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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] |
Rational Trigonometry
''Divine Proportions: Rational Trigonometry to Universal Geometry'' is a 2005 book by the mathematician Norman J. Wildberger on a proposed alternative approach to Euclidean geometry and trigonometry, called rational trigonometry. The book advocates replacing the usual basic quantities of trigonometry, Euclidean distance and angle measure, by squared distance and the square of the sine of the angle, respectively. This is logically equivalent to the standard development (as the replacement quantities can be expressed in terms of the standard ones and vice versa). The author claims his approach holds some advantages, such as avoiding the need for irrational numbers. The book was "essentially self-published" by Wildberger through his publishing company Wild Egg. The formulas and theorems in the book are regarded as correct mathematics but the claims about practical or pedagogical superiority are primarily promoted by Wildberger himself and have received mixed reviews. Overview Th ... [...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] |
Position Resection And Intersection
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] |
Triangulation (surveying)
In surveying, triangulation is the process of determining the location of a point by measuring only angles to it from known points at either end of a fixed baseline by using trigonometry, rather than measuring distances to the point directly as in trilateration. The point can then be fixed as the third point of a triangle with one known side and two known angles. Triangulation can also refer to the accurate surveying of systems of very large triangles, called triangulation networks. This followed from the work of Willebrord Snell in 1615–17, who showed how a point could be located from the angles subtended from ''three'' known points, but measured at the new unknown point rather than the previously fixed points, a problem called resectioning. Surveying error is minimized if a mesh of triangles at the largest appropriate scale is established first. Points inside the triangles can all then be accurately located with reference to it. Such triangulation methods were used for accur ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Solution Of 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 to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laurent Pothenot
{{Disambiguation ...
Laurent may refer to: *Laurent (name), a French masculine given name and a surname **Saint Laurence (aka: Saint ''Laurent''), the martyr Laurent **Pierre Alphonse Laurent, mathematician **Joseph Jean Pierre Laurent, amateur astronomer, discoverer of minor planet (51) Nemausa *Laurent, South Dakota, a proposed town for the Deaf to be named for Laurent Clerc See also *Laurent series, in mathematics, representation of a complex function ''f(z)'' as a power series which includes terms of negative degree, named for Pierre Alphonse Laurent *Saint-Laurent (other) *Laurence (name), feminine form of "Laurent" *Lawrence (other) Lawrence may refer to: Education Colleges and universities * Lawrence Technological University, a university in Southfield, Michigan, United States * Lawrence University, a liberal arts university in Appleton, Wisconsin, United States Preparator ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
George Tyrrell McCaw
George may refer to: People * George (given name) * George (surname) * George (singer), American-Canadian singer George Nozuka, known by the mononym George * George Washington, First President of the United States * George W. Bush, 43rd President of the United States * George H. W. Bush, 41st President of the United States * George V, King of Great Britain, Ireland, the British Dominions and Emperor of India from 1910-1936 * George VI, King of Great Britain, Ireland, the British Dominions and Emperor of India from 1936-1952 * Prince George of Wales * George Papagheorghe also known as Jorge / GEØRGE * George, stage name of Giorgio Moroder * George Harrison, an English musician and singer-songwriter Places South Africa * George, Western Cape ** George Airport United States * George, Iowa * George, Missouri * George, Washington * George County, Mississippi * George Air Force Base, a former U.S. Air Force base located in California Characters * George (Peppa Pig), a 2-year-ol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yard
The yard (symbol: yd) is an English unit of length in both the British imperial and US customary systems of measurement equalling 3 feet or 36 inches. Since 1959 it has been by international agreement standardized as exactly 0.9144 meter. A distance of 1,760 yards is equal to 1 mile. The US survey yard is very slightly longer. Name The term, ''yard'' derives from the Old English , etc., which was used for branches, staves and measuring rods. It is first attested in the late 7th century laws of Ine of Wessex, where the "yard of land" mentioned is the yardland, an old English unit of tax assessment equal to hide. Around the same time the Lindisfarne Gospels account of the messengers from John the Baptist in the Gospel of Matthew used it for a branch swayed by the wind. In addition to the yardland, Old and Middle English both used their forms of "yard" to denote the surveying lengths of or , used in computing acres, a distance now usually known ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cyclic Quadrilateral
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. The word cyclic is from the Ancient Greek (''kuklos''), which means "circle" or "wheel". All triangles have a circumcircle, but not all quadrilaterals do. An example of a quadrilateral that cannot be cyclic is a non-square rhombus. The section characterizations below states what n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inscribed Angle
In geometry, an inscribed angle is the angle formed in the interior of a circle when two chords intersect on the circle. It can also be defined as the angle subtended at a point on the circle by two given points on the circle. Equivalently, an inscribed angle is defined by two chords of the circle sharing an endpoint. The inscribed angle theorem relates the measure of an inscribed angle to that of the central angle subtending the same arc. The inscribed angle theorem appears as Proposition 20 on Book 3 of Euclid's ''Elements''. Theorem Statement The inscribed angle theorem states that an angle ''θ'' inscribed in a circle is half of the central angle 2''θ'' that subtends the same arc on the circle. Therefore, the angle does not change as its vertex is moved to different positions on the circle. Proof Inscribed angles where one chord is a diameter Let ''O'' be the center of a circle, as in the diagram at right. Choose two points on the circle, and call them ''V'' an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
