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Confocal Conic Sections
In geometry, two conic sections are called confocal, if they have the same foci. Because ellipses and hyperbolas possess two foci, there are confocal ellipses, confocal hyperbolas and confocal mixtures of ellipses and hyperbolas. In the mixture of confocal ellipses and hyperbolas, any ellipse intersects any hyperbola orthogonally (at right angles). Parabolas possess only one focus, so, by convention, confocal parabolas have the same focus ''and'' the same axis of symmetry. Consequently, any point not on the axis of symmetry lies on two confocal parabolas which intersect orthogonally (see below). The formal extension of the concept of confocal conics to surfaces leads to confocal quadrics. Confocal ellipses An ellipse which is not a circle is uniquely determined by its foci F_1,\; F_2 and a point not on the major axis (see the definition of an ellipse as a locus of points). The pencil of confocal ellipses with the foci F_1=(c,0),\; F_2=(-c,0) can be described by the equation * \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperboloid
In geometry, a hyperboloid of revolution, sometimes called a circular hyperboloid, is the surface generated by rotating a hyperbola around one of its principal axes. A hyperboloid is the surface obtained from a hyperboloid of revolution by deforming it by means of directional scalings, or more generally, of an affine transformation. A hyperboloid is a quadric surface, that is, a surface defined as the zero set of a polynomial of degree two in three variables. Among quadric surfaces, a hyperboloid is characterized by not being a cone or a cylinder, having a center of symmetry, and intersecting many planes into hyperbolas. A hyperboloid has three pairwise perpendicular axes of symmetry, and three pairwise perpendicular planes of symmetry. Given a hyperboloid, one can choose a Cartesian coordinate system such that the hyperboloid is defined by one of the following equations: : + - = 1, or : + - = -1. The coordinate axes are axes of symmetry of the hyperboloid and the ori ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ellipsoid
An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation. An ellipsoid is a quadric surface; that is, a surface that may be defined as the zero set of a polynomial of degree two in three variables. Among quadric surfaces, an ellipsoid is characterized by either of the two following properties. Every planar cross section is either an ellipse, or is empty, or is reduced to a single point (this explains the name, meaning "ellipse-like"). It is bounded, which means that it may be enclosed in a sufficiently large sphere. An ellipsoid has three pairwise perpendicular axes of symmetry which intersect at a center of symmetry, called the center of the ellipsoid. The line segments that are delimited on the axes of symmetry by the ellipsoid are called the ''principal axes'', or simply axes of the ellipsoid. If the three axes have different lengths, the figure is a triaxial ellipsoid (r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Otto Staude
Otto is a masculine German given name and a surname. It originates as an Old High German short form (variants ''Audo'', '' Odo'', ''Udo'') of Germanic names beginning in ''aud-'', an element meaning "wealth, prosperity". The name is recorded from the 7th century ( Odo, son of Uro, courtier of Sigebert III). It was the name of three 10th-century German kings, the first of whom was Otto I the Great, the first Holy Roman Emperor, founder of the Ottonian dynasty. The Gothic form of the prefix was ''auda-'' (as in e.g. '' Audaþius''), the Anglo-Saxon form was ''ead-'' (as in e.g. '' Eadmund''), and the Old Norse form was '' auð-''. The given name Otis arose from an English surname, which was in turn derived from ''Ode'', a variant form of ''Odo, Otto''. Due to Otto von Bismarck, the given name ''Otto'' was strongly associated with the German Empire in the later 19th century. It was comparatively frequently given in the United States (presumably in German American families) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elliptical Integral
In integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied by Giulio Fagnano and Leonhard Euler (). Their name originates from their originally arising in connection with the problem of finding the arc length of an ellipse. Modern mathematics defines an "elliptic integral" as any function which can be expressed in the form f(x) = \int_^ R \left(t, \sqrt \right) \, dt, where is a rational function of its two arguments, is a polynomial of degree 3 or 4 with no repeated roots, and is a constant. In general, integrals in this form cannot be expressed in terms of elementary functions. Exceptions to this general rule are when has repeated roots, or when contains no odd powers of or if the integral is pseudo-elliptic. However, with the appropriate reduction formula, every elliptic integral can be brought into a form that involves integrals over rational functions and the three Legend ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Charles Graves (bishop)
Charles Graves (6 December 1812 – 17 July 1899) was an Irish mathematician, academic, and clergyman. He was Erasmus Smith's Professor of Mathematics at Trinity College Dublin (1843–1862), and was president of the Royal Irish Academy (1861–1866). He served as dean of the Chapel Royal at Dublin Castle, and later as Bishop of Limerick, Ardfert and Aghadoe. He was the brother of both the jurist and mathematician John Graves, and the writer and clergyman Robert Perceval Graves. Early life Born at 12 Fitzwilliam Square, Dublin, the son of John Crosbie Graves (1776–1835), Chief Police Magistrate for Dublin, by his wife Helena Perceval, the daughter and co-heiress of the Revd Charles Perceval (1751–1795) of Bruhenny, County Cork. Helena enjoyed the patronage of John Freeman-Mitford, 1st Baron Redesdale, who married her second cousin, a daughter of John Perceval, 2nd Earl of Egmont. Educated at Trinity College Dublin, he was elected a Scholar in classics, and in 1834 gra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conformal Map
In mathematics, a conformal map is a function that locally preserves angles, but not necessarily lengths. More formally, let U and V be open subsets of \mathbb^n. A function f:U\to V is called conformal (or angle-preserving) at a point u_0\in U if it preserves angles between directed curves through u_0, as well as preserving orientation. Conformal maps preserve both angles and the shapes of infinitesimally small figures, but not necessarily their size or curvature. The conformal property may be described in terms of the Jacobian derivative matrix of a coordinate transformation. The transformation is conformal whenever the Jacobian at each point is a positive scalar times a rotation matrix (orthogonal with determinant one). Some authors define conformality to include orientation-reversing mappings whose Jacobians can be written as any scalar times any orthogonal matrix. For mappings in two dimensions, the (orientation-preserving) conformal mappings are precisely the locally i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the imaginary numbers. The complex plane allows a geometric interpretation of complex numbers. Under addition, they add like vectors. The multiplication of two complex numbers can be expressed more easily in polar coordinates—the magnitude or ''modulus'' of the product is the product of the two absolute values, or moduli, and the angle or ''argument'' of the product is the sum of the two angles, or arguments. In particular, multiplication by a complex number of modulus 1 acts as a rotation. The complex plane is sometimes known as the Argand plane or Gauss plane. Notational conventions Complex numbers In complex analysis, the complex numbers are customarily represented by the symbol ''z'', which can be separated into its real (''x'') and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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