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Conoid
In geometry a conoid () is a ruled surface, whose rulings (lines) fulfill the additional conditions: :(1) All rulings are parallel to a plane, the '' directrix plane''. :(2) All rulings intersect a fixed line, the ''axis''. The conoid is a right conoid if its axis is perpendicular to its directrix plane. Hence all rulings are perpendicular to the axis. Because of (1) any conoid is a Catalan surface and can be represented parametrically by :\mathbf x(u,v)= \mathbf c(u) + v\mathbf r(u)\ Any curve with fixed parameter is a ruling, describes the ''directrix'' and the vectors are all parallel to the directrix plane. The planarity of the vectors can be represented by :\det(\mathbf r,\mathbf \dot r,\mathbf \ddot r)=0 . If the directrix is a circle, the conoid is called a circular conoid. The term ''conoid'' was already used by Archimedes in his treatise '' On Conoids and Spheroides''. Examples Right circular conoid The parametric representation : \mathbf x(u,v)=(\co ... [...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 * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Whitney Umbrella
frame, Section of the surface In geometry, the Whitney umbrella (or Whitney's umbrella, named after American mathematician Hassler Whitney, and sometimes called a Cayley umbrella) is a specific self-intersecting ruled surface placed in three dimensions. It is the union of all straight lines that pass through points of a fixed parabola and are perpendicular to a fixed straight line which is parallel to the axis of the parabola and lies on its perpendicular bisecting plane. Formulas Whitney's umbrella can be given by the parametric equations in Cartesian coordinates : \left\{\begin{align} x(u, v) &= uv, \\ y(u, v) &= u, \\ z(u, v) &= v^2, \end{align}\right. where the parameters ''u'' and ''v'' range over the real numbers. It is also given by the implicit equation : x^2 - y^2 z = 0. This formula also includes the negative ''z'' axis (which is called the ''handle'' of the umbrella). Properties Whitney's umbrella is a ruled surface and a right conoid. It is important in ... [...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] |
Catalan Surface
In geometry, a Catalan surface, named after the Belgian mathematician Eugène Charles Catalan, is a ruled surface all of whose rulings are parallel to a fixed plane. Equations The vector equation of a Catalan surface is given by :''r'' = ''s''(''u'') + ''v'' ''L''(''u''), where ''r'' = ''s''(''u'') is the space curve and ''L''(''u'') is the unit vector of the ruling at ''u'' = ''u''. All the vectors ''L''(''u'') are parallel to the same plane, called the '' directrix plane'' of the surface. This can be characterized by the condition: the mixed product 'L''(''u''), ''L' ''(''u''), ''L" ''(''u'')= The parametric equations of the Catalan surface ar x=f(u)+vi(u),\quad y=g(u)+vj(u),\quad z=h(u)+vk(u) \, Special cases If all the rulings of a Catalan surface intersect a fixed Line (geometry), line, then the surface is called a conoid. Catalan proved that the helicoid and the plane were the only ruled ''Ruled'' is the fifth full-length LP by The Giraffes. Drums, bass and ... [...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] |
Hyperbolic Paraboloid
In geometry, a paraboloid is a quadric surface that has exactly one axis of symmetry and no center of symmetry. The term "paraboloid" is derived from parabola, which refers to a conic section that has a similar property of symmetry. Every plane section of a paraboloid by a plane parallel to the axis of symmetry is a parabola. The paraboloid is hyperbolic if every other plane section is either a hyperbola, or two crossing lines (in the case of a section by a tangent plane). The paraboloid is elliptic if every other nonempty plane section is either an ellipse, or a single point (in the case of a section by a tangent plane). A paraboloid is either elliptic or hyperbolic. Equivalently, a paraboloid may be defined as a quadric surface that is not a cylinder, and has an implicit equation whose part of degree two may be factored over the complex numbers into two different linear factors. The paraboloid is hyperbolic if the factors are real; elliptic if the factors are complex conj ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Intracellular Parasite
Intracellular parasites are microparasites that are capable of growing and reproducing inside the cells of a host. Types of parasites There are two main types of intracellular parasites: Facultative and Obligate. Facultative intracellular parasites are capable of living and reproducing in or outside of host cells. Obligate intracellular parasites, on the other hand, need a host cell to live and reproduce. Many of these types of cells require specialized host types, and invasion of host cells occurs in different ways. Facultative Facultative intracellular parasites are capable of living and reproducing either inside or outside cells. Bacterial examples include: *''Bartonella henselae'' *'' Francisella tularensis'' *''Listeria monocytogenes'' * ''Salmonella'' Typhi *'' Brucella'' *''Legionella'' *''Mycobacterium'' *'' Nocardia'' *'' Neisseria'' *''Rhodococcus equi'' *''Yersinia'' *''Staphylococcus aureus'' Fungal examples include: *'' Histoplasma capsulatum''. *''Cryptococcus ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros. The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are: plane algebraic curves, which include lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscates and Cassini ovals. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the singular points, the inflection points and the points at infinity. More advanced questions involve the topo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Plücker Conoid
{{disambiguation * 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 di ..., German mathematician and physicist * 29643 Plücker, main-belt asteroid * Plücker Line * Plücker matrix ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
