1.
Sphere
–
A sphere is a perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball. This distance r is the radius of the ball, and the point is the center of the mathematical ball. The longest straight line through the ball, connecting two points of the sphere, passes through the center and its length is twice the radius. While outside mathematics the terms sphere and ball are used interchangeably. The ball and the share the same radius, diameter. The surface area of a sphere is, A =4 π r 2, at any given radius r, the incremental volume equals the product of the surface area at radius r and the thickness of a shell, δ V ≈ A ⋅ δ r. The total volume is the summation of all volumes, V ≈ ∑ A ⋅ δ r. In the limit as δr approaches zero this equation becomes, V = ∫0 r A d r ′, substitute V,43 π r 3 = ∫0 r A d r ′. Differentiating both sides of equation with respect to r yields A as a function of r,4 π r 2 = A. Which is generally abbreviated as, A =4 π r 2, alternatively, the area element on the sphere is given in spherical coordinates by dA = r2 sin θ dθ dφ. In Cartesian coordinates, the element is d S = r r 2 − ∑ i ≠ k x i 2 ∏ i ≠ k d x i, ∀ k. For more generality, see area element, the total area can thus be obtained by integration, A = ∫02 π ∫0 π r 2 sin θ d θ d φ =4 π r 2. In three dimensions, the volume inside a sphere is derived to be V =43 π r 3 where r is the radius of the sphere, archimedes first derived this formula, which shows that the volume inside a sphere is 2/3 that of a circumscribed cylinder. In modern mathematics, this formula can be derived using integral calculus, at any given x, the incremental volume equals the product of the cross-sectional area of the disk at x and its thickness, δ V ≈ π y 2 ⋅ δ x. The total volume is the summation of all volumes, V ≈ ∑ π y 2 ⋅ δ x. In the limit as δx approaches zero this equation becomes, V = ∫ − r r π y 2 d x. At any given x, a right-angled triangle connects x, y and r to the origin, hence, applying the Pythagorean theorem yields, thus, substituting y with a function of x gives, V = ∫ − r r π d x. Which can now be evaluated as follows, V = π − r r = π − π =43 π r 3
Sphere
–
Circumscribed cylinder to a sphere
Sphere
–
A two-dimensional
perspective projection of a sphere
Sphere
Sphere
–
Deck of playing cards illustrating engineering instruments, England, 1702.
King of spades: Spheres
2.
Ellipsoid
–
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. An ellipsoid is a surface, that is a surface that may be defined as the zero set of a polynomial of degree two in three variables. Among quadric surfaces, an ellipsoid is characterized by any of the two following properties, every planar cross section is either an ellipse, or is empty, or is reduced to a single point. It is bounded, which means that it may be enclosed in a large sphere. An ellipsoid has three 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, if the three axes have different lengths, the ellipsoid is said to be tri-axial or scalene, and the axes are uniquely defined. If two of the axes have the length, then the ellipsoid is an ellipsoid of revolution. In this case, the ellipsoid is invariant under a rotation around the third axis, if the third axis is shorter, the ellipsoid is an oblate spheroid, if it is longer, it is prolate spheroid. If the three axes have the length, the ellipsoid is a sphere. The points, and lie on the surface, the line segments from the origin to these points are called the semi-principal axes of the ellipsoid, because a, b, c are half the length of the principal axes. They correspond to the axis and semi-minor axis of an ellipse. If a = b > c, one has an oblate spheroid, if a = b < c, one has a prolate spheroid, if a = b = c, one has a sphere. It is easy to check, The intersection of a plane, remark, The contour of an ellipsoid, seen from a point outside the ellipsoid or from infinity, is in any case a plane section, hence an ellipse. 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 and these parameters may be interpreted as spherical coordinates. More precisely, π /2 − θ is the polar angle, and φ is the azimuth angle of the point of the ellipsoid. More generally, an arbitrarily oriented ellipsoid, centered at v, is defined by the x to the equation T A =1. The eigenvectors of A define the axes of the ellipsoid and the eigenvalues of A are the reciprocals of the squares of the semi-axes
Ellipsoid
–
Tri-axial ellipsoid with distinct semi-axis lengths
Ellipsoid
–
Artist's conception of
Haumea, a Jacobi-ellipsoid
dwarf planet, with its two moons
3.
Equator
–
The Equator usually refers to an imaginary line on the Earths surface equidistant from the North Pole and South Pole, dividing the Earth into the Northern Hemisphere and Southern Hemisphere. The Equator is about 40,075 kilometres long, some 78. 7% lies across water and 21. 3% over land, other planets and astronomical bodies have equators similarly defined. Generally, an equator is the intersection of the surface of a sphere with the plane that is perpendicular to the spheres axis of rotation. The latitude of the Earths equator is by definition 0° of arc, the equator is the only line of latitude which is also a great circle — that is, one whose plane passes through the center of the globe. The plane of Earths equator when projected outwards to the celestial sphere defines the celestial equator, in the cycle of Earths seasons, the plane of the equator passes through the Sun twice per year, at the March and September equinoxes. To an observer on the Earth, the Sun appears to travel North or South over the equator at these times, light rays from the center of the Sun are perpendicular to the surface of the Earth at the point of solar noon on the Equator. Locations on the Equator experience the quickest sunrises and sunsets because the sun moves nearly perpendicular to the horizon for most of the year. The Earth bulges slightly at the Equator, the diameter of the Earth is 12,750 kilometres. Because the Earth spins to the east, spacecraft must also launch to the east to take advantage of this Earth-boost of speed, seasons result from the yearly revolution of the Earth around the Sun and the tilt of the Earths axis relative to the plane of revolution. During the year the northern and southern hemispheres are inclined toward or away from the sun according to Earths position in its orbit, the hemisphere inclined toward the sun receives more sunlight and is in summer, while the other hemisphere receives less sun and is in winter. At the equinoxes, the Earths axis is not tilted toward the sun, instead it is perpendicular to the sun meaning that the day is about 12 hours long, as is the night, across the whole of the Earth. Near the Equator there is distinction between summer, winter, autumn, or spring. The temperatures are usually high year-round—with the exception of high mountains in South America, the temperature at the Equator can plummet during rainstorms. In many tropical regions people identify two seasons, the wet season and the dry season, but many places close to the Equator are on the oceans or rainy throughout the year, the seasons can vary depending on elevation and proximity to an ocean. The Equator lies mostly on the three largest oceans, the Pacific Ocean, the Atlantic Ocean, and the Indian Ocean. The highest point on the Equator is at the elevation of 4,690 metres, at 0°0′0″N 77°59′31″W and this is slightly above the snow line, and is the only place on the Equator where snow lies on the ground. At the Equator the snow line is around 1,000 metres lower than on Mount Everest, the Equator traverses the land of 11 countries, it also passes through two island nations, though without making a landfall in either. Starting at the Prime Meridian and heading eastwards, the Equator passes through, Despite its name, however, its island of Annobón is 155 km south of the Equator, and the rest of the country lies to the north
Equator
–
Left: A monument marking the Equator near the town of
Pontianak, Indonesia Right: Road sign marking the Equator near
Nanyuki, Kenya
Equator
Equator
–
The Equator marked as it crosses
Ilhéu das Rolas, in São Tomé and Príncipe
4.
Geography
–
Geography is a field of science devoted to the study of the lands, the features, the inhabitants, and the phenomena of Earth. The first person to use the word γεωγραφία was Eratosthenes, Geography is an all-encompassing discipline that seeks an understanding of the Earth and its human and natural complexities—not merely where objects are, but how they have changed and come to be. It is often defined in terms of the two branches of geography and physical geography. Geography has been called the world discipline and the bridge between the human and the physical sciences, Geography is a systematic study of the Earth and its features. Traditionally, geography has been associated with cartography and place names, although many geographers are trained in toponymy and cartology, this is not their main preoccupation. Geographers study the space and the temporal database distribution of phenomena, processes, because space and place affect a variety of topics, such as economics, health, climate, plants and animals, geography is highly interdisciplinary. The interdisciplinary nature of the approach depends on an attentiveness to the relationship between physical and human phenomena and its spatial patterns. Names of places. are not geography. know by heart a whole gazetteer full of them would not, in itself and this is a description of the world—that is Geography. In a word Geography is a Science—a thing not of mere names but of argument and reason, of cause, just as all phenomena exist in time and thus have a history, they also exist in space and have a geography. Geography as a discipline can be split broadly into two main fields, human geography and physical geography. The former largely focuses on the environment and how humans create, view, manage. The latter examines the environment, and how organisms, climate, soil, water. The difference between these led to a third field, environmental geography, which combines physical and human geography. Physical geography focuses on geography as an Earth science and it aims to understand the physical problems and the issues of lithosphere, hydrosphere, atmosphere, pedosphere, and global flora and fauna patterns. Physical geography can be divided into broad categories, including, Human geography is a branch of geography that focuses on the study of patterns. It encompasses the human, political, cultural, social, and it requires an understanding of the traditional aspects of physical and human geography, as well as the ways that human societies conceptualize the environment. Integrated geography has emerged as a bridge between the human and the geography, as a result of the increasing specialisation of the two sub-fields. Examples of areas of research in the environmental geography include, emergency management, environmental management, sustainability, geomatics is concerned with the application of computers to the traditional spatial techniques used in cartography and topography
Geography
–
Map of the
Earth
Geography
–
Biogeography
Geography
Geography
5.
Geographic coordinate system
–
A geographic coordinate system is a coordinate system used in geography that enables every location on Earth to be specified by a set of numbers, letters or symbols. The coordinates are chosen such that one of the numbers represents a vertical position. A common choice of coordinates is latitude, longitude and elevation, to specify a location on a two-dimensional map requires a map projection. The invention of a coordinate system is generally credited to Eratosthenes of Cyrene. Ptolemy credited him with the adoption of longitude and latitude. Ptolemys 2nd-century Geography used the prime meridian but measured latitude from the equator instead. Mathematical cartography resumed in Europe following Maximus Planudes recovery of Ptolemys text a little before 1300, in 1884, the United States hosted the International Meridian Conference, attended by representatives from twenty-five nations. Twenty-two of them agreed to adopt the longitude of the Royal Observatory in Greenwich, the Dominican Republic voted against the motion, while France and Brazil abstained. France adopted Greenwich Mean Time in place of local determinations by the Paris Observatory in 1911, the latitude of a point on Earths surface is the angle between the equatorial plane and the straight line that passes through that point and through the center of the Earth. Lines joining points of the same latitude trace circles on the surface of Earth called parallels, as they are parallel to the equator, the north pole is 90° N, the south pole is 90° S. The 0° parallel of latitude is designated the equator, the plane of all geographic coordinate systems. The equator divides the globe into Northern and Southern Hemispheres, the longitude of a point on Earths surface is the angle east or west of a reference meridian to another meridian that passes through that point. All meridians are halves of great ellipses, which converge at the north and south poles, the prime meridian determines the proper Eastern and Western Hemispheres, although maps often divide these hemispheres further west in order to keep the Old World on a single side. The antipodal meridian of Greenwich is both 180°W and 180°E, the combination of these two components specifies the position of any location on the surface of Earth, without consideration of altitude or depth. The grid formed by lines of latitude and longitude is known as a graticule, the origin/zero point of this system is located in the Gulf of Guinea about 625 km south of Tema, Ghana. To completely specify a location of a feature on, in, or above Earth. Earth is not a sphere, but a shape approximating a biaxial ellipsoid. It is nearly spherical, but has an equatorial bulge making the radius at the equator about 0. 3% larger than the radius measured through the poles, the shorter axis approximately coincides with the axis of rotation
Geographic coordinate system
–
Longitude lines are perpendicular and latitude lines are parallel to the equator.
6.
North
–
North is one of the four cardinal directions or compass points. It is the opposite of south and is perpendicular to east and west, North is a noun, adjective, or adverb indicating direction or geography. The word north is related to the Old High German nord, the Latin word borealis comes from the Greek boreas north wind, north, which, according to Ovid, was personified as the son of the river-god Strymon, the father of Calais and Zetes. Septentrionalis is from septentriones, the seven plow oxen, a name of Ursa Maior, the Greek ἀρκτικός is named for the same constellation, and is the source of the English word Arctic. For example, in Lezgian, kefer can mean both disbelief and north, since to the north of the Muslim Lezgian homeland there are areas formerly inhabited by non-Muslim Caucasian, in many languages of Mesoamerica, north also means up. The direction north is associated with colder climates because most of the worlds land at high latitudes is located in the Northern Hemisphere. By convention, the top side of a map is often north, to go north using a compass for navigation, set a bearing or azimuth of 0° or 360°. North is specifically the direction that, in Western culture, is treated as the fundamental direction, on any rotating non-astronomical object, north denotes the side appearing to rotate counter-clockwise when viewed from afar along the axis of rotation. Magnetic north is of interest because it is the direction indicated as north on a properly functioning magnetic compass, the difference between it and true north is called the magnetic declination. But simple generalizations on the subject should be treated as unsound, maps intended for usage in orienteering by compass will clearly indicate the local declination for easy correction to true north. Maps may also indicate grid north, which is a term referring to the direction northwards along the grid lines of a map projection. The visible rotation of the sky around the visible celestial pole provides a vivid metaphor of that direction corresponding to up. Thus the choice of the north as corresponding to up in the hemisphere, or of south in that role in the southern, is, prior to worldwide communication, anything. On the contrary, it is of interest that Chinese and Islamic culture even considered south as the top end for maps. Up is a metaphor for north, the notion that north should always be up and east at the right was established by the Greek astronomer Ptolemy. While the choice of north over south as prime direction reflects quite arbitrary historical factors and their folk definitions are, respectively, where the sun rises and where it sets. The true folk-astronomical definitions of east and west are the directions, an angle from the prime direction. Being the default direction on the compass, north is referred to frequently in Western popular culture, some examples include, The phrase north of X is often used by Americans to mean more than X or greater than X, i. e
North
–
A 16-point
compass rose with north highlighted and at top.
7.
South
–
South is a noun, adjective, or adverb indicating direction or geography. It is one of the four directions or compass points. South is the polar opposite of north and is perpendicular to east and west, the word south comes from Old English sūþ, from earlier Proto-Germanic *sunþaz, possibly related to the same Proto-Indo-European root that the word sun derived from. By convention, the side of a map is south. To go south using a compass for navigation, set a bearing or azimuth of 180°, alternatively, in the Northern Hemisphere outside the tropics, the Sun will be roughly in the south at midday. True south is the direction towards the end of the axis about which the earth rotates. The South Pole is located in Antarctica, magnetic south is the direction towards the south magnetic pole, some distance away from the south geographic pole. Roald Amundsen, from Norway, was the first to reach the South Pole, on 14 December 1911, the Global South refers to the socially and economically less-developed southern half of the globe. 95% of the Global North has enough food and shelter, in the South, on the other hand, only 5% of the population has enough food and shelter. It lacks appropriate technology, it has no political stability, the economies are disarticulated, in the card game bridge, one of the players is known for scoring purposes as South. South partners with North and plays against East and West, in Greek religion, Notos, was the south wind and bringer of the storms of late summer and autumn. The dictionary definition of south at Wiktionary
South
–
A 16-point
compass rose with south at the bottom.
8.
Geoid
–
The geoid is the shape that the surface of the oceans would take under the influence of Earths gravity and rotation alone, in the absence of other influences such as winds and tides. This surface is extended through the continents, all points on a geoid surface have the same gravity potential energy. The geoid can be defined at any value of gravitational potential such as within the earths crust or far out in space and it does not correspond to the actual surface of Earths crust, but to a surface which can only be known through extensive gravitational measurements and calculations. It is often described as the true figure of the Earth. The surface of the geoid is higher than the reference ellipsoid wherever there is a gravity anomaly. The geoid surface is irregular, unlike the ellipsoid which is a mathematical idealized representation of the physical Earth. Although the physical Earth has excursions of +8,848 m and −429 m, If the ocean surface were isopycnic and undisturbed by tides, currents, or weather, it would closely approximate the geoid. The permanent deviation between the geoid and mean sea level is called ocean surface topography, If the continental land masses were criss-crossed by a series of tunnels or canals, the sea level in these canals would also very nearly coincide with the geoid. This means that when traveling by ship, one does not notice the undulations of the geoid, the vertical is always perpendicular to the geoid. Likewise, spirit levels will always be parallel to the geoid, a long voyage, indicate height variations, even though the ship will always be at sea level. This is because GPS satellites, orbiting about the center of gravity of the Earth, to obtain ones geoidal height, a raw GPS reading must be corrected. Conversely, height determined by spirit leveling from a tidal measurement station, as in land surveying. Modern GPS receivers have a grid implemented inside where they obtain the height over the World Geodetic System ellipsoid from the current position. Then they are able to correct the height above WGS ellipsoid to the height above WGS84 geoid, in that case when the height is not zero on a ship it is due to various other factors such as ocean tides, atmospheric pressure and local sea surface topography. The gravitational field of the earth is neither perfect n If that perfect sphere were then covered in water, instead, the water level would be higher or lower depending on the particular strength of gravity in that location. Spherical harmonics are used to approximate the shape of the geoid. The current best such set of spherical harmonic coefficients is EGM96, the geoid is a particular equipotential surface, and is somewhat involved to compute. The gradient of this also provides a model of the gravitational acceleration
Geoid
–
Three-dimensional visualization of geoid undulations, using
units of gravity.
Geoid
–
Geodesy
9.
Sea level
–
Mean sea level is an average level of the surface of one or more of Earths oceans from which heights such as elevations may be measured. A common and relatively straightforward mean sea-level standard is the midpoint between a low and mean high tide at a particular location. Sea levels can be affected by factors and are known to have varied greatly over geological time scales. The careful measurement of variations in MSL can offer insights into ongoing climate change, the term above sea level generally refers to above mean sea level. Precise determination of a sea level is a difficult problem because of the many factors that affect sea level. Sea level varies quite a lot on several scales of time and this is because the sea is in constant motion, affected by the tides, wind, atmospheric pressure, local gravitational differences, temperature, salinity and so forth. The easiest way this may be calculated is by selecting a location and calculating the mean sea level at that point, for example, a period of 19 years of hourly level observations may be averaged and used to determine the mean sea level at some measurement point. One measures the values of MSL in respect to the land, hence a change in MSL can result from a real change in sea level, or from a change in the height of the land on which the tide gauge operates. In the UK, the Ordnance Datum is the sea level measured at Newlyn in Cornwall between 1915 and 1921. Prior to 1921, the datum was MSL at the Victoria Dock, in Hong Kong, mPD is a surveying term meaning metres above Principal Datum and refers to height of 1. 230m below the average sea level. In France, the Marégraphe in Marseilles measures continuously the sea level since 1883 and it is used for a part of continental Europe and main part of Africa as official sea level. Elsewhere in Europe vertical elevation references are made to the Amsterdam Peil elevation, satellite altimeters have been making precise measurements of sea level since the launch of TOPEX/Poseidon in 1992. A joint mission of NASA and CNES, TOPEX/Poseidon was followed by Jason-1 in 2001, height above mean sea level is the elevation or altitude of an object, relative to the average sea level datum. It is also used in aviation, where some heights are recorded and reported with respect to sea level, and in the atmospheric sciences. An alternative is to base height measurements on an ellipsoid of the entire Earth, in aviation, the ellipsoid known as World Geodetic System 84 is increasingly used to define heights, however, differences up to 100 metres exist between this ellipsoid height and mean tidal height. The alternative is to use a vertical datum such as NAVD88. When referring to geographic features such as mountains on a topographic map, the elevation of a mountain denotes the highest point or summit and is typically illustrated as a small circle on a topographic map with the AMSL height shown in metres, feet or both. In the rare case that a location is below sea level, for one such case, see Amsterdam Airport Schiphol
Sea level
–
This marker indicating sea level is situated between
Jerusalem and the
Dead Sea.
Sea level
–
Sea Level sign seen on cliff (circled in red) at
Badwater Basin,
Death Valley National Park
10.
Normal (geometry)
–
In geometry, a normal is an object such as a line or vector that is perpendicular to a given object. For example, in the case, the normal line to a curve at a given point is the line perpendicular to the tangent line to the curve at the point. In the three-dimensional case a normal, or simply normal. The word normal is used as an adjective, a line normal to a plane, the normal component of a force. The concept of normality generalizes to orthogonality, the concept has been generalized to differentiable manifolds of arbitrary dimension embedded in a Euclidean space. The normal vector space or normal space of a manifold at a point P is the set of the vectors which are orthogonal to the tangent space at P, in the case of differential curves, the curvature vector is a normal vector of special interest. For a convex polygon, a surface normal can be calculated as the cross product of two edges of the polygon. For a plane given by the equation a x + b y + c z + d =0, the vector is a normal. For a hyperplane in n+1 dimensions, given by the equation r = a 0 + α1 a 1 + ⋯ + α n a n, where a0 is a point on the hyperplane and ai for i =1. N are non-parallel vectors lying on the hyperplane, a normal to the hyperplane is any vector in the space of A where A is given by A =. That is, any vector orthogonal to all in-plane vectors is by definition a surface normal. If a surface S is parameterized by a system of coordinates x, with s and t real variables. For a surface S given explicitly as a function f of the independent variables x, y, the first one is obtaining its implicit form F = z − f =0, from which the normal follows readily as the gradient ∇ F. The second way of obtaining the normal follows directly from the gradient of the form, ∇ f, by inspection, ∇ F = k ^ − ∇ f. Note that this is equal to ∇ F = k ^ − ∂ f ∂ x i ^ − ∂ f ∂ y j ^, if a surface does not have a tangent plane at a point, it does not have a normal at that point either. For example, a cone does not have a normal at its tip nor does it have a normal along the edge of its base, however, the normal to the cone is defined almost everywhere. In general, it is possible to define a normal almost everywhere for a surface that is Lipschitz continuous, a normal to a surface does not have a unique direction, the vector pointing in the opposite direction of a surface normal is also a surface normal. For an oriented surface, the normal is usually determined by the right-hand rule
Normal (geometry)
–
A polygon and two of its normal vectors
11.
Altitude
–
Altitude or height is defined based on the context in which it is used. As a general definition, altitude is a measurement, usually in the vertical or up direction. The reference datum also often varies according to the context, although the term altitude is commonly used to mean the height above sea level of a location, in geography the term elevation is often preferred for this usage. Vertical distance measurements in the direction are commonly referred to as depth. In aviation, the altitude can have several meanings, and is always qualified by explicitly adding a modifier. Parties exchanging altitude information must be clear which definition is being used, aviation altitude is measured using either mean sea level or local ground level as the reference datum. When flying at a level, the altimeter is always set to standard pressure. On the flight deck, the instrument for measuring altitude is the pressure altimeter. There are several types of altitude, Indicated altitude is the reading on the altimeter when it is set to the local barometric pressure at mean sea level. In UK aviation radiotelephony usage, the distance of a level, a point or an object considered as a point, measured from mean sea level. Absolute altitude is the height of the aircraft above the terrain over which it is flying and it can be measured using a radar altimeter. Also referred to as radar height or feet/metres above ground level, true altitude is the actual elevation above mean sea level. It is indicated altitude corrected for temperature and pressure. Height is the elevation above a reference point, commonly the terrain elevation. Pressure altitude is used to indicate flight level which is the standard for reporting in the U. S. in Class A airspace. Pressure altitude and indicated altitude are the same when the setting is 29.92 Hg or 1013.25 millibars. Density altitude is the altitude corrected for non-ISA International Standard Atmosphere atmospheric conditions, aircraft performance depends on density altitude, which is affected by barometric pressure, humidity and temperature. On a very hot day, density altitude at an airport may be so high as to preclude takeoff and these types of altitude can be explained more simply as various ways of measuring the altitude, Indicated altitude – the altitude shown on the altimeter
Altitude
–
Vertical distance comparison
12.
Reference ellipsoid
–
In geodesy, a reference ellipsoid is a mathematically defined surface that approximates the geoid, the truer figure of the Earth, or other planetary body. Current practice uses the word alone in preference to the full term oblate ellipsoid of revolution or the older term oblate spheroid. In the rare instances where a more general shape is required as a model the term used is triaxial ellipsoid. A great many ellipsoids have been used with various sizes and centres, the shape of an ellipsoid is determined by the shape parameters of that ellipse which generates the ellipsoid when it is rotated about its minor axis. The semi-major axis of the ellipse, a, is identified as the radius of the ellipsoid. For the Earth, f is around 1/300 corresponding to a difference of the major and minor semi-axes of approximately 21 km, some precise values are given in the table below and also in Figure of the Earth. A great many other parameters are used in geodesy but they can all be related to one or two of the set a, b and f, a primary use of reference ellipsoids is to serve as a basis for a coordinate system of latitude, longitude, and elevation. For this purpose it is necessary to identify a zero meridian, for other bodies a fixed surface feature is usually referenced, which for Mars is the meridian passing through the crater Airy-0. It is possible for many different coordinate systems to be defined upon the same reference ellipsoid, the longitude measures the rotational angle between the zero meridian and the measured point. By convention for the Earth, Moon, and Sun it is expressed in degrees ranging from −180° to +180° For other bodies a range of 0° to 360° is used. The latitude measures how close to the poles or equator a point is along a meridian, and is represented as an angle from −90° to +90°, the common or geodetic latitude is the angle between the equatorial plane and a line that is normal to the reference ellipsoid. Depending on the flattening, it may be different from the geocentric latitude. For non-Earth bodies the terms planetographic and planetocentric are used instead, see geodetic system for more detail. If these coordinates, i. e. N is the radius of curvature in the prime vertical, in contrast, extracting φ, λ and h from the rectangular coordinates usually requires iteration. A straightforward method is given in an OSGB publication and also in web notes, more sophisticated methods are outlined in geodetic system. Currently the most common reference used, and that used in the context of the Global Positioning System, is the one defined by WGS84. Traditional reference ellipsoids or geodetic datums are defined regionally and therefore non-geocentric, modern geodetic datums are established with the aid of GPS and will therefore be geocentric, e. g. WGS84. Reference ellipsoids are also useful for mapping of other planetary bodies including planets, their satellites, asteroids
Reference ellipsoid
–
Flattened sphere
13.
Global Positioning System
–
The Global Positioning System is a space-based radionavigation system owned by the United States government and operated by the United States Air Force. The GPS system operates independently of any telephonic or internet reception, the GPS system provides critical positioning capabilities to military, civil, and commercial users around the world. The United States government created the system, maintains it, however, the US government can selectively deny access to the system, as happened to the Indian military in 1999 during the Kargil War. The U. S. Department of Defense developed the system and it became fully operational in 1995. Roger L. Easton of the Naval Research Laboratory, Ivan A, getting of The Aerospace Corporation, and Bradford Parkinson of the Applied Physics Laboratory are credited with inventing it. Announcements from Vice President Al Gore and the White House in 1998 initiated these changes, in 2000, the U. S. Congress authorized the modernization effort, GPS III. In addition to GPS, other systems are in use or under development, mainly because of a denial of access. The Russian Global Navigation Satellite System was developed contemporaneously with GPS, GLONASS can be added to GPS devices, making more satellites available and enabling positions to be fixed more quickly and accurately, to within two meters. There are also the European Union Galileo positioning system and Chinas BeiDou Navigation Satellite System, special and general relativity predict that the clocks on the GPS satellites would be seen by the Earths observers to run 38 microseconds faster per day than the clocks on the Earth. The GPS calculated positions would quickly drift into error, accumulating to 10 kilometers per day, the relativistic time effect of the GPS clocks running faster than the clocks on earth was corrected for in the design of GPS. The Soviet Union launched the first man-made satellite, Sputnik 1, two American physicists, William Guier and George Weiffenbach, at Johns Hopkinss Applied Physics Laboratory, decided to monitor Sputniks radio transmissions. Within hours they realized that, because of the Doppler effect, the Director of the APL gave them access to their UNIVAC to do the heavy calculations required. The next spring, Frank McClure, the deputy director of the APL, asked Guier and Weiffenbach to investigate the inverse problem — pinpointing the users location and this led them and APL to develop the TRANSIT system. In 1959, ARPA also played a role in TRANSIT, the first satellite navigation system, TRANSIT, used by the United States Navy, was first successfully tested in 1960. It used a constellation of five satellites and could provide a navigational fix approximately once per hour, in 1967, the U. S. Navy developed the Timation satellite, which proved the feasibility of placing accurate clocks in space, a technology required by GPS. In the 1970s, the ground-based OMEGA navigation system, based on comparison of signal transmission from pairs of stations. Limitations of these systems drove the need for a more universal navigation solution with greater accuracy, during the Cold War arms race, the nuclear threat to the existence of the United States was the one need that did justify this cost in the view of the United States Congress. This deterrent effect is why GPS was funded and it is also the reason for the ultra secrecy at that time
Global Positioning System
–
Artist's conception of GPS Block II-F satellite in Earth orbit.
Global Positioning System
–
Geodesy
Global Positioning System
–
Civilian GPS receivers ("
GPS navigation device ") in a marine application.
Global Positioning System
–
Automotive navigation system in a taxicab.
14.
Phi (letter)
–
Phi is the 21st letter of the Greek alphabet. In Ancient Greek, it represented a voiceless bilabial plosive. In modern Greek, it represents a voiceless fricative and is correspondingly romanized as f. Its origin is uncertain but it may be that phi originated as the letter qoppa, in traditional Greek numerals, phi has a value of 500 or 500 000. The Cyrillic letter Ef descends from phi, phi is also used as a symbol for the golden ratio and on other occasions in math and science. This use is separately encoded as the Unicode glyph ϕ, the modern Greek pronunciation of the letter is sometimes encountered in English when the letter is being used in this sense. The lower-case letter φ is often used to represent the following, Magnetic flux in physics The golden ratio 1 +52 ≈1.618033988749894848204586834. in mathematics, art, Eulers totient function φ in number theory, also called Eulers phi function. The cyclotomic polynomial functions Φn of algebra, in algebra, group or ring homomorphisms In probability theory, ϕ = −½e−x2/2 is the probability density function of the normal distribution. In probability theory, φX = E is the function of a random variable X. An angle, typically the second angle mentioned, after θ, especially, The argument of a complex number. The phase of a wave in signal processing, in spherical coordinates, mathematicians usually refer to phi as the polar angle. The convention in physics is to use phi as the azimuthal angle, one of the dihedral angles in the backbones of proteins in a Ramachandran plot Internal or effective angle of friction. The work function of a surface, in solid-state physics, a shorthand representation for an aromatic functional group in organic chemistry. The ratio of free energy destabilizations of protein mutants in phi value analysis, in cartography, geodesy and navigation, latitude. In aircraft flight mechanics as the symbol for bank angle, in combustion engineering, fuel–air equivalence ratio. The ratio between the fuel air ratio to the stoichiometric fuel air ratio. The Veblen function in set theory Porosity in geology and hydrology, strength reduction factor in structural engineering, used to account for statistical variabilities in materials and construction methods. The symbol for a voiceless fricative in the International Phonetic Alphabet In economics
Phi (letter)
–
Greek alphabet
15.
Degree (angle)
–
A degree, usually denoted by °, is a measurement of a plane angle, defined so that a full rotation is 360 degrees. It is not an SI unit, as the SI unit of measure is the radian. Because a full rotation equals 2π radians, one degree is equivalent to π/180 radians, the original motivation for choosing the degree as a unit of rotations and angles is unknown. One theory states that it is related to the fact that 360 is approximately the number of days in a year. Ancient astronomers noticed that the sun, which follows through the path over the course of the year. Some ancient calendars, such as the Persian calendar, used 360 days for a year, the use of a calendar with 360 days may be related to the use of sexagesimal numbers. The earliest trigonometry, used by the Babylonian astronomers and their Greek successors, was based on chords of a circle, a chord of length equal to the radius made a natural base quantity. One sixtieth of this, using their standard sexagesimal divisions, was a degree, Aristarchus of Samos and Hipparchus seem to have been among the first Greek scientists to exploit Babylonian astronomical knowledge and techniques systematically. Timocharis, Aristarchus, Aristillus, Archimedes, and Hipparchus were the first Greeks known to divide the circle in 360 degrees of 60 arc minutes, eratosthenes used a simpler sexagesimal system dividing a circle into 60 parts. Furthermore, it is divisible by every number from 1 to 10 except 7 and this property has many useful applications, such as dividing the world into 24 time zones, each of which is nominally 15° of longitude, to correlate with the established 24-hour day convention. Finally, it may be the case more than one of these factors has come into play. For many practical purposes, a degree is a small enough angle that whole degrees provide sufficient precision. When this is not the case, as in astronomy or for geographic coordinates, degree measurements may be written using decimal degrees, with the symbol behind the decimals. Alternatively, the sexagesimal unit subdivisions can be used. One degree is divided into 60 minutes, and one minute into 60 seconds, use of degrees-minutes-seconds is also called DMS notation. These subdivisions, also called the arcminute and arcsecond, are represented by a single and double prime. For example,40. 1875° = 40° 11′ 15″, or, using quotation mark characters, additional precision can be provided using decimals for the arcseconds component. The older system of thirds, fourths, etc. which continues the sexagesimal unit subdivision, was used by al-Kashi and other ancient astronomers, but is rarely used today
Degree (angle)
–
One degree (shown in red) and eighty nine (shown in blue)
16.
Arcminute
–
A minute of arc, arcminute, arc minute, or minute arc is a unit of angular measurement equal to 1/60 of one degree. Since one degree is 1/360 of a turn, one minute of arc is 1/21600 of a turn, a second of arc, arcsecond, or arc second is 1/60 of an arcminute, 1/3600 of a degree, 1/1296000 of a turn, and π/648000 of a radian. To express even smaller angles, standard SI prefixes can be employed, the number of square arcminutes in a complete sphere is 4 π2 =466560000 π ≈148510660 square arcminutes. The standard symbol for marking the arcminute is the prime, though a single quote is used where only ASCII characters are permitted. One arcminute is thus written 1′ and it is also abbreviated as arcmin or amin or, less commonly, the prime with a circumflex over it. The standard symbol for the arcsecond is the prime, though a double quote is commonly used where only ASCII characters are permitted. One arcsecond is thus written 1″ and it is also abbreviated as arcsec or asec. In celestial navigation, seconds of arc are used in calculations. This notation has been carried over into marine GPS receivers, which normally display latitude and longitude in the format by default. An arcsecond is approximately the angle subtended by a U. S. dime coin at a distance of 4 kilometres, a milliarcsecond is about the size of a dime atop the Eiffel Tower as seen from New York City. A microarcsecond is about the size of a period at the end of a sentence in the Apollo mission manuals left on the Moon as seen from Earth, since antiquity the arcminute and arcsecond have been used in astronomy. The principal exception is Right ascension in equatorial coordinates, which is measured in units of hours, minutes. These small angles may also be written in milliarcseconds, or thousandths of an arcsecond, the unit of distance, the parsec, named from the parallax of one arcsecond, was developed for such parallax measurements. It is the distance at which the radius of the Earths orbit would subtend an angle of one arcsecond. The ESA astrometric space probe Gaia is hoped to measure star positions to 20 microarcseconds when it begins producing catalog positions sometime after 2016, there are about 1.3 trillion µas in a turn. Currently the best catalog positions of stars actually measured are in terms of milliarcseconds, apart from the Sun, the star with the largest angular diameter from Earth is R Doradus, a red supergiant with a diameter of 0.05 arcsecond. The dwarf planet Pluto has proven difficult to resolve because its angular diameter is about 0.1 arcsecond, space telescopes are not affected by the Earths atmosphere but are diffraction limited. For example, the Hubble space telescope can reach a size of stars down to about 0. 1″
Arcminute
–
Comparison of angular diameter of the Sun, Moon, planets and the International Space Station. To get a true representation of the sizes, view the image at a distance of 103 times the width of the "Moon: max." circle. For example, if this circle is 10 cm wide on your monitor, view it from 10.3 m away.
17.
Theodolite
–
A theodolite /θiːˈɒdəlaɪt/ is a precision instrument for measuring angles in the horizontal and vertical planes. Theodolites are used mainly for surveying applications, and have adapted for specialized purposes such as meteorology. A modern theodolite consists of a telescope mounted within two perpendicular axes, the horizontal or trunnion axis and the zenith axis. A theodolite measures vertical angles as angles between the zenith, forwards or plunged—typically approximately 90 and 270 degrees, when the telescope is pointed at a target object, the angle of each of these axes can be measured with great precision, typically to seconds of arc. A theodolite may be either transit or non-transit, in a transit theodolite, the telescope can be inverted in the vertical plane, whereas the rotation in the same plane is restricted to a semi-circle in a non-transit theodolite. Some types of transit theodolites do not allow the measurement of vertical angles, the builders level is sometimes mistaken for a transit theodolite, but it measures neither horizontal nor vertical angles. It uses a level to set a telescope level to define a line of sight along a horizontal plane. A theodolite is mounted on its head by means of a forced centering plate or tribrach containing four thumbscrews, or in modern theodolites. Before use, a theodolite must be placed vertically above the point to be measured using a plumb bob. The instrument is then set level using leveling footscrews and circular, both axes of a theodolite are equipped with graduated circles that can be read through magnifying lenses. The vertical circle which transits about the horizontal axis should read 90° when the axis is horizontal, or 270° when the instrument is in its second position. Half of the difference between the two positions is called the index error, the horizontal and vertical axes of a theodolite must be perpendicular, if not then a horizontal axis error exists. This can be tested by aligning the tubular spirit bubble parallel to a line between two footscrews and setting the bubble central, a horizontal axis error is present if the bubble runs off central when the tubular spirit bubble is reversed. To adjust, the operator removes 1/2 the amount the bubble has run off using the screw, then re-level, test. If not, then a collimation error exists, index error, horizontal axis error and collimation error are regularly determined by calibration and are removed by mechanical adjustment. Their existence is taken account in the choice of measurement procedure in order to eliminate their effect on the measurement results of the theodolite. The term diopter was used in old texts as a synonym for theodolite. This derives from an astronomical instrument called a dioptra
Theodolite
–
An optical theodolite, manufactured in the Soviet Union in 1958 and used for topographic surveying
Theodolite
–
Diagram of an Optical Theodolite
Theodolite
–
Sectioned theodolite showing the complexity of the optical paths
Theodolite
–
Eight-inch theodolite, c. 1898
18.
Figure of the Earth
–
The expression figure of the Earth has various meanings in geodesy according to the way it is used and the precision with which the Earths size and shape is to be defined. The actual topographic surface is most apparent with its variety of land forms and this is, in fact, the surface on which actual Earth measurements are made. The topographic surface is generally the concern of topographers and hydrographers, the Pythagorean concept of a spherical Earth offers a simple surface that is mathematically easy to deal with. Many astronomical and navigational computations use it as a representing the Earth. The idea of a planar or flat surface for Earth, however, is sufficient for surveys of small areas, as the local topography is far more significant than the curvature. Plane-table surveys are made for small areas, and no account is taken of the curvature of the Earth. A survey of a city would likely be computed as though the Earth were a surface the size of the city. For such small areas, exact positions can be determined relative to each other without considering the size, in the mid- to late 20th century, research across the geosciences contributed to drastic improvements in the accuracy of the figure of the Earth. The primary utility of this improved accuracy was to provide geographical and gravitational data for the guidance systems of ballistic missiles. This funding also drove the expansion of geoscientific disciplines, fostering the creation, the models for the figure of the Earth vary in the way they are used, in their complexity, and in the accuracy with which they represent the size and shape of the Earth. The simplest model for the shape of the entire Earth is a sphere, the Earths radius is the distance from Earths center to its surface, about 6,371 kilometers. The concept of a spherical Earth dates back to around the 6th century BC, the first scientific estimation of the radius of the earth was given by Eratosthenes about 240 BC, with estimates of the accuracy of Eratosthenes’s measurement ranging from 2% to 15%. The Earth is only approximately spherical, so no single value serves as its natural radius, distances from points on the surface to the center range from 6,353 km to 6,384 km. Several different ways of modeling the Earth as a sphere each yield a mean radius of 6,371 kilometers, 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. The difference 21 kilometers correspond to the polar radius being approximately 0. 3% shorter than the equator radius, since the Earth is flattened at the poles and bulges at the equator, geodesy represents the shape of the earth with an oblate spheroid. The oblate spheroid, or oblate ellipsoid, is an ellipsoid of revolution obtained by rotating an ellipse about its shorter axis and it is the regular geometric shape that most nearly approximates the shape of the Earth. A spheroid describing the figure of the Earth or other body is called a reference ellipsoid. The reference ellipsoid for Earth is called an Earth ellipsoid, an ellipsoid of revolution is uniquely defined by two numbers, two dimensions, or one dimension and a number representing the difference between the two dimensions
Figure of the Earth
–
The curvature of
Earth as seen in
Valencia,
Spain (Playa de la Malvarrosa)
Figure of the Earth
–
An
oblate spheroid
19.
Geodesy
–
Geodesists also study geodynamical phenomena such as crustal motion, tides, and polar motion. For this they design global and national networks, using space and terrestrial techniques while relying on datums. Geodesy — from the Ancient Greek word γεωδαισία geodaisia — is primarily concerned with positioning within the temporally varying gravity field, such geodetic operations are also applied to other astronomical bodies in the solar system. It is also the science of measuring and understanding the earths geometric shape, orientation in space and this applies to the solid surface, the liquid surface and the Earths atmosphere. For this reason, the study of the Earths gravity field is called physical geodesy by some, the geoid is essentially the figure of the Earth abstracted from its topographical features. It is an idealized surface of sea water, the mean sea level surface in the absence of currents, air pressure variations etc. The geoid, unlike the 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 ellipsoid is called the geoidal undulation. It varies globally between ±110 m, when referred to the GRS80 ellipsoid, 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 axis, is a purely geometrical one. The mechanical ellipticity of the Earth can be determined to high precision by observation of satellite orbit perturbations and its relationship with the geometrical flattening is indirect. The relationship depends on the density distribution, or, in simplest terms. The 1980 Geodetic Reference System posited a 6,378,137 m semi-major axis and this system was adopted at the XVII General Assembly of the International Union of Geodesy and Geophysics. It is essentially the basis for geodetic positioning by the Global Positioning System and is also in widespread use outside the geodetic community. The locations of points in space are most conveniently described by three cartesian or rectangular coordinates, X, Y and Z. Since the advent of satellite positioning, such systems are typically geocentric. The X-axis lies within the Greenwich observatorys meridian plane, the coordinate transformation between these two systems is described to good approximation by sidereal time, which takes into account variations in the Earths axial rotation. A more accurate description also takes polar motion into account, a closely monitored by geodesists
Geodesy
–
An old geodetic pillar (1855) at
Ostend,
Belgium
Geodesy
–
Geodesy
Geodesy
–
A
Munich archive with
lithography plates of maps of
Bavaria
Geodesy
–
Geodetic Control Mark (example of a deep benchmark)
20.
Meridian (geography)
–
A meridian is the half of an imaginary great circle on the Earths surface, terminated by the North Pole and the South Pole, connecting points of equal longitude. The position of a point along the meridian is given by its latitude indicating how many degrees north or south of the Equator the point is, each meridian is perpendicular to all circles of latitude. Each is also the length, being half of a great circle on the Earths surface. Most maps show the lines of longitude, the position of the prime meridian has changed a few times throughout history, mainly due to the transit observatory being built next door to the previous one. Such changes had no significant practical effect, historically, the average error in the determination of longitude was much larger than the change in position. The adoption of WGS84 as the system has moved the geodetic prime meridian 102.478 metres east of its last astronomic position. The position of the current geodetic prime meridian is not identified at all by any kind of sign or marking in Greenwich, but can be located using a GPS receiver. The term meridian comes from the Latin meridies, meaning midday, the same Latin stem gives rise to the terms a. m. and p. m. used to disambiguate hours of the day when utilizing the 12-hour clock. Therefore, a compass needle will be parallel to the magnetic meridian, the angle between the magnetic and the true meridian is the magnetic declination, which is relevant for navigating with a compass. Searchable PDF prepared by the author, C. A. White, resources page of the U. S. Department of the Interior, Bureau of Land Management Meridian
Meridian (geography)
–
The prime meridian at Greenwich, England. The meridian is actually 102.5 meters east of this point since the adoption of
WGS84.
21.
Prime Meridian
–
A prime meridian is a meridian in a geographical coordinate system at which longitude is defined to be 0°. Together, a meridian and its antimeridian form a great circle. This great circle divides the sphere, e. g. the Earth, if one uses directions of East and West from a defined prime meridian, then they can be called Eastern Hemisphere and Western Hemisphere. The most widely used modern meridian is the IERS Reference Meridian and it is derived but deviates slightly from the Greenwich Meridian, which was selected as an international standard in 1884. The notion of longitude was developed by the Greek Eratosthenes in Alexandria, and Hipparchus in Rhodes, but it was Ptolemy who first used a consistent meridian for a world map in his Geographia. The main point is to be comfortably west of the tip of Africa as negative numbers were not yet in use. His prime meridian corresponds to 18°40 west of Winchester today, at that time the chief method of determining longitude was by using the reported times of lunar eclipses in different countries. Ptolemys Geographia was first printed with maps at Bologna in 1477, but there was still a hope that a natural basis for a prime meridian existed. The Tordesillas line was settled at 370 leagues west of Cape Verde. This is shown in Diogo Ribeiros 1529 map, in 1541, Mercator produced his famous 41 cm terrestrial globe and drew his prime meridian precisely through Fuertaventura in the Canaries. His later maps used the Azores, following the magnetic hypothesis, but by the time that Ortelius produced the first modern atlas in 1570, other islands such as Cape Verde were coming into use. In his atlas longitudes were counted from 0° to 360°, not 180°W to 180°E as is usual today and this practice was followed by navigators well into the 18th century. In 1634, Cardinal Richelieu used the westernmost island of the Canaries, Ferro, 19°55 west of Paris, the geographer Delisle decided to round this off to 20°, so that it simply became the meridian of Paris disguised. In the early 18th century the battle was on to improve the determination of longitude at sea, between 1765 and 1811, Nevil Maskelyne published 49 issues of the Nautical Almanac based on the meridian of the Royal Observatory, Greenwich. Maskelynes tables not only made the lunar method practicable, they made the Greenwich meridian the universal reference point. In 1884, at the International Meridian Conference in Washington, D. C.22 countries voted to adopt the Greenwich meridian as the meridian of the world. The French argued for a line, mentioning the Azores and the Bering Strait. In October 1884 the Greenwich Meridian was selected by delegates to the International Meridian Conference held in Washington, united States to be the common zero of longitude and standard of time reckoning throughout the world
Prime Meridian
–
Gerardus Mercator in his
Atlas Cosmographicae (1595) uses a prime meridian somewhere close to
25°W, passing just to the west of
Santa Maria Island in the
Atlantic. His 180th meridian runs along the
Strait of Anián (Bering Strait)
Prime Meridian
–
Ptolemy's 1st projection, redrawn under
Maximus Planudes around 1300, using a prime meridian west of Africa
Prime Meridian
–
Diogo Ribeiro's map of 1529, now in the Vatican library
22.
North Pole
–
The North Pole, also known as the Geographic North Pole or Terrestrial North Pole, is defined as the point in the Northern Hemisphere where the Earths axis of rotation meets its surface. The North Pole is the northernmost point on the Earth, lying diametrically opposite the South Pole and it defines geodetic latitude 90° North, as well as the direction of true north. At the North Pole all directions point south, all lines of longitude converge there, along tight latitude circles, counterclockwise is east and clockwise is west. The North Pole is at the center of the Northern Hemisphere, while the South Pole lies on a continental land mass, the North Pole is located in the middle of the Arctic Ocean amid waters that are almost permanently covered with constantly shifting sea ice. This makes it impractical to construct a permanent station at the North Pole, however, the Soviet Union, and later Russia, constructed a number of manned drifting stations on a generally annual basis since 1937, some of which have passed over or very close to the Pole. Since 2002, the Russians have also established a base, Barneo. This operates for a few weeks during early spring, studies in the 2000s predicted that the North Pole may become seasonally ice-free because of Arctic ice shrinkage, with timescales varying from 2016 to the late 21st century or later. The sea depth at the North Pole has been measured at 4,261 m by the Russian Mir submersible in 2007 and at 4,087 m by USS Nautilus in 1958. The nearest land is said to be Kaffeklubben Island, off the northern coast of Greenland about 700 km away. The nearest permanently inhabited place is Alert in the Qikiqtaaluk Region, Nunavut, Canada, around the beginning of the 20th century astronomers noticed a small apparent variation of latitude, as determined for a fixed point on Earth from the observation of stars. Part of this variation could be attributed to a wandering of the Pole across the Earths surface, the wandering has several periodic components and an irregular component. The component with a period of about 435 days is identified with the eight-month wandering predicted by Euler and is now called the Chandler wobble after its discoverer and it is desirable to tie the system of Earth coordinates to fixed landforms. Of course, given plate tectonics and isostasy, there is no system in all geographic features are fixed. Yet the International Earth Rotation and Reference Systems Service and the International Astronomical Union have defined a framework called the International Terrestrial Reference System. As early as the 16th century, many eminent people correctly believed that the North Pole was in a sea and it was therefore hoped that passage could be found through ice floes at favorable times of the year. Several expeditions set out to find the way, generally with whaling ships, one of the earliest expeditions to set out with the explicit intention of reaching the North Pole was that of British naval officer William Edward Parry, who in 1827 reached latitude 82°45′ North. In 1871 the Polaris expedition, a US attempt on the Pole led by Charles Francis Hall, another British Royal Navy attempt on the pole, part of the British Arctic Expedition, by Commander Albert H. Markham reached a then-record 83°2026 North in May 1876 before turning back. An 1879–1881 expedition commanded by US naval officer George W. DeLong ended tragically when their ship, over half the crew, including DeLong, were lost
North Pole
–
North Pole scenery
North Pole
–
An
Azimuthal projection showing the Arctic Ocean and the North Pole. The map also shows the
75th parallel north and
60th parallel north.
North Pole
–
Gerardus Mercator 's map of the North Pole from 1595
North Pole
–
C.G. Zorgdragers map of the North Pole from 1720
23.
South Pole
–
The South Pole, also known as the Geographic South Pole or Terrestrial South Pole, is one of the two points where the Earths axis of rotation intersects its surface. It is the southernmost point on the surface of the Earth, situated on the continent of Antarctica, it is the site of the United States Amundsen–Scott South Pole Station, which was established in 1956 and has been permanently staffed since that year. The Geographic South Pole should not be confused with the South Magnetic Pole, the South Pole is at the center of the Southern Hemisphere. For most purposes, the Geographic South Pole is defined as the point of the two points where the Earths axis of rotation intersects its surface. However, the Earths axis of rotation is actually subject to very small wobbles, the geographic coordinates of the South Pole are usually given simply as 90°S, since its longitude is geometrically undefined and irrelevant. When a longitude is desired, it may be given as 0°, at the South Pole, all directions face north. For this reason, directions at the Pole are given relative to grid north, along tight latitude circles, clockwise is east, and counterclockwise is west, opposite to the North Pole. The Geographic South Pole is located on the continent of Antarctica. It sits atop a featureless, barren, windswept and icy plateau at an altitude of 2,835 metres above sea level, and is located about 1,300 km from the nearest open sea at Bay of Whales. The ice is estimated to be about 2,700 metres thick at the Pole, the polar ice sheet is moving at a rate of roughly 10 metres per year in a direction between 37° and 40° west of grid north, down towards the Weddell Sea. Therefore, the position of the station and other artificial features relative to the geographic pole gradually shift over time. The Geographic South Pole is marked by a stake in the ice alongside a small sign, these are repositioned each year in a ceremony on New Years Day to compensate for the movement of the ice. The sign records the respective dates that Roald Amundsen and Robert F. Scott reached the Pole, followed by a quotation from each man. A new marker stake is designed and fabricated each year by staff at the site, the Ceremonial South Pole is an area set aside for photo opportunities at the South Pole Station. It is located around 180 metres from the Geographic South Pole, Amundsens Tent, The tent was erected by the Norwegian expedition led by Roald Amundsen on its arrival on 14 December 1911. It is currently buried beneath the snow and ice in the vicinity of the Pole and it has been designated a Historic Site or Monument, following a proposal by Norway to the Antarctic Treaty Consultative Meeting. In 1820, several expeditions claimed to have been the first to have sighted Antarctica, with the very first being the Russian expedition led by Faddey Bellingshausen and Mikhail Lazarev. The first landing was probably just over a year later when American Captain John Davis, the basic geography of the Antarctic coastline was not understood until the mid-to-late 19th century
South Pole
–
The Geographic South Pole. (The flag used on the flagpole is interchangeable.)
South Pole
–
1. South Geographic Pole 2.
South Magnetic Pole (2007) 3.
South Geomagnetic Pole (2005) 4.
South Pole of Inaccessibility
South Pole
–
The Ceremonial South Pole in 1998. (Background structures shown have since been replaced or altered.)
South Pole
–
Argentine soldiers saluting the flag after erecting the pole in 1965
24.
Arctic Circle
–
The Arctic Circle is the most northerly of the abstract five major circles of latitude as shown on maps of the Earth. The region north of this circle is known as the Arctic, the position of the Arctic Circle is not fixed, as of 7 April 2017, it runs 66°33′46. 6″ north of the Equator. Its latitude depends on the Earths axial tilt, which fluctuates within a margin of 2° over a 40, 000-year period, consequently, the Arctic Circle is currently drifting northwards at a speed of about 15 m per year. The word arctic comes from the Greek word ἀρκτικός and that from the word ἄρκτος, directly on the Arctic Circle these events occur, in principle, exactly once per year, at the June and December solstices, respectively. That is true at sea level, those limits increase with elevation above sea level, tens of thousands of years ago, waves of people migrated from eastern Siberia across the Bering Strait into North America to settle. Much later, in the period, there has been migration into some Arctic areas by Europeans. The largest communities north of the Arctic Circle are situated in Russia and Norway, Murmansk, Norilsk, Tromsø, rovaniemi in Finland is the largest settlement in the immediate vicinity of the Arctic Circle, lying slightly south of the line. In contrast, the largest North American community north of the Arctic Circle, of the Canadian and United States Arctic communities, Barrow, Alaska is the largest settlement with about 4,000 inhabitants. The Arctic Circle is roughly 16,000 kilometres, the area north of the Circle is about 20,000,000 km2 and covers roughly 4% of Earths surface. The Arctic Circle passes through the Arctic Ocean, the Scandinavian Peninsula, North Asia, Northern America, the land within the Arctic Circle is divided among eight countries, Norway, Sweden, Finland, Russia, the United States, Canada, Denmark, and Iceland. In the interior, summers can be warm, while winters are extremely cold
Arctic Circle
–
A sign along the
Dalton Highway marking the location of the Arctic Circle in Alaska.
Arctic Circle
–
Map of the
Arctic, with the Arctic Circle in blue and the July 10 °C mean
isotherm in red
Arctic Circle
–
Arctic Circle line in
Rovaniemi,
Finland
Arctic Circle
–
Aurora Borealis above Arctic Circle sign along the
Dempster Highway in
Yukon,
Canada
25.
Tropic of Capricorn
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The Tropic of Capricorn is the circle of latitude that contains the subsolar point on the December solstice. It is thus the southernmost latitude where the Sun can be directly overhead and its northern equivalent is the Tropic of Cancer. The Tropic of Capricorn is one of the five major circles of latitude that mark maps of the Earth. As of 3 April 2017, its latitude is 23°26′13. 4″ south of the equator, the Tropic of Capricorn is the dividing line between the Southern Temperate Zone to the south and the tropics to the north. The northern hemisphere equivalent of the Tropic of Capricorn is the Tropic of Cancer, the position of the Tropic of Capricorn is not fixed, but rather it varies in a complex manner over time, see under circles of latitude for information. In southern Africa, where rainfall is more reliable, farming is possible, vegetation here is almost non-existent, though on the eastern slopes of the Andes rainfall is adequate for rainfed agriculture. In modern times the sun appears in the constellation Sagittarius during this time, the change is due to precession of the equinoxes. The word tropic itself comes from the Greek trope, meaning turn, change in direction or circumstances, referring to the fact that the sun appears to turn back at the solstices
Tropic of Capricorn
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Tropic of Capricorn in
1794 Dunn Map of the World
Tropic of Capricorn
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World map showing the Tropic of Capricorn
Tropic of Capricorn
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Monument marking the Tropic of Capricorn just north of
Antofagasta, Chile
Tropic of Capricorn
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Longreach, Queensland, Australia
26.
Antarctic Circle
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The Antarctic Circle is the most southerly of the five major circles of latitude that mark maps of the Earth. The region south of this circle is known as the Antarctic, the position of the Antarctic Circle is not fixed, as of 9 April 2017, it runs 66°33′46. 6″ south of the Equator. Its latitude depends on the Earths axial tilt, which fluctuates within a margin of 2° over a 40, 000-year period, consequently, the Antarctic Circle is currently drifting southwards at a speed of about 15 m per year. Directly on the Antarctic Circle these events occur, in principle, exactly once per year, at the December and June solstices, respectively. That is true at sea level, those limits increase with elevation above sea level, in previous centuries some semi-permanent whaling stations were established on the continent, and some whalers would live there for a year or more. At least three children have been born in Antarctica, albeit in stations north of the Antarctic Circle, the Antarctic Circle is roughly 17,662 kilometres long. The area south of the Circle is about 20,000,000 km2, the continent of Antarctica covers much of the area within the Antarctic Circle
Antarctic Circle
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An iceberg near the Antarctic Circle north of Detaille Island
Antarctic Circle
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Map of the
Antarctic with the Antarctic Circle in blue.
27.
Ecliptic
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The ecliptic is the apparent path of the Sun on the celestial sphere, and is the basis for the ecliptic coordinate system. It also refers to the plane of this path, which is coplanar with the orbit of Earth around the Sun, the motions as described above are simplifications. Due to the movement of Earth around the Earth–Moon center of mass, due to further perturbations by the other planets of the Solar System, the Earth–Moon barycenter wobbles slightly around a mean position in a complex fashion. The ecliptic is actually the apparent path of the Sun throughout the course of a year, because Earth takes one year to orbit the Sun, the apparent position of the Sun also takes the same length of time to make a complete circuit of the ecliptic. With slightly more than 365 days in one year, the Sun moves a little less than 1° eastward every day, again, this is a simplification, based on a hypothetical Earth that orbits at uniform speed around the Sun. The actual speed with which Earth orbits the Sun varies slightly during the year, for example, the Sun is north of the celestial equator for about 185 days of each year, and south of it for about 180 days. The variation of orbital speed accounts for part of the equation of time, if the equator is projected outward to the celestial sphere, forming the celestial equator, it crosses the ecliptic at two points known as the equinoxes. The Sun, in its apparent motion along the ecliptic, crosses the equator at these points, one from south to north. The crossing from south to north is known as the equinox, also known as the first point of Aries. The crossing from north to south is the equinox or descending node. Likewise, the ecliptic itself is not fixed, the gravitational perturbations of the other bodies of the Solar System cause a much smaller motion of the plane of Earths orbit, and hence of the ecliptic, known as planetary precession. The combined action of two motions is called general precession, and changes the position of the equinoxes by about 50 arc seconds per year. Once again, this is a simplification, periodic motions of the Moon and apparent periodic motions of the Sun cause short-term small-amplitude periodic oscillations of Earths axis, and hence the celestial equator, known as nutation. Obliquity of the ecliptic is the used by astronomers for the inclination of Earths equator with respect to the ecliptic. It is about 23. 4° and is currently decreasing 0.013 degrees per hundred years due to planetary perturbations, the angular value of the obliquity is found by observation of the motions of Earth and other planets over many years. From 1984, the Jet Propulsion Laboratorys DE series of computer-generated ephemerides took over as the ephemeris of the Astronomical Almanac. Obliquity based on DE200, which analyzed observations from 1911 to 1979, was calculated, jPLs fundamental ephemerides have been continually updated. J. Laskar computed an expression to order T10 good to 0″. 04/1000 years over 10,000 years, all of these expressions are for the mean obliquity, that is, without the nutation of the equator included
Ecliptic
Ecliptic
28.
Axial tilt
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In astronomy, axial tilt, also known as obliquity, is the angle between an objects rotational axis and its orbital axis, or, equivalently, the angle between its equatorial plane and orbital plane. At an obliquity of zero, the two axes point in the direction, i. e. the rotational axis is perpendicular to the orbital plane. Over the course of an orbit, the obliquity usually does not change considerably, and this causes one pole to be directed more toward the Sun on one side of the orbit, and the other pole on the other side — the cause of the seasons on the Earth. Earths obliquity oscillates between 22.1 and 24.5 degrees on a 41, 000-year cycle, the mean obliquity is currently 23°26′13. 4″. There are two methods of specifying tilt. The IAU also uses the rule to define a positive pole for the purpose of determining orientation. Using this convention, Venus is tilted 177° and it is denoted by the Greek letter ε. Earth currently has a tilt of about 23. 4°. This value remains about the relative to a stationary orbital plane throughout the cycles of axial precession. But the ecliptic due to planetary perturbations, and the obliquity of the ecliptic is not a fixed quantity. At present, it is decreasing at a rate of about 47″ per century, Earths obliquity may have been reasonably accurately measured as early as 1100 BC in India and China. The ancient Greeks had good measurements of the obliquity since about 350 BC, about 830 AD, the Caliph Al-Mamun of Baghdad directed his astronomers to measure the obliquity, and the result was used in the Arab world for many years. It was widely believed, during the Middle Ages, that both precession and Earths obliquity oscillated around a value, with a period of 672 years. Earths axis remains tilted in the direction with reference to the background stars throughout a year. This means that one pole will be directed away from the Sun at one side of the orbit and this is the cause of Earths seasons. Summer occurs in the Northern hemisphere when the pole is directed toward the Sun. Variations in Earths axial tilt can influence the seasons and is likely a factor in climate change. The exact angular value of the obliquity is found by observation of the motions of Earth, from 1984, the Jet Propulsion Laboratorys DE series of computer-generated ephemerides took over as the fundamental ephemeris of the Astronomical Almanac
Axial tilt
Axial tilt
29.
Cross section (geometry)
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In geometry and science, a cross section is the intersection of a body in three-dimensional space with a plane, or the analog in higher-dimensional space. Cutting an object into slices creates many parallel cross sections, conic sections – circles, ellipses, parabolas, and hyperbolas – are formed by cross-sections of a cone at various different angles, as seen in the diagram at left. Any planar cross-section passing through the center of an ellipsoid forms an ellipse on its surface, a cross-section of a cylinder is a circle if the cross-section is parallel to the cylinders base, or an ellipse with non-zero eccentricity if it is neither parallel nor perpendicular to the base. If the cross-section is perpendicular to the base it consists of two line segments unless it is just tangent to the cylinder, in which case it is a single line segment. A cross section of a polyhedron is a polygon, if instead the cross section is taken for a fixed value of the density, the result is an iso-density contour. For the normal distribution, these contours are ellipses, a cross section can be used to visualize the partial derivative of a function with respect to one of its arguments, as shown at left. In economics, a function f specifies the output that can be produced by various quantities x and y of inputs, typically labor. The production function of a firm or a society can be plotted in three-dimensional space, also in economics, a cardinal or ordinal utility function u gives the degree of satisfaction of a consumer obtained by consuming quantities w and v of two goods. Cross sections are used in anatomy to illustrate the inner structure of an organ. A cross section of a trunk, as shown at left, reveals growth rings that can be used to find the age of the tree. Cavalieris principle states that solids with corresponding sections of equal areas have equal volumes. The cross-sectional area of an object when viewed from an angle is the total area of the orthographic projection of the object from that angle. For example, a cylinder of height h and radius r has A ′ = π r 2 when viewed along its central axis, a sphere of radius r has A ′ = π r 2 when viewed from any angle. For a convex body, each ray through the object from the viewers perspective crosses just two surfaces, descriptive geometry Exploded view drawing Graphical projection Plans
Cross section (geometry)
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Pinus taeda cross section showing annual rings,
Cheraw, South Carolina.
30.
Solstice
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The Solstice occurs twice each year as the Sun reaches its most northerly or southerly excursion relative to the celestial equator on the celestial sphere. The seasons of the year are directly connected to both the solstices and the equinoxes, the term solstice can also be used in a broader sense, as the day when this occurs. The day of the solstice has either the most sunlight of the year or the least sunlight of the year for any other than the equator. Alternative terms, with no ambiguity as to which hemisphere is the context, are June solstice and December solstice, at latitudes outside the tropics, the summer solstice marks the day when the sun appears highest in the sky. Within the tropics, the sun directly overhead from days to months before the solstice and again after the solstice. For an observer on the North Pole, the sun reaches the highest position in the sky once a year in June, the day this occurs is called the June solstice day. Similarly, for an observer on the South Pole, the sun reaches the highest position on December solstice day, when it is the summer solstice at one Pole, it is the winter solstice on the other. The suns westerly motion never ceases as the Earth is continually in rotation, however, the suns motion in declination comes to a stop at the moment of solstice. In that sense, solstice means sun-standing and this modern scientific word descends from a Latin scientific word in use in the late Roman republic of the 1st century BC, solstitium. Pliny uses it a number of times in his Natural History with a meaning that it has today. It contains two Latin-language morphemes, sol, sun, and -stitium, stoppage, the Romans used standing to refer to a component of the relative velocity of the Sun as it is observed in the sky. Relative velocity is the motion of an object from the point of view of an observer in a frame of reference, from a fixed position on the ground, the sun appears to orbit around the Earth. To an observer in a frame of reference, the planet Earth is seen to rotate about an axis. The Earths axis is tilted with respect to the plane of the Earths orbit, an observer on Earth therefore sees a solar path that is the result of both rotation and revolution. At maximum or minimum elevation, the relative motion of the Sun perpendicular to the horizon stops. Outside of the tropics, the maximum occurs at the summer solstice. The path of the Sun, or ecliptic, sweeps north and south between the northern and southern hemispheres, the days are longer around the summer solstice and shorter around the winter solstice. When the Suns path crosses the equator, the length of the nights at latitudes +L° and -L° are of equal length and this is known as an equinox
Solstice
Solstice
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Diagram of the Earth's seasons as seen from the north. Far right:
southern solstice
Solstice
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Diagram of the Earth's seasons as seen from the south. Far left:
northern solstice
Solstice
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Two images showing the amount of reflected sunlight at southern and northern summer solstices respectively (watts / m²).