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Exsecant And Versine
The external secant function (abbreviated exsecant, symbolized exsec) is a trigonometric function defined in terms of the secant (trigonometry), secant function: \operatorname \theta = \sec\theta - 1 = \frac - 1. It was introduced in 1855 by American civil engineering, civil engineer Charles Haslett, who used it in conjunction with the existing versine function, \operatorname\theta = 1 - \cos\theta, for designing and measuring circular arc, circular sections of railroad track. It was adopted by surveying, surveyors and civil engineers in the United States for railroad and geometric design of roads, road design, and since the early 20th century has sometimes been briefly mentioned in American trigonometry textbooks and general-purpose engineering manuals. For completeness, a few books also defined a coexsecant or excosecant function (symbolized coexsec or excsc), \operatorname \theta = \csc\theta - 1, the exsecant of the complementary angle, though it was not used in practice. Wh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclid's Elements
The ''Elements'' ( ) is a mathematics, mathematical treatise written 300 BC by the Ancient Greek mathematics, Ancient Greek mathematician Euclid. ''Elements'' is the oldest extant large-scale deductive treatment of mathematics. Drawing on the works of earlier mathematicians such as Hippocrates of Chios, Eudoxus of Cnidus and Theaetetus (mathematician), Theaetetus, the ''Elements'' is a collection in 13 books of definitions, postulates, propositions and mathematical proofs that covers plane and solid Euclidean geometry, elementary number theory, and Commensurability (mathematics), incommensurable lines. These include Pythagorean theorem, Thales' theorem, the Euclidean algorithm for greatest common divisors, Euclid's theorem that there are infinitely many prime numbers, and the Compass-and-straightedge construction, construction of regular polygons and Regular polyhedra, polyhedra. Often referred to as the most successful textbook ever written, the ''Elements'' has continued to be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Canal
Canals or artificial waterways are waterways or engineered channels built for drainage management (e.g. flood control and irrigation) or for conveyancing water transport vehicles (e.g. water taxi). They carry free, calm surface flow under atmospheric pressure, and can be thought of as artificial rivers. In most cases, a canal has a series of dams and locks that create reservoirs of low speed current flow. These reservoirs are referred to as ''slack water levels'', often just called ''levels''. A canal can be called a navigation canal when it parallels a natural river and shares part of the latter's discharges and drainage basin, and leverages its resources by building dams and locks to increase and lengthen its stretches of slack water levels while staying in its valley. A canal can cut across a drainage divide atop a ridge, generally requiring an external water source above the highest elevation. The best-known example of such a canal is the Panama Can ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Track Transition Curve
A transition curve (also, spiral easement or, simply, spiral) is a spiral-shaped length of highway or track (rail transport), railroad track that is used between sections having different profiles and radii, such as between straightaways (tangents) and curves, or between two different curves. In the horizontal plane, the radius of a transition curve varies continually over its length between the disparate radii of the sections that it joins—for example, from infinite radius at a tangent to the nominal radius of a smooth curve. The resulting spiral provides a gradual, eased transition, preventing undesirable sudden, abrupt changes in centripetal acceleration, lateral (centripetal) acceleration that would otherwise occur without a transition curve. Similarly, on highways, transition curves allow drivers to change steering gradually when entering or exiting curves. Transition curves also serve as a transition in the vertical plane, whereby the elevation of the inside or outside ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler Spiral
An Euler spiral is a curve whose curvature changes linearly with its curve length (the curvature of a circular curve is equal to the reciprocal of the radius). This curve is also referred to as a clothoid or Cornu spiral.Levien, Raph"The Euler spiral: a mathematical history."Rapp. tech (2008). The behavior of Fresnel integrals can be illustrated by an Euler spiral, a connection first made by Marie Alfred Cornu in 1874. Euler's spiral is a type of superspiral that has the property of a monotonic curvature function. The Euler spiral has applications to diffraction computations. They are also widely used in railway engineering, railway and highway engineering to design transition curves between straight and curved sections of railways or roads. A similar application is also found in photonic integrated circuits. The principle of linear variation of the curvature of the transition curve between a tangent and a circular curve defines the geometry of the Euler spiral: *Its curvature be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sagitta (geometry)
In geometry, the sagitta (sometimes abbreviated as sag) of a circular arc is the distance from the midpoint of the arc to the midpoint of its chord. It is used extensively in architecture when calculating the arc necessary to span a certain height and distance and also in optics where it is used to find the depth of a spherical mirror or lens. The name comes directly from Latin ''sagitta'', meaning an "arrow". Formulas In the following equations, s denotes the sagitta (the depth or height of the arc), r equals the radius of the circle, and l the length of the chord spanning the base of the arc. As \tfrac12l and r - s are two sides of a right triangle with r as the hypotenuse, the Pythagorean theorem gives us : r^2 = \left(\tfrac12l\right)^2 + \left(r-s\right)^2. This may be rearranged to give any of the other three: : \begin s &= r - \sqrt, \\ 0mul &= 2\sqrt, \\ pxr &= \frac = \frac+\frac. \end The sagitta may also be calculated from the versine function, for an arc that spa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chord (geometry)
A chord (from the Latin ''chorda'', meaning " bowstring") of a circle is a straight line segment whose endpoints both lie on a circular arc. If a chord were to be extended infinitely on both directions into a line, the object is a ''secant line''. The perpendicular line passing through the chord's midpoint is called '' sagitta'' (Latin for "arrow"). More generally, a chord is a line segment joining two points on any curve, for instance, on an ellipse. A chord that passes through a circle's center point is the circle's ''diameter''. In circles Among properties of chords of a circle are the following: # Chords are equidistant from the center if and only if their lengths are equal. # Equal chords are subtended by equal angles from the center of the circle. # A chord that passes through the center of a circle is called a diameter and is the longest chord of that specific circle. # If the line extensions (secant lines) of chords AB and CD intersect at a point P, then their ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Central Angle
A central angle is an angle whose apex (vertex) is the center O of a circle and whose legs (sides) are radii intersecting the circle in two distinct points A and B. Central angles are subtended by an arc between those two points, and the arc length is the central angle of a circle of radius one (measured in radians). The central angle is also known as the arc's angular distance. The arc length spanned by a central angle on a sphere is called '' spherical distance''. The size of a central angle is or (radians). When defining or drawing a central angle, in addition to specifying the points and , one must specify whether the angle being defined is the convex angle (<180°) or the reflex angle (>180°). Equivalently, one must specify whether the movement from point to point is clockwise or counterclockwise. Formulas If the intersection points and of the legs of the angle with the circle form a diameter, then is a straight angle. (In radians, .) Let be the min ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Common Logarithm
In mathematics, the common logarithm (aka "standard logarithm") is the logarithm with base 10. It is also known as the decadic logarithm, the decimal logarithm and the Briggsian logarithm. The name "Briggsian logarithm" is in honor of the British mathematician Henry Briggs who conceived of and developed the values for the "common logarithm". Historically', the "common logarithm" was known by its Latin name ''logarithmus decimalis'' or ''logarithmus decadis''. The mathematical notation for using the common logarithm is , , or sometimes with a capital ; on calculators, it is printed as "log", but mathematicians usually mean natural logarithm (logarithm with base ≈ 2.71828) rather than common logarithm when writing "log". Before the early 1970s, handheld electronic calculators were not available, and mechanical calculators capable of multiplication were bulky, expensive and not widely available. Instead, tables of base-10 logarithms were used in science, engineering and navi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Surveying
Surveying or land surveying is the technique, profession, art, and science of determining the land, terrestrial Plane (mathematics), two-dimensional or Three-dimensional space#In Euclidean geometry, three-dimensional positions of Point (geometry), points and the Euclidean distance, distances and angles between them. 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 designated 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. A professional in land surveying is called a land surveyor. 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, Satellite navigation, GNSS receivers, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Galileo Galilei
Galileo di Vincenzo Bonaiuti de' Galilei (15 February 1564 – 8 January 1642), commonly referred to as Galileo Galilei ( , , ) or mononymously as Galileo, was an Italian astronomer, physicist and engineer, sometimes described as a polymath. He was born in the city of Pisa, then part of the Duchy of Florence. Galileo has been called the father of observational astronomy, modern-era classical physics, the scientific method, and modern science. Galileo studied speed and velocity, gravity and free fall, the principle of relativity, inertia, projectile motion and also worked in applied science and technology, describing the properties of the pendulum and "hydrostatic balances". He was one of the earliest Renaissance developers of the thermoscope and the inventor of various sector (instrument), military compasses. With an improved telescope he built, he observed the stars of the Milky Way, the phases of Venus, the Galilean moons, four largest satellites of Jupiter, Saturn's r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thomas Fincke
Thomas Fincke (6 January 1561 – 24 April 1656) was a Danish mathematician and physicist, and a professor at the University of Copenhagen for more than 60 years. Biography Thomas Jacobsen Fincke was born in Flensburg in Schleswig. Fincke was the son of Councillor Jacob Fincke and Anna Thorsmede. He completed his primary schooling at Flensburg. From 1577, he studied mathematics, rhetoric and other philosophical studies for five years at the University of Strasbourg. Fincke's lasting achievement is found in his book ''Geometria rotundi'' (1583), in which he introduced the modern names of the trigonometric functions tangent and secant. In 1590, he became professor of mathematics at the University of Copenhagen. In 1603 he also obtained a professorship in medicine. Personal life He was married to Ivaria Jungesdatter Ivers (1574–1614). His son Jacob Fincke (1592–1663) was a professor of physics. His daughters married scientist Caspar Bartholin the Elder (1585–1629), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
