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Nodary
In physics and geometry, the nodary is the curve that is traced by the focus of a hyperbola as it rolls without slipping along the axis, a roulette curve. John Oprea, Differential Geometry and its Applications, MAA 2007. pp. 147–148 The differential equation of the curve is: y^2 + \frac=b^2. Its parametric equation is: :x(u)=a\operatorname(u,k)+(a/k)\big((1-k^2)u - E(u,k)\big) :y(u)=-a\operatorname(u,k)+(a/k)\operatorname(u,k) where k= \cos(\tan^(b/a)) is the elliptic modulus and E(u,k) is the incomplete elliptic integral of the second kind and sn, cn and dn are Jacobi's elliptic functions. The surface of revolution is the nodoid constant mean curvature surface. References Plane curves {{geometry-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of events." Physics is one of the most fundamental scientific disciplines, with its main goal being to understand how the universe behaves. "Physics is one of the most fundamental of the sciences. Scientists of all disciplines use the ideas of physics, including chemists who study the structure of molecules, paleontologists who try to reconstruct how dinosaurs walked, and climatologists who study how human activities affect the atmosphere and oceans. Physics is also the foundation of all engineering and technology. No engineer could design a flat-screen TV, an interplanetary spacecraft, or even a better mousetrap without first understanding the basic laws of physics ... [...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 geom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperbola
In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, that are mirror images of each other and resemble two infinite bows. The hyperbola is one of the three kinds of conic section, formed by the intersection of a plane and a double cone. (The other conic sections are the parabola and the ellipse. A circle is a special case of an ellipse.) If the plane intersects both halves of the double cone but does not pass through the apex of the cones, then the conic is a hyperbola. Hyperbolas arise in many ways: * as the curve representing the reciprocal function y(x) = 1/x in the Cartesian plane, * as the path followed by the shadow of the tip of a sundial, * as the shape of an open orbit (as distinct from a closed elliptical orbit), such as the orbit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Roulette Curve
In the differential geometry of curves, a roulette is a kind of curve, generalizing cycloids, epicycloids, hypocycloids, trochoids, epitrochoids, hypotrochoids, and involutes. Definition Informal definition Roughly speaking, a roulette is the curve described by a point (called the ''generator'' or ''pole'') attached to a given curve as that curve rolls without slipping, along a second given curve that is fixed. More precisely, given a curve attached to a plane which is moving so that the curve rolls, without slipping, along a given curve attached to a fixed plane occupying the same space, then a point attached to the moving plane describes a curve, in the fixed plane called a roulette. Special cases and related concepts In the case where the rolling curve is a line and the generator is a point on the line, the roulette is called an involute of the fixed curve. If the rolling curve is a circle and the fixed curve is a line then the roulette is a trochoid. If, in this case, t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Incomplete Elliptic Integral Of The Second Kind
In integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied by Giulio Fagnano and Leonhard Euler (). Their name originates from their originally arising in connection with the problem of finding the arc length of an ellipse. Modern mathematics defines an "elliptic integral" as any function which can be expressed in the form f(x) = \int_^ R \left(t, \sqrt \right) \, dt, where is a rational function of its two arguments, is a polynomial of degree 3 or 4 with no repeated roots, and is a constant. In general, integrals in this form cannot be expressed in terms of elementary functions. Exceptions to this general rule are when has repeated roots, or when contains no odd powers of or if the integral is pseudo-elliptic. However, with the appropriate reduction formula, every elliptic integral can be brought into a form that involves integrals over rational functions and the three Legen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacobi's Elliptic Functions
In mathematics, the Jacobi elliptic functions are a set of basic elliptic functions. They are found in the description of the motion of a pendulum (see also pendulum (mathematics)), as well as in the design of electronic elliptic filters. While trigonometric functions are defined with reference to a circle, the Jacobi elliptic functions are a generalization which refer to other conic sections, the ellipse in particular. The relation to trigonometric functions is contained in the notation, for example, by the matching notation \operatorname for \sin. The Jacobi elliptic functions are used more often in practical problems than the Weierstrass elliptic functions as they do not require notions of complex analysis to be defined and/or understood. They were introduced by . Carl Friedrich Gauss had already studied special Jacobi elliptic functions in 1797, the lemniscate elliptic functions in particular, but his work was published much later. Overview There are twelve Jacobi elliptic f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nodoid
In differential geometry, a nodoid is a surface of revolution with constant nonzero mean curvature obtained by rolling a hyperbola along a fixed line, tracing the focus Focus, or its plural form foci may refer to: Arts * Focus or Focus Festival, former name of the Adelaide Fringe arts festival in South Australia Film *''Focus'', a 1962 TV film starring James Whitmore * ''Focus'' (2001 film), a 2001 film based ..., and revolving the resulting nodary curve around the line.. References External linksWolfram Demonstrations: Delaunay Nodoids {{geometry-stub Surfaces ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Constant Mean Curvature Surface
In differential geometry, constant-mean-curvature (CMC) surfaces are surfaces with constant mean curvature.Carl Johan Lejdfors, Surfaces of Constant Mean Curvature. Master’s thesis Lund University, Centre for Mathematical Sciences Mathematics 2003:E1/ref> This includes minimal surfaces as a subset, but typically they are treated as special case. Note that these surfaces are generally different from constant Gaussian curvature surfaces, with the important exception of the sphere. History In 1841 Delaunay proved that the only surfaces of revolution with constant mean curvature were the surfaces obtained by rotating the roulettes of the conics. These are the plane, cylinder, sphere, the catenoid, the unduloid and nodoid. In 1853 J. H. Jellet showed that if S is a compact star-shaped surface in \R^3 with constant mean curvature, then it is the standard sphere. Subsequently, A. D. Alexandrov proved that a compact embedded surface in \R^3 with constant mean curvature H \neq 0 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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