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Catenary
In physics and geometry, a catenary (, ) is the curve that an idealized hanging chain or cable assumes under its own weight when supported only at its ends in a uniform gravitational field. The catenary curve has a U-like shape, superficially similar in appearance to a parabola, which it is not. The curve appears in the design of certain types of arches and as a cross section of the catenoid—the shape assumed by a soap film bounded by two parallel circular rings. The catenary is also called the alysoid, chainette, MathWorld or, particularly in the materials sciences, funicular. Rope statics describes catenaries in a classic statics problem involving a hanging rope. Mathematically, the catenary curve is the graph of the hyperbolic cosine function. The surface of revolution of the catenary curve, the catenoid, is a minimal surface, specifically a minimal surface of revolution. A hanging chain will assume a shape of least potential energy which is a catenary. Galile ...
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Catenary Arch
A catenary arch is a type of architectural arch that follows an inverted catenary curve. The catenary curve has been employed in buildings since ancient times. It forms an underlying principle to the overall system of vaults and buttresses in stone vaulted Gothic cathedrals and in Renaissance domes. It is not a parabolic arch. In history The 17th-century scientist Robert Hooke wrote: "''Ut pendet continuum flexile, sic stabit contiguum rigidum inversum''", or, "As hangs a flexible cable so, inverted, stand the touching pieces of an arch." A note written by Thomas Jefferson in 1788 reads, "I have lately received from Italy a treatise on the equilibrium of arches, by the Abbé Mascheroni. It appears to be a very scientific work. I have not yet had time to engage in it; but I find that the conclusions of his demonstrations are, that every part of the catenary is in perfect equilibrium". Structural properties Architecturally, a catenary arch has the ability to withstand the ...
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Overhead Line
An overhead line or overhead wire is an electrical cable that is used to transmit electrical energy to electric locomotives, trolleybuses or trams. It is known variously as: * Overhead catenary * Overhead contact system (OCS) * Overhead equipment (OHE) * Overhead line equipment (OLE or OHLE) * Overhead lines (OHL) * Overhead wiring (OHW) * Traction wire * Trolley wire This article follows the International Union of Railways in using the generic term ''overhead line''. An overhead line consists of one or more wires (or rails, particularly in tunnels) situated over rail tracks, raised to a high electrical potential by connection to feeder stations at regular intervals. The feeder stations are usually fed from a high-voltage electrical grid. Overview Electric trains that collect their current from overhead lines use a device such as a pantograph, bow collector or trolley pole. It presses against the underside of the lowest overhead wire, the contact wire. Current colle ...
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Steel Catenary Riser
A steel catenary riser (SCR) is a common method of connecting a subsea pipeline to a deepwater floating or fixed oil production platform. SCRs are used to transfer fluids like oil, gas, injection water, etc. between the platforms and the pipelines. Description In the offshore industry the word catenary is used as an adjective or noun with a meaning wider than is its historical meaning in mathematics. Thus, an SCR that uses a rigid, steel pipe that has a considerable bending stiffness is described as a catenary. That is because in the scale of depth of the ocean, the bending stiffness of a rigid pipe has little effect on the shape of the suspended span of an SCR. The shape assumed by the SCR is controlled mainly by weight, buoyancy and hydrodynamic forces due to currents and waves. The shape of the SCR is well approximated by stiffened catenary equations. In preliminary considerations, in spite of using conventional, rigid steel pipe, the shape of the SCR can be also approximated with ...
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Catenoid
In geometry, a catenoid is a type of surface, arising by rotating a catenary curve about an axis (a surface of revolution). It is a minimal surface, meaning that it occupies the least area when bounded by a closed space. It was formally described in 1744 by the mathematician Leonhard Euler. Soap film attached to twin circular rings will take the shape of a catenoid. Because they are members of the same associate family of surfaces, a catenoid can be bent into a portion of a helicoid, and vice versa. Geometry The catenoid was the first non-trivial minimal surface in 3-dimensional Euclidean space to be discovered apart from the plane. The catenoid is obtained by rotating a catenary about its directrix. It was found and proved to be minimal by Leonhard Euler in 1744. Early work on the subject was published also by Jean Baptiste Meusnier. There are only two minimal surfaces of revolution ( surfaces of revolution which are also minimal surfaces): the plane and the catenoid. Th ...
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Christiaan Huygens
Christiaan Huygens, Lord of Zeelhem, ( , , ; also spelled Huyghens; la, Hugenius; 14 April 1629 – 8 July 1695) was a Dutch mathematician, physicist, engineer, astronomer, and inventor, who is regarded as one of the greatest scientists of all time and a major figure in the Scientific Revolution. In physics, Huygens made groundbreaking contributions in optics and mechanics, while as an astronomer he is chiefly known for his studies of the rings of Saturn and the discovery of its moon Titan. As an engineer and inventor, he improved the design of telescopes and invented the pendulum clock, a breakthrough in timekeeping and the most accurate timekeeper for almost 300 years. An exceptionally talented mathematician and physicist, Huygens was the first to idealize a physical problem by a set of mathematical parameters, and the first to fully mathematize a mechanistic explanation of an unobservable physical phenomenon.Dijksterhuis, F.J. (2008) Stevin, Huygens and the Dutch ...
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Minimal Surface Of Revolution
In mathematics, a minimal surface of revolution or minimum surface of revolution is a surface of revolution defined from two points in a half-plane, whose boundary is the axis of revolution of the surface. It is generated by a curve that lies in the half-plane and connects the two points; among all the surfaces that can be generated in this way, it is the one that minimizes the surface area. A basic problem in the calculus of variations is finding the curve between two points that produces this minimal surface of revolution. Relation to minimal surfaces A minimal surface of revolution is a subtype of minimal surface. A minimal surface is defined not as a surface of minimal area, but as a surface with a mean curvature of 0. Since a mean curvature of 0 is a necessary condition of a surface of minimal area, all minimal surfaces of revolution are minimal surfaces, but not all minimal surfaces are minimal surfaces of revolution. As a point forms a circle when rotated about an axis, ...
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Parabola
In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves. One description of a parabola involves a point (the focus) and a line (the directrix). The focus does not lie on the directrix. The parabola is the locus of points in that plane that are equidistant from both the directrix and the focus. Another description of a parabola is as a conic section, created from the intersection of a right circular conical surface and a plane parallel to another plane that is tangential to the conical surface. The line perpendicular to the directrix and passing through the focus (that is, the line that splits the parabola through the middle) is called the "axis of symmetry". The point where the parabola intersects its axis of symmetry is called the " vertex" and is the point where the parabola is most sharply cu ...
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Hyperbolic Cosine
In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points form a circle with a unit radius, the points form the right half of the unit hyperbola. Also, similarly to how the derivatives of and are and respectively, the derivatives of and are and respectively. Hyperbolic functions occur in the calculations of angles and distances in hyperbolic geometry. They also occur in the solutions of many linear differential equations (such as the equation defining a catenary), cubic equations, and Laplace's equation in Cartesian coordinates. Laplace's equations are important in many areas of physics, including electromagnetic theory, heat transfer, fluid dynamics, and special relativity. The basic hyperbolic functions are: * hyperbolic sine "" (), * hyperbolic cosine "" (),''Collins Concise Dictionary'', p. 328 from which are derived: * hyperbolic tangent "" (), * h ...
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Joachim Jungius
Joachim Jungius (born Joachim Junge; 22 October 1587 – 23 September 1657) was a German mathematician, logician and philosopher of science. Life Jungius was a native of Lübeck. He studied metaphysics at the Universities of Rostock and Giessen, where in 1608 he earned his degree. Beginning in 1609, he was a professor of mathematics at the University of Giessen, and in 1614–15, with Wolfgang Ratke (1571–1635) and Christoph Helvig (1581–1617), he took part in studies of educational reform. In 1616, he returned to Rostock in order to study medicine, later obtaining his medical doctorate from the University of Padua with Santorio Santorio in 1619. From 1619 to 1623, he practiced medicine in Lübeck. In 1622 at Rostock, he founded an early scientific society known as ''Societas Ereunetica sive Zetetica''. From 1624 to 1628, Jungius worked as a professor of mathematics at Rostock, his service here being briefly interrupted in 1625, when he spent time as profess ...
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Two New Sciences
The ''Discourses and Mathematical Demonstrations Relating to Two New Sciences'' ( it, Discorsi e dimostrazioni matematiche intorno a due nuove scienze ) published in 1638 was Galileo Galilei's final book and a scientific testament covering much of his work in physics over the preceding thirty years. It was written partly in Italian and partly in Latin. After his ''Dialogue Concerning the Two Chief World Systems'', the Roman Inquisition had banned the publication of any of Galileo's works, including any he might write in the future. After the failure of his initial attempts to publish ''Two New Sciences'' in France, Germany, and Poland, it was published by Lodewijk Elzevir who was working in Leiden, South Holland, where the writ of the Inquisition was of less consequence (see House of Elzevir). Fra Fulgenzio Micanzio, the official theologian of the Republic of Venice, had initially offered to help Galileo publish in Venice the new work, but he pointed out that publishing the ''T ...
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Minimal Surface
In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below). The term "minimal surface" is used because these surfaces originally arose as surfaces that minimized total surface area subject to some constraint. Physical models of area-minimizing minimal surfaces can be made by dipping a wire frame into a soap solution, forming a soap film, which is a minimal surface whose boundary is the wire frame. However, the term is used for more general surfaces that may self-intersect or do not have constraints. For a given constraint there may also exist several minimal surfaces with different areas (for example, see minimal surface of revolution): the standard definitions only relate to a local optimum, not a global optimum. Definitions Minimal surfaces can be defined in several equivalent ways in R3. The fact that they are equivalent serves to demonstrate how minimal surface theory lies at ...
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Curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that appeared more than 2000 years ago in Euclid's ''Elements'': "The urvedline is ��the first species of quantity, which has only one dimension, namely length, without any width nor depth, and is nothing else than the flow or run of the point which ��will leave from its imaginary moving some vestige in length, exempt of any width." This definition of a curve has been formalized in modern mathematics as: ''A curve is the image of an interval to a topological space by a continuous function''. In some contexts, the function that defines the curve is called a ''parametrization'', and the curve is a parametric curve. In this article, these curves are sometimes called ''topological curves'' to distinguish them from more constrained curves su ...
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