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Minimal Surfaces 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, f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bulle Caténoïde
Bulle (; frp, Bulo ) is a municipality in the district of Gruyère in the canton of Fribourg in Switzerland. In January 2006 Bulle incorporated the formerly independent municipality of La Tour-de-Trême. History Ancient times Bulle is first mentioned in the 9th century as ''Butulum''. In 1200 it was mentioned as ''Bollo''. The municipality was formerly known by its German name ''Boll''; however, that name is no longer used. Very little is known about the early history of the Bulle area. In 1995, a large grave mound from the early Hallstatt period was partially excavated. The grave mound lies about from the hill on which the church was later built. Middle ages During the Early Middle Ages it was the home of a parish church that covered a large parish. This Church of St. Eusebius was probably built in the 6th or 7th century by the Bishop of Lausanne. The church is mentioned several times between 852 and 875. In the 9th century, the parish was split into several inde ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wireframe Model
A wire-frame model, also wireframe model, is a visual representation of a three-dimensional (3D) physical object used in 3D computer graphics. It is created by specifying each edge of the physical object where two mathematically continuous smooth surfaces meet, or by connecting an object's constituent vertices using (straight) lines or curves. The object is projected into screen space and rendered by drawing lines at the location of each edge. The term "wire frame" comes from designers using metal wire to represent the three-dimensional shape of solid objects. 3D wire frame computer models allow for the construction and manipulation of solids and solid surfaces. 3D solid modeling efficiently draws higher quality representations of solids than conventional line drawing. Using a wire-frame model allows for the visualization of the underlying design structure of a 3D model. Traditional two-dimensional views and drawings/renderings can be created by the appropriate rotatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carl Wolfgang Benjamin Goldschmidt
Carl Wolfgang Benjamin Goldschmidt (4 August 1807 – 15 February 1851) was a German astronomer, mathematician, and physicist of Jewish descent who was a professor of astronomy at the University of Göttingen. He is also known as Benjamin Goldschmidt, C. W. B. Goldschmidt, Carl Goldschmidt, and Karl Goldschmidt. Goldschmidt, who suffered from an enlargement of the heart, died in his sleep and was found on the morning of 15 February 1851. Mathematical works A student of Carl Friedrich Gauss and an assistant to Gauss at the university observatory, Goldschmidt frequently collaborated with Gauss on various mathematical and scientific works. Goldschmidt was in turn a professor of Gauss's protégé Bernhard Riemann. Data gathered by Gauss and Goldschmidt on the growth of the logarithmic integral compared to the distribution of prime numbers was cited by Riemann in "On the Number of Primes Less Than a Given Magnitude", Riemann's seminal paper on the prime-counting function. In 1831, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
German People
, native_name_lang = de , region1 = , pop1 = 72,650,269 , region2 = , pop2 = 534,000 , region3 = , pop3 = 157,000 3,322,405 , region4 = , pop4 = 21,000 3,000,000 , region5 = , pop5 = 125,000 982,226 , region6 = , pop6 = 900,000 , region7 = , pop7 = 142,000 840,000 , region8 = , pop8 = 9,000 500,000 , region9 = , pop9 = 357,000 , region10 = , pop10 = 310,000 , region11 = , pop11 = 36,000 250,000 , region12 = , pop12 = 25,000 200,000 , region13 = , pop13 = 233,000 , region14 = , pop14 = 211,000 , region15 = , pop15 = 203,000 , region16 = , pop16 = 201,000 , region17 = , pop17 = 101,000 148,00 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs. The society is one of the four parts of the Joint Policy Board for Mathematics and a member of the Conference Board of the Mathematical Sciences. History The AMS was founded in 1888 as the New York Mathematical Society, the brainchild of Thomas Fiske, who was impressed by the London Mathematical Society on a visit to England. John Howard Van Amringe was the first president and Fiske became secretary. The society soon decided to publish a journal, but ran into some resistance, due to concerns about competing with the American Journal of Mathematics. The result was the ''Bulletin of the American Mathematical Society'', with Fiske as editor-in-chief. The de facto journal, as intended, was influential in in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. The cat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. Galileo Galil ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boundary Conditions
In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to the differential equation which also satisfies the boundary conditions. Boundary value problems arise in several branches of physics as any physical differential equation will have them. Problems involving the wave equation, such as the determination of normal modes, are often stated as boundary value problems. A large class of important boundary value problems are the Sturm–Liouville problems. The analysis of these problems involves the eigenfunctions of a differential operator. To be useful in applications, a boundary value problem should be well posed. This means that given the input to the problem there exists a unique solution, which depends continuously on the input. Much theoretical work in the field of partial differential equ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differentiable Function
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its domain. A differentiable function is smooth (the function is locally well approximated as a linear function at each interior point) and does not contain any break, angle, or cusp. If is an interior point in the domain of a function , then is said to be ''differentiable at'' if the derivative f'(x_0) exists. In other words, the graph of has a non-vertical tangent line at the point . is said to be differentiable on if it is differentiable at every point of . is said to be ''continuously differentiable'' if its derivative is also a continuous function over the domain of the function f. Generally speaking, is said to be of class if its first k derivatives f^(x), f^(x), \ldots, f^(x) exist and are continuous over the domain of the func ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Of A Function
In mathematics, the graph of a function f is the set of ordered pairs (x, y), where f(x) = y. In the common case where x and f(x) are real numbers, these pairs are Cartesian coordinates of points in two-dimensional space and thus form a subset of this plane. In the case of functions of two variables, that is functions whose domain consists of pairs (x, y), the graph usually refers to the set of ordered triples (x, y, z) where f(x,y) = z, instead of the pairs ((x, y), z) as in the definition above. This set is a subset of three-dimensional space; for a continuous real-valued function of two real variables, it is a surface. In science, engineering, technology, finance, and other areas, graphs are tools used for many purposes. In the simplest case one variable is plotted as a function of another, typically using rectangular axes; see '' Plot (graphics)'' for details. A graph of a function is a special case of a relation. In the modern foundations of mathematics, and, typicall ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartesian Coordinate
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] |
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. The cat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
