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Devil's Curve
In geometry, a Devil's curve, also known as the Devil on Two Sticks, is a curve defined in the Cartesian plane by an equation of the form : y^2(y^2 - b^2) = x^2(x^2 - a^2). The polar equation of this curve is of the form :r = \sqrt = \sqrt. Devil's curves were discovered in 1750 by Gabriel Cramer, who studied them extensively. The name comes from the shape its central lemniscate takes when graphed. The shape is named after the juggling game diabolo The diabolo ( ; commonly misspelled ''diablo'') is a juggling or circus prop consisting of an axle () and two cups (hourglass/egg timer shaped) or discs derived from the Chinese yo-yo. This object is spun using a string attached to two hand ..., which was named after the Devil and which involves two sticks, a string, and a spinning prop in the likeness of the lemniscate. For , b, , a, it is vertical. Is , b, = , a, , the shape becomes a circle. The vertical hourglass intersects the y-axis at b,-b, 0 . The horizonta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Devils Curve A=0
A devil is the personification of evil as it is conceived in many and various cultures and religious traditions. Devil or Devils may also refer to: * Satan * Devil in Christianity * Demon * Folk devil Art, entertainment, and media Film and television * ''The Devil'' (1908 film), a 1908 film directed by D. W. Griffith * ''The Devil'' (1915 film), an American film starring Bessie Barriscale * ''The Devil'' (1918 Hungarian film), a Hungarian film directed by Michael Curtiz * ''The Devil'' (1918 German film), a German silent mystery film * ''The Devil'' (1921 film), an American film starring George Arliss * ''To Bed or Not to Bed'' (also known as ''The Devil''), a 1963 Italian film * ''The Devils'' (film), a 1971 British film directed by Ken Russell * ''The Devil'' (1972 film), a Polish film * ''The Devil'' (TV series), a 2007 South Korean television series ** ''Devil'' (TV series), a 2008 Japanese television series remake of the South Korean series * ''Devil'' (2010 fi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is called a ''geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts. During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Carl Friedrich Gauss' ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in a Euclidean space. This implies that surfaces can be studied ''intrinsically'', that is, as stand-alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry. Later in the 19th century, it appeared that geometries ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (geometry), point. This is the definition that appeared more than 2000 years ago in Euclid's Elements, Euclid's ''Elements'': "The [curved] line 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 (mathematics), image of an interval (mathematics), 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 artic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartesian Plane
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in the same unit of length. Each reference coordinate line is called a ''coordinate axis'' or just ''axis'' (plural ''axes'') of the system, and the point where they meet is its ''origin'', at ordered pair . The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin. One can use the same principle to specify the position of any point in three-dimensional space by three Cartesian coordinates, its signed distances to three mutually perpendicular planes (or, equivalently, by its perpendicular projection onto three mutually perpendicular lines). In general, ''n'' Cartesian coordinates (an element of real ''n''-space) specify the point in an ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polar Equation
In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to the origin of a Cartesian coordinate system) is called the ''pole'', and the ray from the pole in the reference direction is the ''polar axis''. The distance from the pole is called the ''radial coordinate'', ''radial distance'' or simply ''radius'', and the angle is called the ''angular coordinate'', ''polar angle'', or ''azimuth''. Angles in polar notation are generally expressed in either degrees or radians (2 rad being equal to 360°). Grégoire de Saint-Vincent and Bonaventura Cavalieri independently introduced the concepts in the mid-17th century, though the actual term "polar coordinates" has been attributed to Gregorio Fontana in the 18th century. The initial motivation for the introduction of the polar system was the study of circular ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gabriel Cramer
Gabriel Cramer (; 31 July 1704 – 4 January 1752) was a Genevan mathematician. He was the son of physician Jean Cramer and Anne Mallet Cramer. Biography Cramer showed promise in mathematics from an early age. At 18 he received his doctorate and at 20 he was co-chairHe did not get the chair of philosophy he had been a candidate for; but the University of Geneva was so impressed by him that it created a chair of mathematics for him and for his friend Jean-Louis Calandrini; the two alternated as chairs. of mathematics at the University of Geneva. In 1728 he proposed a solution to the St. Petersburg Paradox that came very close to the concept of expected utility theory given ten years later by Daniel Bernoulli. He published his best-known work in his forties. This included his treatise on algebraic curves (1750). It contains the earliest demonstration that a curve of the ''n''-th degree is determined by ''n''(''n'' + 3)/2 points on it, in general position. (See Cramer's theorem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lemniscate
In algebraic geometry, a lemniscate is any of several figure-eight or -shaped curves. The word comes from the Latin "''lēmniscātus''" meaning "decorated with ribbons", from the Greek λημνίσκος meaning "ribbons",. or which alternatively may refer to the wool from which the ribbons were made. Curves that have been called a lemniscate include three quartic plane curves: the hippopede or lemniscate of Booth, the lemniscate of Bernoulli, and the lemniscate of Gerono. The study of lemniscates (and in particular the hippopede) dates to ancient Greek mathematics, but the term "lemniscate" for curves of this type comes from the work of Jacob Bernoulli in the late 17th century. History and examples Lemniscate of Booth The consideration of curves with a figure-eight shape can be traced back to Proclus, a Greek Neoplatonist philosopher and mathematician who lived in the 5th century AD. Proclus considered the cross-sections of a torus by a plane parallel to the axis of the tor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diabolo
The diabolo ( ; commonly misspelled ''diablo'') is a juggling or circus prop consisting of an axle () and two cups (hourglass/egg timer shaped) or discs derived from the Chinese yo-yo. This object is spun using a string attached to two hand sticks ("batons" or "wands"). A large variety of tricks are possible with the diabolo, including tosses, and various types of interaction with the sticks, string, and various parts of the user's body. Multiple diabolos can be spun on a single string. Like the Western yo-yo (which has an independent origin), it maintains its spinning motion through a rotating effect based on conservation of angular momentum. History Origin The Diabolo is derived from the Chinese yo-yo encountered by Europeans during the colonial era. However, the origin of the Chinese yo-yo is unknown. The earliest mention of the Chinese yo-yo is in the late Ming dynasty Wanli period (1572–1620), with its details well recorded in the book ''Dijing Jingwulue'' by th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
