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Helicoid
The helicoid, also known as helical surface, after the plane and the catenoid, is the third minimal surface to be known. Description It was described by Euler in 1774 and by Jean Baptiste Meusnier in 1776. Its name derives from its similarity to the helix: for every point on the helicoid, there is a helix contained in the helicoid which passes through that point. Since it is considered that the planar range extends through negative and positive infinity, close observation shows the appearance of two parallel or mirror planes in the sense that if the slope of one plane is traced, the co-plane can be seen to be bypassed or skipped, though in actuality the co-plane is also traced from the opposite perspective. The helicoid is also a ruled surface (and a right conoid), meaning that it is a trace of a line. Alternatively, for any point on the surface, there is a line on the surface passing through it. Indeed, Catalan proved in 1842 that the helicoid and the plane were the only ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalized Helicoid
In geometry, a generalized helicoid is a surface in Euclidean space generated by rotating and simultaneously displacing a curve, the ''profile curve'', along a line, its ''axis''. Any point of the given curve is the starting point of a circular helix. If the profile curve is contained in a plane through the axis, it is called the meridian of the generalized helicoid. Simple examples of generalized helicoids are the helicoids. The meridian of a helicoid is a line which intersects the axis orthogonally. Essential types of generalized helicoids are * ruled generalized helicoids. Their profile curves are lines and the surfaces are ruled surfaces. *circular generalized helicoids. Their profile curves are circles. In mathematics helicoids play an essential role as minimal surfaces. In the technical area generalized helicoids are used for staircases, slides, screws, and pipes. Analytical representation Screw motion of a point Moving a point on a screwtype curve means, the point i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Helicoid
The helicoid, also known as helical surface, after the plane and the catenoid, is the third minimal surface to be known. Description It was described by Euler in 1774 and by Jean Baptiste Meusnier in 1776. Its name derives from its similarity to the helix: for every point on the helicoid, there is a helix contained in the helicoid which passes through that point. Since it is considered that the planar range extends through negative and positive infinity, close observation shows the appearance of two parallel or mirror planes in the sense that if the slope of one plane is traced, the co-plane can be seen to be bypassed or skipped, though in actuality the co-plane is also traced from the opposite perspective. The helicoid is also a ruled surface (and a right conoid), meaning that it is a trace of a line. Alternatively, for any point on the surface, there is a line on the surface passing through it. Indeed, Catalan proved in 1842 that the helicoid and the plane were the only ... [...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. Th ... [...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 algebrai ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 trans ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Archimedes Screw
The Archimedes screw, also known as the Archimedean screw, hydrodynamic screw, water screw or Egyptian screw, is one of the earliest hydraulic machines. Using Archimedes screws as water pumps (Archimedes screw pump (ASP) or screw pump) dates back many centuries. As a machine used for transferring water from a low-lying body of water into irrigation ditches, water is pumped by turning a screw-shaped surface inside a pipe. In the modern world, Archimedes screw pumps are widely used in wastewater treatment plants and for dewatering low-lying regions. Archimedes Screws Turbines (ASTs) are a new form of small hydroelectric powerplant that can be applied even in low head sites. Archimedes screw generators operate in a wide range of flows (0.01 m^3/s to 14.5 m^3/s) and heads (0.1 m to 10 m), including low heads and moderate flow rates that is not ideal for traditional turbines and not occupied by high performance technologies. The Archimedes screw is a reversible hydraulic machine, an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jean Baptiste Meusnier
Jean Baptiste Marie Charles Meusnier de la Place ( Tours, 19 June 1754 — le Pont de Cassel, near Mainz, 13 June 1793) was a French mathematician, engineer and Revolutionary general. He is best known for Meusnier's theorem on the curvature of surfaces, which he formulated while he was at the École Royale du Génie (Royal School of Engineering). He also discovered the helicoid. He worked with Lavoisier on the decomposition of water and the evolution of hydrogen. Dirigible balloon Meusnier is sometimes portrayed as the inventor of the dirigible, because of an uncompleted project he conceived in 1784, not long after the first balloon flights of the Montgolfiers, and presented to the French Academy of Sciences. This concerned an elliptical balloon (''ballonet'') 84 metres long, with a capacity of 1,700 cubic metres, powered by three propellers driven by 80 men. The basket, in the form of a boat, was suspended from the canopy on a system of three ropes. Jacques Charles ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Helix
A helix () is a shape like a corkscrew or spiral staircase. It is a type of smooth space curve with tangent lines at a constant angle to a fixed axis. Helices are important in biology, as the DNA molecule is formed as two intertwined helices, and many proteins have helical substructures, known as alpha helices. The word ''helix'' comes from the Greek word ''ἕλιξ'', "twisted, curved". A "filled-in" helix – for example, a "spiral" (helical) ramp – is a surface called '' helicoid''. Properties and types The ''pitch'' of a helix is the height of one complete helix turn Turn may refer to: Arts and entertainment Dance and sports * Turn (dance and gymnastics), rotation of the body * Turn (swimming), reversing direction at the end of a pool * Turn (professional wrestling), a transition between face and heel * Turn, ..., measured parallel to the axis of the helix. A double helix consists of two (typically congruent) helices with the same axis, differing by a tran ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dini's Surface
In geometry, Dini's surface is a surface with constant negative curvature that can be created by twisting a pseudosphere. It is named after Ulisse Dini and described by the following parametric equations: : \begin x&=a \cos u \sin v \\ y&=a \sin u \sin v \\ z&=a \left(\cos v +\ln \tan \frac \right) + bu \end Another description is a generalized helicoid constructed from the tractrix In geometry, a tractrix (; plural: tractrices) is the curve along which an object moves, under the influence of friction, when pulled on a horizontal plane by a line segment attached to a pulling point (the ''tractor'') that moves at a right angl .... See also * Breather surface References {{reflist Surfaces ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
