|
Translation Surface (differential Geometry)
In differential geometry a translation surface is a surface that is generated by translations: * For two space curves c_1, c_2 with a common point P, the curve c_1 is shifted such that point P is moving on c_2. By this procedure curve c_1 generates a surface: the ''translation surface''. If both curves are contained in a common plane, the translation surface is planar (part of a plane). This case is generally ignored. Simple ''examples'': #Right circular cylinder: c_1 is a circle (or another cross section) and c_2 is a line. #The ''elliptic'' paraboloid \; z=x^2+y^2\; can be generated by \ c_1:\; (x,0,x^2)\ and \ c_2:\;(0,y,y^2)\ (both curves are parabolas). #The ''hyperbolic'' paraboloid z=x^2-y^2 can be generated by c_1: (x,0,x^2) (parabola) and c_2:(0,y,-y^2) (downwards open parabola). Translation surfaces are popular in descriptive geometry and architecture, because they can be modelled easily. In differential geometry minimal surfaces are represented by translati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Surfaces
A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. Surface or surfaces may also refer to: Mathematics *Surface (mathematics), a generalization of a plane which needs not be flat *Surface (differential geometry), a differentiable two-dimensional manifold *Surface (topology), a two-dimensional manifold * Algebraic surface, an algebraic variety of dimension two *Coordinate surfaces *Fractal surface, generated using a stochastic algorithm *Polyhedral surface * Surface area *Surface integral Arts and entertainment * Surface (band), an American R&B and pop trio ** ''Surface'' (Surface album), 1986 *Surfaces (band), American musical duo * ''Surface'' (Circle album), 1998 * "Surface" (Aero Chord song), 2014 * ''Surface'' (2005 TV series), an American science fiction show, 2005–2006 * ''Surface'' (2022 TV series), an American psychological thriller miniseries that began streaming in 2022 *'' The Surface'', an American film, 2 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kohlhammer Verlag
W. Kohlhammer Verlag GmbH, or Kohlhammer Verlag, is a German publishing house headquartered in Stuttgart. History Kohlhammer Verlag was founded in Stuttgart on 30 April 1866 by . Kohlhammer had taken over the businesses of his late father-in-law, a 120-year-old printer and a profitable . The printing business, operating out of the back of a commercial building at 14 Urbanstrasse, became W. Kohlhammer Verlag and was funded by proceeds from the bathhouse until it was closed in 1890. Kohlhammer purchased the ''Deutsche Feuerwehrzeitung'' in 1882 and printed that publication until 1923. In 1872 Kohlhammer started a weekly newspaper, the ''Neue Deutsche Familienblatt'' that by 1914 had a circulation of 185,000. Contemporary Employees of Kohlhammer joined those of other Stuttgart-based companies in early 2016 to petition the mayor to abate traffic congestion hindering their operations inside the city. In 2017, Kohlhammer Verlag employed about 400 people in Stuttgart, Würzburg and Aug ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Second Fundamental Form
In differential geometry, the second fundamental form (or shape tensor) is a quadratic form on the tangent plane of a smooth surface in the three-dimensional Euclidean space, usually denoted by \mathrm (read "two"). Together with the first fundamental form, it serves to define extrinsic invariants of the surface, its principal curvatures. More generally, such a quadratic form is defined for a smooth immersed submanifold in a Riemannian manifold. Surface in R3 Motivation The second fundamental form of a parametric surface in was introduced and studied by Gauss. First suppose that the surface is the graph of a twice continuously differentiable function, , and that the plane is tangent to the surface at the origin. Then and its partial derivatives with respect to and vanish at (0,0). Therefore, the Taylor expansion of ''f'' at (0,0) starts with quadratic terms: : z=L\frac + Mxy + N\frac + \text\,, and the second fundamental form at the origin in the coordinates is the qu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Derivative
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Partial derivatives are used in vector calculus and differential geometry. The partial derivative of a function f(x, y, \dots) with respect to the variable x is variously denoted by It can be thought of as the rate of change of the function in the x-direction. Sometimes, for z=f(x, y, \ldots), the partial derivative of z with respect to x is denoted as \tfrac. Since a partial derivative generally has the same arguments as the original function, its functional dependence is sometimes explicitly signified by the notation, such as in: :f'_x(x, y, \ldots), \frac (x, y, \ldots). The symbol used to denote partial derivatives is ∂. One of the first known uses of this symbol in mathematics is by Marquis de Condorcet from 1770, who used it for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Central Projection
In mathematics, a projection is a mapping of a set (or other mathematical structure) into a subset (or sub-structure), which is equal to its square for mapping composition, i.e., which is idempotent. The restriction to a subspace of a projection is also called a ''projection'', even if the idempotence property is lost. An everyday example of a projection is the casting of shadows onto a plane (sheet of paper): the projection of a point is its shadow on the sheet of paper, and the projection (shadow) of a point on the sheet of paper is that point itself (idempotency). The shadow of a three-dimensional sphere is a closed disk. Originally, the notion of projection was introduced in Euclidean geometry to denote the projection of the three-dimensional Euclidean space onto a plane in it, like the shadow example. The two main projections of this kind are: * The projection from a point onto a plane or central projection: If ''C'' is a point, called the center of projection, then th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parallel Projection
In three-dimensional geometry, a parallel projection (or axonometric projection) is a projection of an object in three-dimensional space onto a fixed plane, known as the ''projection plane'' or '' image plane'', where the ''rays'', known as '' lines of sight'' or ''projection lines'', are parallel to each other. It is a basic tool in descriptive geometry. The projection is called '' orthographic'' if the rays are perpendicular (orthogonal) to the image plane, and ''oblique'' or ''skew'' if they are not. Overview A parallel projection is a particular case of ''projection'' in mathematics and ''graphical projection'' in technical drawing. Parallel projections can be seen as the limit of a central or perspective projection, in which the rays pass through a fixed point called the ''center'' or ''viewpoint'', as this point is moved towards infinity. Put differently, a parallel projection corresponds to a perspective projection with an infinite focal length (the distance between th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jig (tool)
A jig is a type of custom-made tool used to control the location and/or motion of parts or other tools. Description A jig's primary purpose is to provide repeatability, accuracy, and interchangeability in the manufacturing of products.. An example of a jig is when a key is duplicated; the original is used as a jig so the new key can have the same path as the old one. Since the advent of automation and computer numerical controlled (CNC) machines, jigs are often not required because the tool path is digitally programmed and stored in memory. Jigs may be made for reforming plastics. Jigs or templates have been known long before the industrial age. There are many types of jigs, and each one is custom-tailored to do a specific job. Drill jig A ''drill jig'' is a type of jig that expedites repetitive hole center location on multiple interchangeable parts by acting as a template to guide the twist drill or other boring device into the precise location of each intended hole center ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ruled Surface
In geometry, a surface is ruled (also called a scroll) if through every point of there is a straight line that lies on . Examples include the plane, the lateral surface of a cylinder or cone, a conical surface with elliptical directrix, the right conoid, the helicoid, and the tangent developable of a smooth curve in space. A ruled surface can be described as the set of points swept by a moving straight line. For example, a cone is formed by keeping one point of a line fixed whilst moving another point along a circle. A surface is ''doubly ruled'' if through every one of its points there are two distinct lines that lie on the surface. The hyperbolic paraboloid and the hyperboloid of one sheet are doubly ruled surfaces. The plane is the only surface which contains at least three distinct lines through each of its points . The properties of being ruled or doubly ruled are preserved by projective maps, and therefore are concepts of projective geometry. In algebraic geometry, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
