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Plücker's Conoid
In geometry, Plücker's conoid is a ruled surface named after the German mathematician Julius Plücker. It is also called a conical wedge or cylindroid; however, the latter name is ambiguous, as "cylindroid" may also refer to an elliptic cylinder. Plücker's conoid is the surface defined by the function of two variables: : z=\frac. This function has an essential singularity at the origin. By using cylindrical coordinates in space, we can write the above function into parametric equations : x=v\cos u,\quad y=v\sin u,\quad z=\sin 2u. Thus Plücker's conoid is a right conoid, which can be obtained by rotating a horizontal line about the with the oscillatory motion (with period 2''π'') along the segment of the axis (Figure 4). A generalization of Plücker's conoid is given by the parametric equations : x=v \cos u,\quad y=v \sin u,\quad z= \sin nu. where denotes the number of folds in the surface. The difference is that the period of the oscillatory motion along the i ... [...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 geometries ... [...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] |
Julius Plücker
Julius Plücker (16 June 1801 – 22 May 1868) was a German mathematician and physicist. He made fundamental contributions to the field of analytical geometry and was a pioneer in the investigations of cathode rays that led eventually to the discovery of the electron. He also vastly extended the study of Lamé curves. Biography Early years Plücker was born at Elberfeld (now part of Wuppertal). After being educated at Düsseldorf and at the universities of Bonn, Heidelberg and Berlin he went to Paris in 1823, where he came under the influence of the great school of French geometers, whose founder, Gaspard Monge, had only recently died. In 1825 he returned to Bonn, and in 1828 was made professor of mathematics. In the same year he published the first volume of his ''Analytisch-geometrische Entwicklungen'', which introduced the method of "abridged notation". In 1831 he published the second volume, in which he clearly established on a firm and independent basis projective ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elliptic Cylinder
A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base. A cylinder may also be defined as an infinite curvilinear surface in various modern branches of geometry and topology. The shift in the basic meaning—solid versus surface (as in ball and sphere)—has created some ambiguity with terminology. The two concepts may be distinguished by referring to solid cylinders and cylindrical surfaces. In the literature the unadorned term cylinder could refer to either of these or to an even more specialized object, the ''right circular cylinder''. Types The definitions and results in this section are taken from the 1913 text ''Plane and Solid Geometry'' by George Wentworth and David Eugene Smith . A ' is a surface consisting of all the points on all the lines which are parallel to a given line and which pass through a fixed plane curve in a p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Essential Singularity
In complex analysis, an essential singularity of a function is a "severe" singularity near which the function exhibits odd behavior. The category ''essential singularity'' is a "left-over" or default group of isolated singularities that are especially unmanageable: by definition they fit into neither of the other two categories of singularity that may be dealt with in some manner – removable singularities and poles. In practice some include non-isolated singularities too; those do not have a residue. Formal description Consider an open subset U of the complex plane \mathbb. Let a be an element of U, and f\colon U\setminus\\to \mathbb a holomorphic function. The point a is called an ''essential singularity'' of the function f if the singularity is neither a pole nor a removable singularity. For example, the function f(z)=e^ has an essential singularity at z=0. Alternative descriptions Let \;a\; be a complex number, assume that f(z) is not defined at \;a\; but is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Origin (mathematics)
In mathematics, the origin of a Euclidean space is a special point, usually denoted by the letter ''O'', used as a fixed point of reference for the geometry of the surrounding space. In physical problems, the choice of origin is often arbitrary, meaning any choice of origin will ultimately give the same answer. This allows one to pick an origin point that makes the mathematics as simple as possible, often by taking advantage of some kind of geometric symmetry. Cartesian coordinates In a Cartesian coordinate system, the origin is the point where the axes of the system intersect.. The origin divides each of these axes into two halves, a positive and a negative semiaxis. Points can then be located with reference to the origin by giving their numerical coordinates—that is, the positions of their projections along each axis, either in the positive or negative direction. The coordinates of the origin are always all zero, for example (0,0) in two dimensions and (0,0,0) in three. Ot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cylindrical Coordinates
A cylindrical coordinate system is a three-dimensional coordinate system that specifies point positions by the distance from a chosen reference axis ''(axis L in the image opposite)'', the direction from the axis relative to a chosen reference direction ''(axis A)'', and the distance from a chosen reference plane perpendicular to the axis ''(plane containing the purple section)''. The latter distance is given as a positive or negative number depending on which side of the reference plane faces the point. The ''origin'' of the system is the point where all three coordinates can be given as zero. This is the intersection between the reference plane and the axis. The axis is variously called the ''cylindrical'' or ''longitudinal'' axis, to differentiate it from the ''polar axis'', which is the ray that lies in the reference plane, starting at the origin and pointing in the reference direction. Other directions perpendicular to the longitudinal axis are called ''radial lines''. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Right Conoid
In geometry, a right conoid is a ruled surface generated by a family of straight lines that all intersect perpendicularly to a fixed straight line, called the ''axis'' of the right conoid. Using a Cartesian coordinate system in three-dimensional space, if we take the to be the axis of a right conoid, then the right conoid can be represented by the parametric equations: :x=v\cos u :y=v\sin u :z=h(u) where is some function for representing the ''height'' of the moving line. Examples A typical example of right conoids is given by the parametric equations : x=v\cos u, y=v\sin u, z=2\sin u The image on the right shows how the coplanar lines generate the right conoid. Other right conoids include: *Helicoid: x=v\cos u, y=v\sin u, z=cu. *Whitney umbrella: x=vu, y=v, z=u^2. *Wallis's conical edge: x=v\cos u, y=v \sin u, z=c\sqrt. *Plücker's conoid: x=v\cos u, y=v\sin u, z=c\sin nu. *hyperbolic paraboloid: x=v, y=u, z=uv (with x-axis and y-axis as its axes). See also * Cono ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Plucker Conoid (n=2)
Plucker is an offline Web and free e-book reader for Palm OS based handheld devices, Windows Mobile (Pocket PC) devices, and other PDAs. Plucker contains POSIX tools, scripts, and "conduits" which work on Linux, Mac OS X, Microsoft Windows, and Unix. Web pages can be processed, compressed, and transferred to the PDA for viewing by the Plucker viewer. Features * Clickable images (with pan and zoom) * Italic and narrow support * High-resolution fonts * Multiple concurrent documents (more than one copy of the same material) * Configurable display options (rotation, configurable toolbars) * Advanced stylus options (gestures and hardware button navigation) * zlib and PalmDoc compression (in native and ARM optimized versions) * Python, C++, and Perl distillers and scripts * A Microsoft Windows graphical installer Through the use of intelligent "distillers" written in many common languages (currently Python, C++ and Perl with third-party versions written in Java), content can be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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