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Helicoid
The helicoid, also known as helical surface, is a smooth Surface (differential geometry), surface embedded in three-dimensional space. It is the surface traced by an infinite line that is simultaneously being rotated and lifted along its Rotation around a fixed axis, fixed axis of rotation. It is the third minimal surface to be known, after the Plane (geometry), plane and the catenoid. Description It was described by Euler in 1774 and by Jean Baptiste Meusnier in 1776. Its Nomenclature, name derives from its similarity to the helix: for every Point (geometry), point on the helicoid, there is a helix contained in the helicoid which passes through that point. 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, Eugène Charles Catalan, Catalan proved in 1842 that the helicoid and the plane were the only ruled minimal surfaces. A helic ... [...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, is a smooth Surface (differential geometry), surface embedded in three-dimensional space. It is the surface traced by an infinite line that is simultaneously being rotated and lifted along its Rotation around a fixed axis, fixed axis of rotation. It is the third minimal surface to be known, after the Plane (geometry), plane and the catenoid. Description It was described by Euler in 1774 and by Jean Baptiste Meusnier in 1776. Its Nomenclature, name derives from its similarity to the helix: for every Point (geometry), point on the helicoid, there is a helix contained in the helicoid which passes through that point. 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, Eugène Charles Catalan, Catalan proved in 1842 that the helicoid and the plane were the only ruled minimal surfaces. A helic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ruled Surface
In geometry, a Differential geometry of surfaces, surface in 3-dimensional Euclidean space is ruled (also called a scroll) if through every Point (geometry), point of , there is a straight line that lies on . Examples include the plane (mathematics), plane, the lateral surface of a cylinder (geometry), cylinder or cone (geometry), cone, a conical surface with ellipse, elliptical directrix (rational normal scroll), 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 thr ... [...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 Leonhard Euler ( ; ; ; 15 April 170718 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and t ... [...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 \R^3. 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 (mathematics), surface that is generated by Translation (geometry), 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. Through 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 geo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Surface (differential Geometry)
In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric. Surfaces have been extensively studied from various perspectives: ''extrinsically'', relating to their embedding in Euclidean space and ''intrinsically'', reflecting their properties determined solely by the distance within the surface as measured along curves on the surface. One of the fundamental concepts investigated is the Gaussian curvature, first studied in depth by Carl Friedrich Gauss, who showed that curvature was an intrinsic property of a surface, independent of its isometric embedding in Euclidean space. Surfaces naturally arise as graphs of functions of a pair of variables, and sometimes appear in parametric form or as loci associated to space curves. An important role in their study has been played by Lie groups (in the spirit of the Erlangen program), namely the symmetry groups of ... [...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 documented hydraulic machines. It was so-named after the Greek mathematician Archimedes who first described it around 234 BC, although the device had been developed in Egypt earlier in the century. It is a reversible hydraulic machine that can be operated both as a pump or a power generator. As a machine used for lifting water from a low-lying body of water into irrigation ditches, water is lifted 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. Run in reverse, Archimedes screw turbines act as a new form of small hydroelectric powerplant that can be applied even in low head sites. Such 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mean Curvature
In mathematics, the mean curvature H of a surface S is an ''extrinsic'' measure of curvature that comes from differential geometry and that locally describes the curvature of an embedded surface in some ambient space such as Euclidean space. The concept was used by Sophie Germain in her work on elasticity theory. Jean Baptiste Marie Meusnier used it in 1776, in his studies of minimal surfaces. It is important in the analysis of minimal surfaces, which have mean curvature zero, and in the analysis of physical interfaces between fluids (such as soap films) which, for example, have constant mean curvature in static flows, by the Young–Laplace equation. Definition Let p be a point on the surface S inside the three dimensional Euclidean space . Each plane through p containing the normal line to S cuts S in a (plane) curve. Fixing a choice of unit normal gives a signed curvature to that curve. As the plane is rotated by an angle \theta (always containing the normal line) that cur ... [...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 and '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Helix
A helix (; ) is a shape like a cylindrical coil spring or the thread of a machine screw. 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 , "twisted, curved". A "filled-in" helix – for example, a "spiral" (helical) ramp – is a surface called a '' helicoid''. Properties and types The pitch of a helix is the height of one complete helix 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 translation along the axis. A circular helix (i.e. one with constant radius) has constant band curvature and constant torsion. The slope of a circular helix is commonly defined as the ratio of the circumference of the circular cylinder that it ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
