 B 3 − B 2 A 3 , A 3 B 1 − B 3 A 1 , A 1 B 2 − B 1 A 2 ] {\displaystyle \mathbf {A} \times \mathbf {B} =[A_{2}B_{3}-B_{2}A_{3},A_{3}B_{1}-B_{3}A_{1},A_{1}B_{2}-B_{1}A_{2}]} , and can also be written in components, using Einstein summation convention as ( A × B ) i = ε i j k A j B k {\displaystyle (\mathbf {A} \times \mathbf {B} )_{i}=\varepsilon _{ijk}A_{j}B_{k}} where ε i j k {\displaystyle \varepsilon _{ijk}}

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5642-516: The definition of canonical projections, π i : R 3 → R {\displaystyle \pi _{i}:\mathbb {R} ^{3}\rightarrow \mathbb {R} } , where 1 ≤ i ≤ 3 {\displaystyle 1\leq i\leq 3} . For example, π 1 ( x 1 , x 2 , x 3 ) = x {\displaystyle \pi _{1}(x_{1},x_{2},x_{3})=x} . This then allows

5733-493: The definition of the standard basis B Standard = { E 1 , E 2 , E 3 } {\displaystyle {\mathcal {B}}_{\text{Standard}}=\{E_{1},E_{2},E_{3}\}} defined by π i ( E j ) = δ i j {\displaystyle \pi _{i}(E_{j})=\delta _{ij}} where δ i j {\displaystyle \delta _{ij}}

5824-648: The diameter at the equator is about 43 km (27 mi) greater than at the poles. Sites near the Equator, such as the Guiana Space Centre in Kourou , French Guiana , are good locations for spaceports as they have the fastest rotational speed of any latitude, 460 m (1,509 ft)/sec. The added velocity reduces the fuel needed to launch spacecraft eastward (in the direction of Earth's rotation) to orbit, while simultaneously avoiding costly maneuvers to flatten inclination during missions such as

5915-457: The ellipse is (4)   m 1 /m 2 = ν C where ν C is the Chandler resonance frequency. The result is in good agreement with the observations. From Figure 2 together with eq.(4), one obtains ν C = 0.83 , corresponding to a Chandler resonance period of (5)   τ C = 441 sidereal days = 1.20 sidereal years p 0 = 2.2 hPa , λ 0 = −170° the latitude of maximum pressure, and t 0 = −0.07 years = −25 days . It

6006-632: The equator generally have a tropical rainforest climate , also known as an equatorial climate, though cold ocean currents cause some regions to have tropical monsoon climates with a dry season in the middle of the year, and the Somali Current generated by the Asian monsoon due to continental heating via the high Tibetan Plateau causes Greater Somalia to have an arid climate despite its equatorial location. Average annual temperatures in equatorial lowlands are around 31 °C (88 °F) during

6097-473: The equator is at the elevation of 4,690 metres (15,387 ft), at 0°0′0″N 77°59′31″W  /  0.00000°N 77.99194°W  / 0.00000; -77.99194  ( highest point on the equator ) , found on the southern slopes of Volcán Cayambe [summit 5,790 metres (18,996 ft)] in Ecuador . This is slightly above the snow line and is the only place on the equator where snow lies on

6188-435: The global scale mass transport between the oceans and the continents. Major earthquakes cause abrupt polar motion by altering the volume distribution of the Earth's solid mass. These shifts are quite small in magnitude relative to the long-term core/mantle and isostatic rebound components of polar motion. In the absence of external torques, the vector of the angular momentum M of a rotating system remains constant and

6279-488: The gravitational attraction of the Moon and Sun. They are also called nutations , except for the slowest, which is the precession of the equinoxes . Polar motion is observed routinely by space geodesy methods such as very-long-baseline interferometry , lunar laser ranging and satellite laser ranging . The annual component is rather constant in amplitude, and its frequency varies by not more than 1 to 2%. The amplitude of

6370-744: The ground. At the equator, the snow line is around 1,000 metres (3,300 ft) lower than on Mount Everest and as much as 2,000 metres (6,600 ft) lower than the highest snow line in the world, near the Tropic of Capricorn on Llullaillaco . There is a widespread maritime tradition of holding ceremonies to mark a sailor's first crossing of the equator. In the past, these ceremonies have been notorious for their brutality, especially in naval practice. Milder line-crossing ceremonies, typically featuring King Neptune , are also held for passengers' entertainment on some civilian ocean liners and cruise ships. Three-dimensional space In geometry ,

6461-421: The hyperboloid of one sheet and the hyperbolic paraboloid are ruled surfaces , meaning that they can be made up from a family of straight lines. In fact, each has two families of generating lines, the members of each family are disjoint and each member one family intersects, with just one exception, every member of the other family. Each family is called a regulus . Another way of viewing three-dimensional space

6552-460: The identity ‖ A × B ‖ = ‖ A ‖ ⋅ ‖ B ‖ ⋅ | sin ⁡ θ | . {\displaystyle \left\|\mathbf {A} \times \mathbf {B} \right\|=\left\|\mathbf {A} \right\|\cdot \left\|\mathbf {B} \right\|\cdot \left|\sin \theta \right|.} The space and product form an algebra over

6643-625: The land of eleven sovereign states . Indonesia is the country straddling the greatest length of the equatorial line across both land and sea. Starting at the Prime Meridian and heading eastwards, the Equator passes through: The Equator also passes through the territorial seas of three countries: Maldives (south of Gaafu Dhaalu Atoll ), Kiribati (south of Buariki Island ), and the United States (south of Baker Island ). Despite its name, no part of Equatorial Guinea lies on

6734-428: The most compelling and useful way to model the world as it is experienced, it is only one example of a 3-manifold. In this classical example, when the three values refer to measurements in different directions ( coordinates ), any three directions can be chosen, provided that these directions do not lie in the same plane . Furthermore, if these directions are pairwise perpendicular , the three values are often labeled by

6825-421: The observed amplitude (in units of mas), and ν the frequency (in units of reciprocal sidereal years) of the Chandler wobble. In order to generate the Chandler wobble, recurring excitation is necessary. Seismic activity, groundwater movement, snow load, or atmospheric interannual dynamics have been suggested as such recurring forces, e.g. Atmospheric excitation seems to be the most likely candidate. Others propose

6916-413: The order of decades. Finally, a secular polar drift of about 0.10   m per year in the direction of 80° west has been observed which is due to mass redistribution within the Earth's interior by continental drift, and/or slow motions within mantle and core which gives rise to changes of the moment of inertia. The annual variation was discovered by Karl Friedrich Küstner in 1885 by exact measurements of

7007-534: The other four are the two polar circles (the Arctic Circle and the Antarctic Circle ) and the two tropical circles (the Tropic of Cancer and the Tropic of Capricorn ). The equator is the only line of latitude which is also a great circle —meaning, one whose plane passes through the center of the globe. The plane of Earth's equator, when projected outwards to the celestial sphere , defines

7098-414: The pole's average location over the year 1900. It consists of three major components: a free oscillation called Chandler wobble with a period of about 435 days, an annual oscillation, and an irregular drift in the direction of the 80th meridian west, which has lately been less extremely west. The slow drift, about 20 m since 1900, is partly due to motions in the Earth's core and mantle, and partly to

7189-555: The position of any point in three-dimensional space is given by an ordered triple of real numbers , each number giving the distance of that point from the origin measured along the given axis, which is equal to the distance of that point from the plane determined by the other two axes. Other popular methods of describing the location of a point in three-dimensional space include cylindrical coordinates and spherical coordinates , though there are an infinite number of possible methods. For more, see Euclidean space . Below are images of

7280-472: The product of n − 1 vectors to produce a vector perpendicular to all of them. But if the product is limited to non-trivial binary products with vector results, it exists only in three and seven dimensions . It can be useful to describe three-dimensional space as a three-dimensional vector space V {\displaystyle V} over the real numbers. This differs from R 3 {\displaystyle \mathbb {R} ^{3}} in

7371-435: The range of validity of the empirical formula eq.(2), there is reasonable agreement with eq.(7). From eqs.(2) and (7), one finds the number p 0 ∼ 0.2 hPa . The observed maximum value of m yields m max ≥ 230 mas . Together with eq.(8), one obtains (9)   τ D = 1/ν D ≥ 100 years The number of the maximum pressure amplitude is tiny, indeed. It clearly indicates the resonance amplification of Chandler wobble in

7462-459: The redistribution of water mass as the Greenland ice sheet melts, and to isostatic rebound , i.e. the slow rise of land that was formerly burdened with ice sheets or glaciers. The drift is roughly along the 80th meridian west . Since about 2000, the pole has found a less extreme drift, which is roughly along the central meridian. This less dramatically westward drift of motion is attributed to

7553-536: The seasonally varying spatial difference of the Earth's surface pressure. In northern winter, there is a pressure high over the North Atlantic Ocean and a pressure low over Siberia with temperature differences of the order of 50°, and vice versa in summer, thus an unbalanced mass distribution on the surface of the Earth. The position of the vector m of the annual component describes an ellipse (Figure 2). The calculated ratio between major and minor axis of

7644-530: The space R 3 {\displaystyle \mathbb {R} ^{3}} is sometimes referred to as a coordinate space. Physically, it is conceptually desirable to use the abstract formalism in order to assume as little structure as possible if it is not given by the parameters of a particular problem. For example, in a problem with rotational symmetry, working with the more concrete description of three-dimensional space R 3 {\displaystyle \mathbb {R} ^{3}} assumes

7735-418: The surface area of the sphere is A = 4 π r 2 . {\displaystyle A=4\pi r^{2}.} Another type of sphere arises from a 4-ball, whose three-dimensional surface is the 3-sphere : points equidistant to the origin of the euclidean space R . If a point has coordinates, P ( x , y , z , w ) , then x + y + z + w = 1 characterizes those points on

7826-404: The terms width /breadth , height /depth , and length . Books XI to XIII of Euclid's Elements dealt with three-dimensional geometry. Book XI develops notions of orthogonality and parallelism of lines and planes, and defines solids including parallelpipeds, pyramids, prisms, spheres, octahedra, icosahedra and dodecahedra. Book XII develops notions of similarity of solids. Book XIII describes

7917-446: The unit 3-sphere centered at the origin. This 3-sphere is an example of a 3-manifold: a space which is 'looks locally' like 3-D space. In precise topological terms, each point of the 3-sphere has a neighborhood which is homeomorphic to an open subset of 3-D space. In three dimensions, there are nine regular polytopes: the five convex Platonic solids and the four nonconvex Kepler-Poinsot polyhedra . A surface generated by revolving

8008-455: The variation of the latitude of stars, while S.C. Chandler found the free nutation in 1891. Both periods superpose, giving rise to a beat frequency with a period of about 5 to 8 years (see Figure 1). This polar motion should not be confused with the changing direction of the Earth's rotation axis relative to the stars with different periods, caused mostly by the torques on the Geoid due to

8099-445: The whole of Earth. Near the equator, this means the variation in the strength of solar radiation is different relative to the time of year than it is at higher latitudes: maximum solar radiation is received during the equinoxes, when a place at the equator is under the subsolar point at high noon, and the intermediate seasons of spring and autumn occur at higher latitudes; and the minimum occurs during both solstices, when either pole

8190-491: The work of Hermann Grassmann and Giuseppe Peano , the latter of whom first gave the modern definition of vector spaces as an algebraic structure. In mathematics, analytic geometry (also called Cartesian geometry) describes every point in three-dimensional space by means of three coordinates. Three coordinate axes are given, each perpendicular to the other two at the origin , the point at which they cross. They are usually labeled x , y , and z . Relative to these axes,

8281-473: Was not until Josiah Willard Gibbs that these two products were identified in their own right, and the modern notation for the dot and cross product were introduced in his classroom teaching notes, found also in the 1901 textbook Vector Analysis written by Edwin Bidwell Wilson based on Gibbs' lectures. Also during the 19th century came developments in the abstract formalism of vector spaces, with

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