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Bipolar Coordinates
Bipolar coordinates are a two-dimensional orthogonal coordinate system based on the Apollonian circles.Eric W. Weisstein, Concise Encyclopedia of Mathematics CD-ROM, ''Bipolar Coordinates'', CD-ROM edition 1.0, May 20, 1999 Confusingly, the same term is also sometimes used for two-center bipolar coordinates. There is also a third system, based on two poles ( biangular coordinates). The term "bipolar" is further used on occasion to describe other curves having two singular points (foci), such as ellipses, hyperbolas, and Cassini ovals. However, the term ''bipolar coordinates'' is reserved for the coordinates described here, and never used for systems associated with those other curves, such as elliptic coordinates. Definition The system is based on two foci ''F''1 and ''F''2. Referring to the figure at right, the ''σ''-coordinate of a point ''P'' equals the angle ''F''1 ''P'' ''F''2, and the ''τ''-coordinate equals the natural logarithm of the ratio of the di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bipolar Sigma Isosurfaces
Bipolar may refer to: Astronomy * Bipolar nebula, a distinctive nebular formation * Bipolar outflow, two continuous flows of gas from the poles of a star Mathematics * Bipolar coordinates, a two-dimensional orthogonal coordinate system * Bipolar set, a derivative of a polar set * Bipolar theorem, a theorem in convex analysis which provides necessary and sufficient conditions for a cone to be equal to its bipolar Medicine * Bipolar disorder, a mental disorder that causes periods of depression and periods of elevated mood ** Bipolar I disorder, a bipolar spectrum disorder characterized by the occurrence of at least one manic or mixed episode ** Bipolar II disorder, a bipolar spectrum disorder characterized by at least one episode of hypomania and at least one episode of major depression ** Bipolar disorder not otherwise specified, a diagnosis for bipolar disorder when it does not fall within the other established sub-types * Bipolar neuron, a type of neuron which has two extensio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theresa M
Teresa (also Theresa, Therese; french: Thérèse) is a feminine given name. It originates in the Iberian Peninsula in late antiquity. Its derivation is uncertain, it may be derived from Greek θερίζω (''therízō'') "to harvest or reap", or from θέρος (''theros'') "summer". It is first recorded in the form ''Therasia'', the name of Therasia of Nola, an aristocrat of the 4th century. Its popularity outside of Iberia increased because of saint Teresa of Ávila, and more recently Thérèse of Lisieux and Mother Teresa. In the United States it was ranked as the 852nd most popular name for girls born in 2008, down from 226th in 1992 (it ranked 65th in 1950, and 102nd in 1900). Spelled "Teresa," it was the 580th most popular name for girls born in 2008, down from 206th in 1992 (it ranked 81st in 1950, and 220th in 1900). People In aristocracy: *Teresa of Portugal (other) ** Theresa, Countess of Portugal (1080–1130), mother of Afonso Henriques, the first K ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Toroidal Coordinates
Toroidal coordinates are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional bipolar coordinate system about the axis that separates its two foci. Thus, the two foci F_1 and F_2 in bipolar coordinates become a ring of radius a in the xy plane of the toroidal coordinate system; the z-axis is the axis of rotation. The focal ring is also known as the reference circle. Definition The most common definition of toroidal coordinates (\tau, \sigma, \phi) is : x = a \ \frac \cos \phi : y = a \ \frac \sin \phi : z = a \ \frac together with \mathrm(\sigma)=\mathrm(z). The \sigma coordinate of a point P equals the angle F_ P F_ and the \tau coordinate equals the natural logarithm of the ratio of the distances d_ and d_ to opposite sides of the focal ring : \tau = \ln \frac. The coordinate ranges are -\pi<\sigma\le\pi, and Coordinate surfaces Surfaces of con ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bispherical Coordinates
Bispherical coordinates are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional bipolar coordinate system about the axis that connects the two foci. Thus, the two foci F_ and F_ in bipolar coordinates remain points (on the z-axis, the axis of rotation) in the bispherical coordinate system. Definition The most common definition of bispherical coordinates (\tau, \sigma, \phi) is :\begin x &= a \ \frac \cos \phi, \\ y &= a \ \frac \sin \phi, \\ z &= a \ \frac, \end where the \sigma coordinate of a point P equals the angle F_ P F_ and the \tau coordinate equals the natural logarithm of the ratio of the distances d_ and d_ to the foci : \tau = \ln \frac The coordinates ranges are -∞ < < ∞, 0 ≤ ≤ and 0 ≤ ≤ 2. Coordinate surfaces Surfaces of constant correspond to intersecting tori of different r ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bipolar Cylindrical Coordinates
Bipolar cylindrical coordinates are a three-dimensional orthogonal coordinate system that results from projecting the two-dimensional bipolar coordinate system in the perpendicular z-direction. The two lines of foci F_ and F_ of the projected Apollonian circles are generally taken to be defined by x=-a and x=+a, respectively, (and by y=0) in the Cartesian coordinate system. The term "bipolar" is often used to describe other curves having two singular points (foci), such as ellipses, hyperbolas, and Cassini ovals. However, the term ''bipolar coordinates'' is never used to describe coordinates associated with those curves, e.g., elliptic coordinates. Basic definition The most common definition of bipolar cylindrical coordinates (\sigma, \tau, z) is : x = a \ \frac : y = a \ \frac : z = \ z where the \sigma coordinate of a point P equals the angle F_ P F_ and the \tau coordinate equals the natural logarithm of the ratio of the distances d_ and d_ to the focal lines ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polar Plotter
A polar plotter also known as polargraph or Kritzler is a plotter which uses two-center bipolar coordinates to produce vector drawings using a pen suspended from strings connected to two pulleys at the top of the plotting surface. This gives it two degrees of freedom and allows it to scale to fairly large drawings simply by moving the motors further apart and using longer strings. Some polar plotters will integrate a raising mechanism for the pen which allows lines to be broken while drawing. The system has been used by a number of artists and makers, including: * Jürg Lehni & Uli Franke (2002) * Ben Leduc-Mills (2010) * Alex Weber (2011) * Harvey Moon * Sandy Nobel (2012) * Maslow CNC References {{Reflist External links Polargraph Drawing MachineDer Kritzlerpolargraph.co.ukMelt (Open Source Polargraph Controller) Plotters ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Electric Field
An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field for a system of charged particles. Electric fields originate from electric charges and time-varying electric currents. Electric fields and magnetic fields are both manifestations of the electromagnetic field, one of the four fundamental interactions (also called forces) of nature. Electric fields are important in many areas of physics, and are exploited in electrical technology. In atomic physics and chemistry, for instance, the electric field is the attractive force holding the atomic nucleus and electrons together in atoms. It is also the force responsible for chemical bonding between atoms that result in molecules. The electric field is defined as a vector field that associates to each point in space the electrostatic ( Coulomb) for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Separation Of Variables
In mathematics, separation of variables (also known as the Fourier method) is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs on a different side of the equation. Ordinary differential equations (ODE) Suppose a differential equation can be written in the form :\frac f(x) = g(x)h(f(x)) which we can write more simply by letting y = f(x): :\frac=g(x)h(y). As long as ''h''(''y'') ≠ 0, we can rearrange terms to obtain: : = g(x) \, dx, so that the two variables ''x'' and ''y'' have been separated. ''dx'' (and ''dy'') can be viewed, at a simple level, as just a convenient notation, which provides a handy mnemonic aid for assisting with manipulations. A formal definition of ''dx'' as a differential (infinitesimal) is somewhat advanced. Alternative notation Those who dislike Leibniz's notation may prefer to write this as :\frac \frac = g(x), but that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Helmholtz Equation
In mathematics, the eigenvalue problem for the Laplace operator is known as the Helmholtz equation. It corresponds to the linear partial differential equation \nabla^2 f = -k^2 f, where is the Laplace operator (or "Laplacian"), is the eigenvalue, and is the (eigen)function. When the equation is applied to waves, is known as the wave number. The Helmholtz equation has a variety of applications in physics, including the wave equation and the diffusion equation, and it has uses in other sciences. Motivation and uses The Helmholtz equation often arises in the study of physical problems involving partial differential equations (PDEs) in both space and time. The Helmholtz equation, which represents a time-independent form of the wave equation, results from applying the technique of separation of variables to reduce the complexity of the analysis. For example, consider the wave equation \left(\nabla^2-\frac\frac\right) u(\mathbf,t)=0. Separation of variables begins by assumi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laplace's Equation
In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as \nabla^2\! f = 0 or \Delta f = 0, where \Delta = \nabla \cdot \nabla = \nabla^2 is the Laplace operator,The delta symbol, Δ, is also commonly used to represent a finite change in some quantity, for example, \Delta x = x_1 - x_2. Its use to represent the Laplacian should not be confused with this use. \nabla \cdot is the divergence operator (also symbolized "div"), \nabla is the gradient operator (also symbolized "grad"), and f (x, y, z) is a twice-differentiable real-valued function. The Laplace operator therefore maps a scalar function to another scalar function. If the right-hand side is specified as a given function, h(x, y, z), we have \Delta f = h. This is called Poisson's equation, a generalization of Laplace's equation. Laplace's equation and Poisson's equation are the simplest exa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Differential Equations
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to how is thought of as an unknown number to be solved for in an algebraic equation like . However, it is usually impossible to write down explicit formulas for solutions of partial differential equations. There is, correspondingly, a vast amount of modern mathematical and scientific research on methods to numerically approximate solutions of certain partial differential equations using computers. Partial differential equations also occupy a large sector of pure mathematical research, in which the usual questions are, broadly speaking, on the identification of general qualitative features of solutions of various partial differential equations, such as existence, uniqueness, regularity, and stability. Among the many open questions are the e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
