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Spherical Law Of Cosines
In spherical trigonometry, the law of cosines (also called the cosine rule for sides) is a theorem relating the sides and angles of spherical triangles, analogous to the ordinary law of cosines from plane trigonometry. Given a unit sphere, a "spherical triangle" on the surface of the sphere is defined by the great circles connecting three points , and on the sphere (shown at right). If the lengths of these three sides are (from to (from to ), and (from to ), and the angle of the corner opposite is , then the (first) spherical law of cosines states:Romuald Ireneus 'Scibor-MarchockiSpherical trigonometry ''Elementary-Geometry Trigonometry'' web page (1997).W. Gellert, S. Gottwald, M. Hellwich, H. Kästner, and H. Küstner, ''The VNR Concise Encyclopedia of Mathematics'', 2nd ed., ch. 12 (Van Nostrand Reinhold: New York, 1989). \cos c = \cos a \cos b + \sin a \sin b \cos C\, Since this is a unit sphere, the lengths , and are simply equal to the angles (in radians) subte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spherical Trigonometry
Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the edge (geometry), sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, geodesics are great circles. Spherical trigonometry is of great importance for calculations in astronomy, geodesy, and navigation. The origins of spherical trigonometry in Greek mathematics and the major developments in Islamic mathematics are discussed fully in History of trigonometry and Mathematics in medieval Islam. The subject came to fruition in Early Modern times with important developments by John Napier, Jean Baptiste Joseph Delambre, Delambre and others, and attained an essentially complete form by the end of the nineteenth century with the publication of Todhunter's textbook ''Spherical trigonometry for the use of colleges and Schools''. Since then, significant developments have been the application of vector methods, quaternion m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binet–Cauchy Identity
In algebra, the Binet–Cauchy identity, named after Jacques Philippe Marie Binet and Augustin-Louis Cauchy, states that \left(\sum_^n a_i c_i\right) \left(\sum_^n b_j d_j\right) = \left(\sum_^n a_i d_i\right) \left(\sum_^n b_j c_j\right) + \sum_ (a_i b_j - a_j b_i ) (c_i d_j - c_j d_i ) for every choice of real or complex numbers (or more generally, elements of a commutative ring). Setting and , it gives Lagrange's identity, which is a stronger version of the Cauchy–Schwarz inequality for the Euclidean space \R^n. The Binet-Cauchy identity is a special case of the Cauchy–Binet formula for matrix determinants. The Binet–Cauchy identity and exterior algebra When , the first and second terms on the right hand side become the squared magnitudes of dot and cross products respectively; in dimensions these become the magnitudes of the dot and wedge products. We may write it (a \cdot c)(b \cdot d) = (a \cdot d)(b \cdot c) + (a \wedge b) \cdot (c \wedge d) where , , , and a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Half-side Formula
In spherical trigonometry, the half side formula relates the angles and lengths of the sides of spherical triangles, which are triangles drawn on the surface of a sphere and so have curved sides and do not obey the formulas for plane triangles. For a triangle \triangle ABC on a sphere, the half-side formula is. \begin \tan \tfrac12 a &= \sqrt \end where are the angular lengths (measure of central angle, arc lengths normalized to a sphere of unit radius) of the sides opposite angles respectively, and S = \tfrac12 (A+B+ C) is half the sum of the angles. Two more formulas can be obtained for b and c by permuting the labels A, B, C. The polar dual relationship for a spherical triangle is the ''half-angle formula'', \begin \tan \tfrac12 A &= \sqrt \end where semiperimeter s = \tfrac12 (a + b + c) is half the sum of the sides. Again, two more formulas can be obtained by permuting the labels A, B, C. Half-tangent variant The same relationships can be written as rational equ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Versine
The versine or versed sine is a trigonometric function found in some of the earliest (Sanskrit ''Aryabhatia'', Section I) trigonometric tables. The versine of an angle is 1 minus its . There are several related functions, most notably the coversine and haversine. The latter, half a versine, is of particular importance in the haversine formula of navigation. Overview The versine[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regiomontanus
Johannes Müller von Königsberg (6 June 1436 – 6 July 1476), better known as Regiomontanus (), was a mathematician, astrologer and astronomer of the German Renaissance, active in Vienna, Buda and Nuremberg. His contributions were instrumental in the development of Copernican heliocentrism in the decades following his death. Regiomontanus wrote under the Latinized name of ''Ioannes de Monteregio'' (or ''Monte Regio''; ''Regio Monte''); the toponym ''Regiomontanus'' was first used by Philipp Melanchthon in 1534. He is named after Königsberg in Lower Franconia, not the larger Königsberg (modern Kaliningrad) in Prussia. Life Although little is known of Regiomontanus' early life, it is believed that at eleven years of age, he became a student at the University of Leipzig, Saxony. In 1451 he continued his studies at Alma Mater Rudolfina, the university in Vienna, in the Duchy of Austria, where he became a pupil and friend of Georg von Peuerbach. In 1452 he was awarded hi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Horizontal Coordinate System
The horizontal coordinate system is a celestial coordinate system that uses the observer's local horizon as the fundamental plane to define two angles of a spherical coordinate system: altitude and ''azimuth''. Therefore, the horizontal coordinate system is sometimes called the az/el system, the alt/az system, or the alt-azimuth system, among others. In an altazimuth mount of a telescope, the instrument's two axes follow altitude and azimuth. Definition This celestial coordinate system divides the sky into two hemispheres: The upper hemisphere, where objects are above the horizon and are visible, and the lower hemisphere, where objects are below the horizon and cannot be seen, since the Earth obstructs views of them. The great circle separating the hemispheres is called the ''celestial horizon'', which is defined as the great circle on the celestial sphere whose plane is normal to the local gravity vector (the vertical direction). In practice, the horizon can be define ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Georg Von Peuerbach
Georg von Peuerbach (also Purbach, Peurbach; ; 30 May 1423 – 8 April 1461) was an Austrian astronomer, poet, mathematician and instrument maker, best known for his streamlined presentation of Ptolemaic astronomy in the ''Theoricae Novae Planetarum.'' Peuerbach was instrumental in making astronomy, mathematics and literature simple and accessible for Europeans during the Renaissance and beyond. Biography Peuerbach's life remains relatively unknown until he enrolled at the University of Vienna in 1446. He was born in the Austrian town of Peuerbach in upper Austria. A horoscope published eighty-nine years after his death places his date of birth specifically on 30 May 1423, though other evidence only indicates that he was born sometime after 1421. He received his Bachelor of Arts in 1448. Georg's intellect was discovered by a priest of his hometown, Dr. Heinrich Barucher. Dr. Barucher recognized Peuerbach's academic abilities from a young age and put him in contact with the Augus ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nilakantha Somayaji
Keļallur Nīlakaṇṭha Somayāji (14 June 1444 – 1544), also referred to as Keļallur Comatiri, was a mathematician and astronomer of the Kerala school of astronomy and mathematics. One of his most influential works was the comprehensive astronomical treatise '' Tantrasamgraha'' completed in 1501. He had also composed an elaborate commentary on Aryabhatiya called the ''Aryabhatiya Bhasya''. In this Bhasya, Nilakantha had discussed infinite series expansions of trigonometric functions and problems of algebra and spherical geometry. ''Grahapariksakrama'' is a manual on making observations in astronomy based on instruments of the time. Early life Nilakantha was born into a Brahmin family which came from South Malabar in Kerala. Biographical details Nilakantha Somayaji was one of the very few authors of the scholarly traditions of India who had cared to record details about his own life and times. In one of his works titled '' Siddhanta-star'' and also in his ow ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Al-Battani
Al-Battani (before 858929), archaically Latinized as Albategnius, was a Muslim astronomer, astrologer, geographer and mathematician, who lived and worked for most of his life at Raqqa, now in Syria. He is considered to be the greatest and most famous of the astronomers of the medieval Islamic world. Al-Battānī's writings became instrumental in the development of science and astronomy in the west. His (), is the earliest extant (astronomical table) made in the Ptolemaic tradition that is hardly influenced by Hindu or Sasanian astronomy. Al-Battānī refined and corrected Ptolemy's ''Almagest'', but also included new ideas and astronomical tables of his own. A handwritten Latin version by the Italian astronomer Plato Tiburtinus was produced between 1134 and 1138, through which medieval astronomers became familiar with al-Battānī. In 1537, a Latin translation of the was printed in Nuremberg. An annotated version, also in Latin, published in three separate volumes betwee ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Muhammad Ibn Musa Al-Khwarizmi
Muhammad ibn Musa al-Khwarizmi , or simply al-Khwarizmi, was a mathematician active during the Islamic Golden Age, who produced Arabic-language works in mathematics, astronomy, and geography. Around 820, he worked at the House of Wisdom in Baghdad, the contemporary capital city of the Abbasid Caliphate. One of the most prominent scholars of the period, his works were widely influential on later authors, both in the Islamic world and Europe. His popularizing treatise on algebra, compiled between 813 and 833 as ''Al-Jabr'' (''The Compendious Book on Calculation by Completion and Balancing''),Oaks, J. (2009), "Polynomials and Equations in Arabic Algebra", ''Archive for History of Exact Sciences'', 63(2), 169–203. presented the first systematic solution of linear and quadratic equations. One of his achievements in algebra was his demonstration of how to solve quadratic equations by completing the square, for which he provided geometric justifications. Because al-Khwarizmi was t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Big O Notation
Big ''O'' notation is a mathematical notation that describes the asymptotic analysis, limiting behavior of a function (mathematics), function when the Argument of a function, argument tends towards a particular value or infinity. Big O is a member of a #Related asymptotic notations, family of notations invented by German mathematicians Paul Gustav Heinrich Bachmann, Paul Bachmann, Edmund Landau, and others, collectively called Bachmann–Landau notation or asymptotic notation. The letter O was chosen by Bachmann to stand for '':wikt:Ordnung#German, Ordnung'', meaning the order of approximation. In computer science, big O notation is used to Computational complexity theory, classify algorithms according to how their run time or space requirements grow as the input size grows. In analytic number theory, big O notation is often used to express a bound on the difference between an arithmetic function, arithmetical function and a better understood approximation; one well-known exam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maclaurin Series
Maclaurin or MacLaurin is a surname. Notable people with the surname include: * Colin Maclaurin (1698–1746), Scottish mathematician * Normand MacLaurin (1835–1914), Australian politician and university administrator * Henry Normand MacLaurin (1878–1915), Australian general * Ian MacLaurin, Baron MacLaurin of Knebworth (b. 1937) * Richard Cockburn Maclaurin (1870–1920), US physicist and educator See also * Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ... in mathematics, a special case of which is the ''Maclaurin series'' * Maclaurin (crater), a crater on the Moon * McLaurin (other) * MacLaren (surname) * McLaren (other) {{surname, Maclaurin Clan MacLaren ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
