|
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] |
Differential Geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry as far back as antiquity. It also relates to astronomy, the geodesy of the Earth, and later the study of hyperbolic geometry by Lobachevsky. The simplest examples of smooth spaces are the plane and space curves and surfaces in the three-dimensional Euclidean space, and the study of these shapes formed the basis for development of modern differential geometry during the 18th and 19th centuries. Since the late 19th century, differential geometry has grown into a field concerned more generally with geometric structures on differentiable manifolds. A geometric structure is one which defines some notion of size, distance, shape, volume, or other rigidifying str ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Surface Of Revolution
A surface of revolution is a surface in Euclidean space created by rotating a curve (the generatrix) around an axis of rotation. Examples of surfaces of revolution generated by a straight line are cylindrical and conical surfaces depending on whether or not the line is parallel to the axis. A circle that is rotated around any diameter generates a sphere of which it is then a great circle, and if the circle is rotated around an axis that does not intersect the interior of a circle, then it generates a torus which does not intersect itself (a ring torus). Properties The sections of the surface of revolution made by planes through the axis are called ''meridional sections''. Any meridional section can be considered to be the generatrix in the plane determined by it and the axis. The sections of the surface of revolution made by planes that are perpendicular to the axis are circles. Some special cases of hyperboloids (of either one or two sheets) and elliptic paraboloids are s ... [...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] |
Focus (geometry)
In geometry, focuses or foci (), singular focus, are special points with reference to which any of a variety of curves is constructed. For example, one or two foci can be used in defining conic sections, the four types of which are the circle, ellipse, parabola, and hyperbola. In addition, two foci are used to define the Cassini oval and the Cartesian oval, and more than two foci are used in defining an ''n''-ellipse. Conic sections Defining conics in terms of two foci An ellipse can be defined as the locus of points for which the sum of the distances to two given foci is constant. A circle is the special case of an ellipse in which the two foci coincide with each other. Thus, a circle can be more simply defined as the locus of points each of which is a fixed distance from a single given focus. A circle can also be defined as the circle of Apollonius, in terms of two different foci, as the locus of points having a fixed ratio of distances to the two foci. A parabola ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
.svg "Click for more on -> Hyperbola")
