1.
Locus (mathematics)
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In geometry, a locus is a set of points, whose location satisfies or is determined by one or more specified conditions. Until the beginning of 20th century, a shape was not considered as an infinite set of points, rather. Thus a circle in the Euclidean plane was defined as the locus of a point that is at a distance of a fixed point. In contrast to the view, the old formulation avoids considering infinite collections. Once set theory became the universal basis over which the mathematics is built. Examples from plane geometry include, The set of points equidistant from two points is a perpendicular bisector to the segment connecting the two points. The set of points equidistant from two lines cross is the angle bisector. All conic sections are loci, Parabola, the set of points equidistant from a single point, Circle, the set of points for which the distance from a single point is constant. The set of points for each of which the ratio of the distances to two given foci is a constant is referred to as a Circle of Apollonius. Hyperbola, the set of points for each of which the value of the difference between the distances to two given foci is a constant. Ellipse, the set of points for each of which the sum of the distances to two given foci is a constant, the circle is the special case in which the two foci coincide with each other. Other examples of loci appear in areas of mathematics. For example, in dynamics, the Mandelbrot set is a subset of the complex plane that may be characterized as the connectedness locus of a family of polynomial maps. Proof that all the points on the given shape satisfy the conditions and we find the locus of the points P that have a given ratio of distances k = d1/d2 to two given points. In this example we choose k=3, A and B as the fixed points and it is the circle of Apollonius defined by these values of k, A, and B. A triangle ABC has a side with length c. We determine the locus of the third vertex C such that the medians from A and C are orthogonal and we choose an orthonormal coordinate system such that A, B. C is the third vertex

2.
Degree (angle)
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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

3.
Navigation
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Navigation is a field of study that focuses on the process of monitoring and controlling the movement of a craft or vehicle from one place to another. The field of navigation includes four categories, land navigation, marine navigation, aeronautic navigation. It is also the term of art used for the specialized knowledge used by navigators to perform navigation tasks, all navigational techniques involve locating the navigators position compared to known locations or patterns. Navigation, in a sense, can refer to any skill or study that involves the determination of position and direction. In this sense, navigation includes orienteering and pedestrian navigation, for information about different navigation strategies that people use, visit human navigation. In the European medieval period, navigation was considered part of the set of seven mechanical arts, early Pacific Polynesians used the motion of stars, weather, the position of certain wildlife species, or the size of waves to find the path from one island to another. Maritime navigation using scientific instruments such as the mariners astrolabe first occurred in the Mediterranean during the Middle Ages, the perfecting of this navigation instrument is attributed to Portuguese navigators during early Portuguese discoveries in the Age of Discovery. Open-seas navigation using the astrolabe and the compass started during the Age of Discovery in the 15th century, the Portuguese began systematically exploring the Atlantic coast of Africa from 1418, under the sponsorship of Prince Henry. In 1488 Bartolomeu Dias reached the Indian Ocean by this route, in 1492 the Spanish monarchs funded Christopher Columbuss expedition to sail west to reach the Indies by crossing the Atlantic, which resulted in the Discovery of America. In 1498, a Portuguese expedition commanded by Vasco da Gama reached India by sailing around Africa, soon, the Portuguese sailed further eastward, to the Spice Islands in 1512, landing in China one year later. The fleet of seven ships sailed from Sanlúcar de Barrameda in Southern Spain in 1519, crossed the Atlantic Ocean, some ships were lost, but the remaining fleet continued across the Pacific making a number of discoveries including Guam and the Philippines. By then, only two galleons were left from the original seven, the Victoria led by Elcano sailed across the Indian Ocean and north along the coast of Africa, to finally arrive in Spain in 1522, three years after its departure. The Trinidad sailed east from the Philippines, trying to find a path back to the Americas. He arrived in Acapulco on October 8,1565, the term stems from 1530s, from Latin navigationem, from navigatus, pp. of navigare to sail, sail over, go by sea, steer a ship, from navis ship and the root of agere to drive. Roughly, the latitude of a place on Earth is its angular distance north or south of the equator, latitude is usually expressed in degrees ranging from 0° at the Equator to 90° at the North and South poles. The height of Polaris in degrees above the horizon is the latitude of the observer, similar to latitude, the longitude of a place on Earth is the angular distance east or west of the prime meridian or Greenwich meridian. Longitude is usually expressed in degrees ranging from 0° at the Greenwich meridian to 180° east and west, sydney, for example, has a longitude of about 151° east. New York City has a longitude of 74° west, for most of history, mariners struggled to determine longitude

4.
Azimuth
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An azimuth is an angular measurement in a spherical coordinate system. An example is the position of a star in the sky, the star is the point of interest, the reference plane is the horizon or the surface of the sea, and the reference vector points north. The azimuth is the angle between the vector and the perpendicular projection of the star down onto the horizon. Azimuth is usually measured in degrees, the concept is used in navigation, astronomy, engineering, mapping, mining and artillery. In land navigation, azimuth is usually denoted alpha, α, azimuth has also been more generally defined as a horizontal angle measured clockwise from any fixed reference plane or easily established base direction line. Today, the plane for an azimuth is typically true north, measured as a 0° azimuth. Moving clockwise on a 360 degree circle, east has azimuth 90°, south 180°, there are exceptions, some navigation systems use south as the reference vector. Any direction can be the vector, as long as it is clearly defined. Quite commonly, azimuths or compass bearings are stated in a system in which either north or south can be the zero, the reference direction, stated first, is always north or south, and the turning direction, stated last, is east or west. The directions are chosen so that the angle, stated between them, is positive, between zero and 90 degrees. If the bearing happens to be exactly in the direction of one of the cardinal points and this is the reason why the X and Y axis in the above formula are swapped. If the azimuth becomes negative, one can always add 360°, the formula in radians would be slightly easier, α = atan2 Caveat, Most computer libraries reverse the order of the atan2 parameters. We are standing at latitude φ1, longitude zero, we want to find the azimuth from our viewpoint to Point 2 at latitude φ2, longitude L. The difference is usually small, if Point 2 is not more than 100 km away. Various websites will calculate geodetic azimuth, e. g. GeoScience Australia site, formulas for calculating geodetic azimuth are linked in the distance article. Normal-section azimuth is simpler to calculate, Bomford says Cunninghams formula is exact for any distance, replace φ2 with declination and longitude difference with hour angle, and change the sign. There is a variety of azimuthal map projections. They all have the property that directions from a point are preserved

5.
Great circle
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A great circle, also known as an orthodrome or Riemannian circle, of a sphere is the intersection of the sphere and a plane that passes through the center point of the sphere. This partial case of a circle of a sphere is opposed to a circle, the intersection of the sphere. Any diameter of any great circle coincides with a diameter of the sphere, and therefore all great circles have the same circumference as each other, a great circle is the largest circle that can be drawn on any given sphere. Every circle in Euclidean 3-space is a circle of exactly one sphere. For most pairs of points on the surface of a sphere, there is a great circle through the two points. The exception is a pair of points, for which there are infinitely many great circles. The minor arc of a circle between two points is the shortest surface-path between them. In this sense, the arc is analogous to “straight lines” in Euclidean geometry. The length of the arc of a great circle is taken as the distance between two points on a surface of a sphere in Riemannian geometry. The great circles are the geodesics of the sphere, in higher dimensions, the great circles on the n-sphere are the intersection of the n-sphere with 2-planes that pass through the origin in the Euclidean space Rn+1. To prove that the arc of a great circle is the shortest path connecting two points on the surface of a sphere, one can apply calculus of variations to it. Consider the class of all paths from a point p to another point q. Introduce spherical coordinates so that p coincides with the north pole. Any curve on the sphere that does not intersect either pole, except possibly at the endpoints, can be parametrized by θ = θ, ϕ = ϕ, a ≤ t ≤ b provided we allow φ to take on arbitrary real values. The infinitesimal arc length in these coordinates is d s = r θ ′2 + ϕ ′2 sin 2 θ d t. So the length of a curve γ from p to q is a functional of the curve given by S = r ∫ a b θ ′2 + ϕ ′2 sin 2 θ d t. Note that S is at least the length of the meridian from p to q, S ≥ r ∫ a b | θ ′ | d t ≥ r | θ − θ |. Since the starting point and ending point are fixed, S is minimized if and only if φ =0, so the curve must lie on a meridian of the sphere φ = φ0 = constant

6.
Rhumb line
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In navigation, a rhumb line, rhumb, or loxodrome is an arc crossing all meridians of longitude at the same angle, that is, a path with constant bearing as measured relative to true or magnetic north. A rhumb line can be contrasted with a circle, which is the path of shortest distance between two points on the surface of a sphere. On a great circle, the bearing to the point does not remain constant. In other words, a circle is locally straight with zero geodesic curvature. Meridians of longitude and parallels of latitude provide special cases of the rhumb line, on a north–south passage the rhumb line course coincides with a great circle, as it does on an east–west passage along the equator. On a Mercator projection map, any line is a straight line. But theoretically a loxodrome can extend beyond the edge of the map. Rhumb lines which cut meridians at oblique angles are loxodromic curves which spiral towards the poles, on a Mercator projection the north and south poles occur at infinity and are therefore never shown. However the full loxodrome on an infinitely high map would consist of infinitely many line segments between the two edges, on a stereographic projection map, a loxodrome is an equiangular spiral whose center is the north or south pole. All loxodromes spiral from one pole to the other, near the poles, they are close to being logarithmic spirals, so they wind around each pole an infinite number of times but reach the pole in a finite distance. The pole-to-pole length of a loxodrome is the length of the meridian divided by the cosine of the bearing away from true north, loxodromes are not defined at the poles. Three views of a pole-to-pole loxodrome The word loxodrome comes from Ancient Greek λοξός loxós, oblique + δρόμος drómos, the word rhumb may come from Spanish or Portuguese rumbo/rumo and Greek ῥόμβος rhómbos, from rhémbein. A ship sailing towards the point of the compass describes such a line which cuts all the meridians at the same angle. In Mercators Projection the Loxodromic lines are evidently straight, a misunderstanding could arise because the term “rhumb” had no precise meaning when it came into use. Therefore “rhumb” was applicable to the lines on portolans when portolans were in use. As Leo Bagrow states. the word is applied to the sea-charts of this period. Cartometric investigation has revealed that no projection was used in the early charts, λ̂ for constant φ traces out a parallel of latitude, while φ̂ for constant λ traces out a meridian of longitude. The unit vector β ^ = λ ^ + φ ^ has a constant angle β with the unit vector φ̂ for any λ and φ, since their scalar product is β ^ ⋅ φ ^ = cos β

7.
Geography
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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