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Focal Conics
In geometry, focal conics are a pair of curves consisting of either *an ellipse and a hyperbola, where the hyperbola is contained in a plane, which is orthogonal to the plane containing the ellipse. The vertices of the hyperbola are the foci of the ellipse and its foci are the vertices of the ellipse (see diagram). or *two parabolas, which are contained in two orthogonal planes and the vertex of one parabola is the focus of the other and vice versa. Focal conics play an essential role answering the question: "Which right circular cones contain a given ellipse or hyperbola or parabola (see below)". Focal conics are used as directrices for generating Dupin cyclides as canal surfaces in two ways. Focal conics can be seen as degenerate focal surfaces: Dupin cyclides are the only surfaces, where focal surfaces collapse to a pair of curves, namely focal conics. In Physical chemistry focal conics are used for describing geometrical properties of liquid crystals. One should not mix ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Liquid Crystal
Liquid crystal (LC) is a state of matter whose properties are between those of conventional liquids and those of solid crystals. For example, a liquid crystal may flow like a liquid, but its molecules may be oriented in a crystal-like way. There are many types of LC phases, which can be distinguished by their optical properties (such as textures). The contrasting textures arise due to molecules within one area of material ("domain") being oriented in the same direction but different areas having different orientations. LC materials may not always be in a LC state of matter (just as water may be ice or water vapor). Liquid crystals can be divided into 3 main types: * thermotropic, *lyotropic, and * metallotropic. Thermotropic and lyotropic liquid crystals consist mostly of organic molecules, although a few minerals are also known. Thermotropic LCs exhibit a phase transition into the LC phase as temperature changes. Lyotropic LCs exhibit phase transitions as a function of b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dandelin Sphere
In geometry, the Dandelin spheres are one or two spheres that are tangent both to a plane and to a cone that intersects the plane. The intersection of the cone and the plane is a conic section, and the point at which either sphere touches the plane is a focus of the conic section, so the Dandelin spheres are also sometimes called focal spheres.Taylor, Charles. ''An Introduction to the Ancient and Modern Geometry of Conics''page 196 ("focal spheres") (Deighton, Bell and co., 1881). The Dandelin spheres were discovered in 1822. They are named in honor of the French mathematician |
Semi-latus Rectum
In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special case of the ellipse, though historically it was sometimes called a fourth type. The ancient Greek mathematicians studied conic sections, culminating around 200 BC with Apollonius of Perga's systematic work on their properties. The conic sections in the Euclidean plane have various distinguishing properties, many of which can be used as alternative definitions. One such property defines a non-circular conic to be the set of those points whose distances to some particular point, called a ''focus'', and some particular line, called a ''directrix'', are in a fixed ratio, called the ''eccentricity''. The type of conic is determined by the value of the eccentricity. In analytic geometry, a conic may be defined as a plane algebraic curve o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Eccentricity
In mathematics, the eccentricity of a conic section is a non-negative real number that uniquely characterizes its shape. More formally two conic sections are similar if and only if they have the same eccentricity. One can think of the eccentricity as a measure of how much a conic section deviates from being circular. In particular: * The eccentricity of a circle is zero. * The eccentricity of an ellipse which is not a circle is greater than zero but less than 1. * The eccentricity of a parabola is 1. * The eccentricity of a hyperbola is greater than 1. * The eccentricity of a pair of lines is \infty Definitions Any conic section can be defined as the locus of points whose distances to a point (the focus) and a line (the directrix) are in a constant ratio. That ratio is called the eccentricity, commonly denoted as . The eccentricity can also be defined in terms of the intersection of a plane and a double-napped cone associated with the conic section. If the cone is oriented ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Physical Chemistry
Physical chemistry is the study of macroscopic and microscopic phenomena in chemical systems in terms of the principles, practices, and concepts of physics such as motion, energy, force, time, thermodynamics, quantum chemistry, statistical mechanics, analytical dynamics and chemical equilibria. Physical chemistry, in contrast to chemical physics, is predominantly (but not always) a supra-molecular science, as the majority of the principles on which it was founded relate to the bulk rather than the molecular or atomic structure alone (for example, chemical equilibrium and colloids). Some of the relationships that physical chemistry strives to resolve include the effects of: # Intermolecular forces that act upon the physical properties of materials ( plasticity, tensile strength, surface tension in liquids). # Reaction kinetics on the rate of a reaction. # The identity of ions and the electrical conductivity of materials. # Surface science and electrochemistry of cell membrane ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Focal Surface
For a surface in three dimension the focal surface, surface of centers or evolute is formed by taking the centers of the curvature spheres, which are the tangential spheres whose radii are the reciprocals of one of the principal curvatures at the point of tangency. Equivalently it is the surface formed by the centers of the circles which osculate the curvature lines. As the principal curvatures are the eigenvalues of the second fundamental form, there are two at each point, and these give rise to two points of the focal surface on each normal direction to the surface. Away from umbilical points, these two points of the focal surface are distinct; at umbilical points the two sheets come together. When the surface has a ridge the focal surface has a cuspidal edge, three such edges pass through an elliptical umbilic and only one through a hyperbolic umbilic. At points where the Gaussian curvature is zero, one sheet of the focal surface will have a point at infinity corresponding ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Canal Surface
In geometry and topology, a channel or canal surface is a surface formed as the envelope of a family of spheres whose centers lie on a space curve, its '' directrix''. If the radii of the generating spheres are constant, the canal surface is called a pipe surface. Simple examples are: * right circular cylinder (pipe surface, directrix is a line, the axis of the cylinder) * torus (pipe surface, directrix is a circle), * right circular cone (canal surface, directrix is a line (the axis), radii of the spheres not constant), * surface of revolution (canal surface, directrix is a line), Canal surfaces play an essential role in descriptive geometry, because in case of an orthographic projection its contour curve can be drawn as the envelope of circles. *In technical area canal surfaces can be used for ''blending surfaces'' smoothly. Envelope of a pencil of implicit surfaces Given the pencil of implicit surfaces :\Phi_c: f(,c)=0 , c\in _1,c_2/math>, two neighboring surfaces \Phi_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
