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The Tienstra formula is used to solve the resection problem in surveying, by which the location of a given point is determined by observations of angles to known landmarks from the unknown point.
Surveying or land surveying is the technique, profession, and science of determining the terrestrial 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 building corners or the surface location of subsurface features, or other purposes required by government or civil law, such as property sales.
J.M.Tienstra (1895-1951) was a professor of the Delft university of Technology where he taught the use of barycentric coordinates in solving the resection problem. It seems most probable that his name became attached to the procedure for this reason, though when, and by whom, the formula was first proposed is unknown.
In geometry, the barycentric coordinate system is a coordinate system in which the location of a point of a simplex is specified as the center of mass, or barycenter, of usually unequal masses placed at its vertices. Coordinates also extend outside the simplex, where one or more coordinates become negative. The system was introduced in 1827 by August Ferdinand Möbius.
The resection problem consists in finding the location of an observer by measuring the angles subtended by lines of sight from the observer to three known points. Tienstra’s formula provides the most compact and elegant solution to this problem.
In mathematics, the trigonometric functions are functions of an angle. They relate the angles of a triangle to the lengths of its sides. Trigonometric functions are important in the study of triangles and modeling periodic phenomena, among many other applications.
A right triangle or right-angled triangle is a triangle in which one angle is a right angle. The relation between the sides and angles of a right triangle is the basis for trigonometry.
In mathematics, the Lambert W function, also called the omega function or product logarithm, is a set of functions, namely the branches of the inverse relation of the function f(z) = zez, where ez is the exponential function, and z is any complex number. In other words
In trigonometry, the law of sines, sine law, sine formula, or sine rule is an equation relating the lengths of the sides of a triangle to the sines of its angles. According to the law,
In geometry, the incircle or inscribed circle of a triangle is the largest circle contained in the triangle; it touches the three sides. The center of the incircle is a triangle center called the triangle's incenter.
In mathematics and physics, a brachistochrone curve, or curve of fastest descent, is the one lying on plane between a point A and a lower point B, where B is not directly below A, on which a bead slides frictionlessly under the influence of a uniform gravitational field to a given end point in the shortest time. The problem was posed by Johann Bernoulli in 1696.
In Euclidean geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral whose vertices all lie on a single circle. This circle is called the circumcircle or circumscribed circle, and the vertices are said to be concyclic. The center of the circle and its radius are called the circumcenter and the circumradius respectively. Other names for these quadrilaterals are concyclic quadrilateral and chordal quadrilateral, the latter since the sides of the quadrilateral are chords of the circumcircle. Usually the quadrilateral is assumed to be convex, but there are also crossed cyclic quadrilaterals. The formulas and properties given below are valid in the convex case.
In algebra, a cubic function is a function of the form
The Basel problem is a problem in mathematical analysis with relevance to number theory, first posed by Pietro Mengoli in 1650 and solved by Leonhard Euler in 1734 and read on 5 December 1735 in The Saint Petersburg Academy of Sciences. Since the problem had withstood the attacks of the leading mathematicians of the day, Euler's solution brought him immediate fame when he was twenty-eight. Euler generalised the problem considerably, and his ideas were taken up years later by Bernhard Riemann in his seminal 1859 paper "On the Number of Primes Less Than a Given Magnitude", in which he defined his zeta function and proved its basic properties. The problem is named after Basel, hometown of Euler as well as of the Bernoulli family who unsuccessfully attacked the problem.
Spherical trigonometry is the branch of spherical geometry that deals with the relationships between trigonometric functions of the sides and angles of the spherical polygons defined by a number of intersecting great circles on the sphere. Spherical trigonometry is of great importance for calculations in astronomy, geodesy and navigation.
In geometry, the cissoid of Diocles is a cubic plane curve notable for the property that it can be used to construct two mean proportionals to a given ratio. In particular, it can be used to double a cube. It can be defined as the cissoid of a circle and a line tangent to it with respect to the point on the circle opposite to the point of tangency. In fact, the family of cissoids is named for this example and some authors refer to it simply as the cissoid. It has a single cusp at the pole, and is symmetric about the diameter of the circle which is the line of tangency of the cusp. The line is an asymptote. It is a member of the conchoid of de Sluze family of curves and in form it resembles a tractrix.
Atmospheric refraction is the deviation of light or other electromagnetic wave from a straight line as it passes through the atmosphere due to the variation in air density as a function of height. This refraction is due to the velocity of light through air, decreasing with increased density. Atmospheric refraction near the ground produces mirages. Such refraction can also raise or lower, or stretch or shorten, the images of distant objects without involving mirages. Turbulent air can make distant objects appear to twinkle or shimmer. The term also applies to the refraction of sound. Atmospheric refraction is considered in measuring the position of both celestial and terrestrial objects.
In geometry, a cevian is any line segment in a triangle with one endpoint on a vertex of the triangle and the other endpoint on the opposite side. Medians, altitudes, and angle bisectors are special cases of cevians. The name cevian comes from the Italian mathematician Giovanni Ceva, who proved a well-known theorem about cevians which also bears his name.
The main trigonometric identities between trigonometric functions are proved, using mainly the geometry of the right triangle. For greater and negative angles, see Trigonometric functions.
Resection is a method for determining an unknown position measuring angles with respect to known positions. Measurements can be made with a compass and topographic map, Theodolite or with a total station using known points of a Geodetic network or landmarks of a map.
Trigonometry is a branch of mathematics that studies relationships involving lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies.
Solution of triangles is the main trigonometric problem of finding the characteristics of a triangle, 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.
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 and the segment CB subtends an angle ; the problem is to determine the position of the point P..
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 P1 and P2. From P1 and P2 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 P1 and P2. See figure; the angles measured are (α1, β1, α2, β2).
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