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Hansen's Problem
Hansen's problem is a problem in planar surveying, named after the astronomer Peter Andreas Hansen (1795–1874), who worked on the geodetic survey of Denmark. There are two known points ''A'' and ''B'', and two unknown points ''P''1 and ''P''2. From ''P''1 and ''P''2 an observer measures the angles made by the lines of sight to each of the other three points. The problem is to find the positions of ''P''1 and ''P''2. See figure; the angles measured are (''α''1, ''β''1, ''α''2, ''β''2). Since it involves observations of angles made at unknown points, the problem is an example of Resection (orientation), resection (as opposed to intersection). Solution method overview Define the following angles: ''γ'' = ''P''1''AP''2, ''δ'' = ''P''1''BP''2, ''φ'' = ''P''2''AB'', ''ψ'' = ''P''1''BA''. As a first step we will solve for ''φ'' and ''ψ''. The sum of these two unknown ang ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Surveying
Surveying or land surveying is the technique, profession, art, and science of determining the terrestrial two-dimensional or three-dimensional positions of points and the distances and angles between them. A land surveying professional is called a land surveyor. These points are usually on the surface of the Earth, and they are often used to establish maps and boundaries for ownership, locations, such as the designed positions of structural components for construction or the surface location of subsurface features, or other purposes required by government or civil law, such as property sales. Surveyors work with elements of geodesy, geometry, trigonometry, regression analysis, physics, engineering, metrology, programming languages, and the law. They use equipment, such as total stations, robotic total stations, theodolites, GNSS receivers, retroreflectors, 3D scanners, LiDAR sensors, radios, inclinometer, handheld tablets, optical and digital levels, subsurface locators, d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peter Andreas Hansen
Peter Andreas Hansen (born 8 December 1795, Tønder, Schleswig, Denmark; died 28 March 1874, Gotha, Thuringia, Germany) was a Danish-born German astronomer. Biography The son of a goldsmith, Hansen learned the trade of a watchmaker at Flensburg, and exercised it at Berlin and Tønder, 1818–1820. He had, however, long been a student of science; and Dr Dircks, a physician practising at Tønder, prevailed with his father to send him in 1820 to Copenhagen, where he won the patronage of H.C. Schumacher and attracted the personal notice of King Frederick VI. The Danish survey was then in progress, and he acted as Schumacher's assistant in work connected with it, chiefly at the new observatory of Altona, from 1821–1825. Thence he passed on to Gotha as director of the Gotha Observatory; nor could he be tempted to relinquish the post by successive invitations to replace F.G.W. Struve at Dorpat in 1829, Friedrich Wilhelm August Argelander at Helsinki in 1837, and F.W. Bessel at Kö ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Resection (orientation)
Position resection and intersection are methods for determining an unknown geographic position ( position finding) by measuring angles with respect to known positions. In ''resection'', the one point with unknown coordinates is occupied and sightings are taken to the known points; in ''intersection'', the two points with known coordinates are occupied and sightings are taken to the unknown point. Measurements can be made with a compass and topographic map (or nautical chart), theodolite or with a total station using known points of a geodetic network or landmarks of a map. Resection versus intersection Resection and its related method, ''intersection'', are used in surveying as well as in general land navigation (including inshore marine navigation using shore-based landmarks). Both methods involve taking azimuths or bearings to two or more objects, then drawing ''lines of position'' along those recorded bearings or azimuths. When intersecting, lines of position are used to fix th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Law Of Sines
In trigonometry, the law of sines, sine law, sine formula, or sine rule is an equation relating the lengths of the sides of any triangle to the sines of its angles. According to the law, \frac \,=\, \frac \,=\, \frac \,=\, 2R, where , and are the lengths of the sides of a triangle, and , and are the opposite angles (see figure 2), while is the radius of the triangle's circumcircle. When the last part of the equation is not used, the law is sometimes stated using the reciprocals; \frac \,=\, \frac \,=\, \frac. The law of sines can be used to compute the remaining sides of a triangle when two angles and a side are known—a technique known as triangulation. It can also be used when two sides and one of the non-enclosed angles are known. In some such cases, the triangle is not uniquely determined by this data (called the ''ambiguous case'') and the technique gives two possible values for the enclosed angle. The law of sines is one of two trigonometric equations commonly a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trigonometric Identity
In trigonometry, trigonometric identities are Equality (mathematics), equalities that involve trigonometric functions and are true for every value of the occurring Variable (mathematics), variables for which both sides of the equality are defined. Geometrically, these are identity (mathematics), identities involving certain functions of one or more angles. They are distinct from Trigonometry#Triangle identities, triangle identities, which are identities potentially involving angles but also involving side lengths or other lengths of a triangle. These identities are useful whenever expressions involving trigonometric functions need to be simplified. An important application is the integral, integration of non-trigonometric functions: a common technique involves first using the Trigonometric substitution, substitution rule with a trigonometric function, and then simplifying the resulting integral with a trigonometric identity. Pythagorean identities The basic relationship betwe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Solving Triangles
Solution of triangles ( la, solutio triangulorum) is the main trigonometric problem of finding the characteristics of a triangle (angles and lengths of sides), when some of these are known. The triangle can be located on a plane or on a sphere. Applications requiring triangle solutions include geodesy, astronomy, construction, and navigation. Solving plane triangles A general form triangle has six main characteristics (see picture): three linear (side lengths ) and three angular (). The classical plane trigonometry problem is to specify three of the six characteristics and determine the other three. A triangle can be uniquely determined in this sense when given any of the following: *Three sides (SSS) *Two sides and the included angle (SAS) *Two sides and an angle not included between them (SSA), if the side length adjacent to the angle is shorter than the other side length. *A side and the two angles adjacent to it (ASA) *A side, the angle opposite to it and an angle adjacent t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Snellius–Pothenot Problem
The Snellius–Pothenot problem is a problem in planar surveying. Given three known points A, B and C, an observer at an unknown point P observes that the segment AC subtends an angle \alpha and the segment CB subtends an angle \beta; the problem is to determine the position of the point P. (See figure; the point denoted C is between A and B as seen from P). Since it involves the observation of known points from an unknown point, the problem is an example of resection. Historically it was first studied by Snellius, who found a solution around 1615. Formulating the equations First equation Denoting the (unknown) angles ''CAP'' as ''x'' and ''CBP'' as ''y'' we get: :x+y = 2 \pi - \alpha - \beta - C by using the sum of the angles formula for the quadrilateral ''PACB''. The variable ''C'' represents the (known) internal angle in this quadrilateral at point ''C''. (Note that in the case where the points ''C'' and ''P'' are on the same side of the line ''AB'', the angle C will b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trigonometry
Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. The Greeks focused on the calculation of chords, while mathematicians in India created the earliest-known tables of values for trigonometric ratios (also called trigonometric functions) such as sine. Throughout history, trigonometry has been applied in areas such as geodesy, surveying, celestial mechanics, and navigation. Trigonometry is known for its many identities. These trigonometric identities are commonly used for rewriting trigonometrical expressions with the aim to simplify an expression, to find a more useful form of an expression, or to solve an equation. History Sumerian astronomers studied angle measure, using a division of circles into 360 degrees. They, and later the Babylonians, studied the ratios of the sides of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Surveying
Surveying or land surveying is the technique, profession, art, and science of determining the terrestrial two-dimensional or three-dimensional positions of points and the distances and angles between them. A land surveying professional is called a land surveyor. These points are usually on the surface of the Earth, and they are often used to establish maps and boundaries for ownership, locations, such as the designed positions of structural components for construction or the surface location of subsurface features, or other purposes required by government or civil law, such as property sales. Surveyors work with elements of geodesy, geometry, trigonometry, regression analysis, physics, engineering, metrology, programming languages, and the law. They use equipment, such as total stations, robotic total stations, theodolites, GNSS receivers, retroreflectors, 3D scanners, LiDAR sensors, radios, inclinometer, handheld tablets, optical and digital levels, subsurface locators, d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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