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Conoid Bench George Nakashima 1977
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)=(\cos ... [...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] |
Simpson's Rule
In numerical integration, Simpson's rules are several approximations for definite integrals, named after Thomas Simpson (1710–1761). The most basic of these rules, called Simpson's 1/3 rule, or just Simpson's rule, reads \int_a^b f(x) \, dx \approx \frac \left (a) + 4f\left(\frac\right) + f(b)\right In German and some other languages, it is named after Johannes Kepler, who derived it in 1615 after seeing it used for wine barrels (barrel rule, ). The approximate equality in the rule becomes exact if is a polynomial up to and including 3rd degree. If the 1/3 rule is applied to ''n'' equal subdivisions of the integration range 'a'', ''b'' one obtains the composite Simpson's 1/3 rule. Points inside the integration range are given alternating weights 4/3 and 2/3. Simpson's 3/8 rule, also called Simpson's second rule, requires one more function evaluation inside the integration range and gives lower error bounds, but does not improve on order of the error. If the ... [...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] |
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] |
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] |
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] |
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] |
Tangent Plane
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More precisely, a straight line is said to be a tangent of a curve at a point if the line passes through the point on the curve and has slope , where ''f'' is the derivative of ''f''. A similar definition applies to space curves and curves in ''n''-dimensional Euclidean space. As it passes through the point where the tangent line and the curve meet, called the point of tangency, the tangent line is "going in the same direction" as the curve, and is thus the best straight-line approximation to the curve at that point. The tangent line to a point on a differentiable curve can also be thought of as a ''tangent line approximation'', the graph of the affine function that best approximates the original function at the given point. Similarly, th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
On Conoids And Spheroids
''On Conoids and Spheroids'' ( grc, Περὶ κωνοειδέων καὶ σφαιροειδέων) is a surviving work by the Greek mathematician and engineer Archimedes ( 287 BC – 212 BC). Consisting of 32 propositions, the work explores properties of and theorems related to the solids generated by revolution of conic sections about their axes, including paraboloids, hyperboloids, and spheroids. The principal result of the work is comparing the volume of any segment cut off by a plane with the volume of a cone A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex. A cone is formed by a set of line segments, half-lines, or lines co ... with equal base and axis. The work is addressed to Dositheus of Pelusium. Footnotes References * External links ON CONOIDS AND SPHEROIDS - The Works of Archimedes* * Ancient Greek mathematical works Eucl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
