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Constant Mean Curvature
In differential geometry, constant-mean-curvature (CMC) surfaces are surfaces with constant mean curvature.Carl Johan Lejdfors, Surfaces of Constant Mean Curvature. Master’s thesis Lund University, Centre for Mathematical Sciences Mathematics 2003:E1/ref> This includes minimal surfaces as a subset, but typically they are treated as special case. Note that these surfaces are generally different from constant Gaussian curvature surfaces, with the important exception of the sphere. History In 1841 Charles-Eugène Delaunay, Delaunay proved that the only surfaces of revolution with constant mean curvature were the surfaces obtained by rotating the Roulette (curve), roulettes of the conics. These are the plane, cylinder, sphere, the catenoid, the unduloid and nodoid. In 1853 J. H. Jellet showed that if S is a compact star-shaped surface in \R^3 with constant mean curvature, then it is the standard sphere. Subsequently, Aleksandr Danilovich Aleksandrov, A. D. Alexandrov proved that a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nodoid
In differential geometry, a nodoid is a surface of revolution with Constant mean curvature surface, constant nonzero mean curvature obtained by rolling a hyperbola along a fixed line, tracing the Focus (geometry), focus, and revolving the resulting nodary curve around the line.. References External linksWolfram Demonstrations: Delaunay Nodoids {{geometry-stub Surfaces ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. If the axis of revolution does not touch the circle, the surface has a ring shape and is called a torus of revolution. If the axis of revolution is tangent to the circle, the surface is a horn torus. If the axis of revolution passes twice through the circle, the surface is a spindle torus. If the axis of revolution passes through the center of the circle, the surface is a degenerate torus, a double-covered sphere. If the revolved curve is not a circle, the surface is called a ''toroid'', as in a square toroid. Real-world objects that approximate a torus of revolution include swim rings, inner tubes and ringette rings. Eyeglass lenses that combine spherical and cylindrical correction are toric lenses. A torus should not be confused with a '' solid torus'', which is formed by r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Air-supported Structures
An air-supported (or air-inflated) structure is any building that derives its structural integrity from the use of internal pressurized air to inflate a pliable material (i.e. structural fabric) envelope, so that air is the main support of the structure, and where access is via airlocks. The first air-supported structure built in history was the radome manufactured at the Cornell Aeronautical Laboratory in 1948 by Walter Bird. The concept was implemented on a large scale by David H. Geiger with the United States pavilion at Expo '70 in Osaka, Japan in 1970. It is usually dome-shaped, since this shape creates the greatest volume for the least amount of material. To maintain structural integrity, the structure must be pressurized such that the internal pressure equals or exceeds any external pressure being applied to the structure (i.e. wind pressure). The structure does not have to be airtight to retain structural integrity—as long as the pressurization system that supplies in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Block Copolymers
In polymer chemistry, a copolymer is a polymer derived from more than one species of monomer. The polymerization of monomers into copolymers is called copolymerization. Copolymers obtained from the copolymerization of two monomer species are sometimes called ''bipolymers''. Those obtained from three and four monomers are called ''terpolymers'' and ''quaterpolymers'', respectively. Copolymers can be characterized by a variety of techniques such as NMR spectroscopy and size-exclusion chromatography to determine the molecular size, weight, properties, and composition of the material. Commercial copolymers include acrylonitrile butadiene styrene (ABS), styrene/butadiene co-polymer (SBR), nitrile rubber, styrene-acrylonitrile, styrene-isoprene-styrene (SIS) and ethylene-vinyl acetate, all of which are formed by chain-growth polymerization. Another production mechanism is step-growth polymerization, which is used to produce the nylon-12/6/66 copolymer of nylon 12, nylon 6 and ny ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triply Periodic Minimal Surface
In differential geometry, a triply periodic minimal surface (TPMS) is a minimal surface in ℝ3 that is invariant under a rank-3 lattice of translations. These surfaces have the symmetries of a crystallographic group. Numerous examples are known with cubic, tetragonal, rhombohedral, and orthorhombic symmetries. Monoclinic and triclinic examples are certain to exist, but have proven hard to parametrise. TPMS are of relevance in natural science. TPMS have been observed as biological membranes, as block copolymers, equipotential surfaces in crystals etc. They have also been of interest in architecture, design and art. Properties Nearly all studied TPMS are free of self-intersections (i.e. embedded in ℝ3): from a mathematical standpoint they are the most interesting (since self-intersecting surfaces are trivially abundant). All connected TPMS have genus ≥ 3, and in every lattice there exist orientable embedded TPMS of every genus ≥3. Embedded TPMS are orientable and divid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Superhydrophobic
Ultrahydrophobic (or superhydrophobic) surfaces are highly hydrophobic, i.e., extremely difficult to wet. The contact angles of a water droplet on an ultrahydrophobic material exceed 150°. This is also referred to as the lotus effect, after the superhydrophobic leaves of the lotus plant. A droplet striking these kinds of surfaces can fully rebound like an elastic ball. Interactions of bouncing drops can be further reduced using special superhydrophobic surfaces that promote symmetry breaking, pancake bouncing or waterbowl bouncing. Theory In 1805, Thomas Young defined the contact angle ''θ'' by analysing the forces acting on a fluid droplet resting on a smooth solid surface surrounded by a gas. :\gamma_\ =\gamma_+\gamma_\cos where :\gamma_\ = Interfacial tension between the solid and gas :\gamma_\ = Interfacial tension between the solid and liquid :\gamma_\ = Interfacial tension between the liquid and gas ''θ'' can be measured using a contact angle goniometer. Wenzel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Young–Laplace Equation
In physics, the Young–Laplace equation () is an algebraic equation that describes the capillary pressure difference sustained across the interface between two static fluids, such as water and air, due to the phenomenon of surface tension or wall tension, although use of the latter is only applicable if assuming that the wall is very thin. The Young–Laplace equation relates the pressure difference to the shape of the surface or wall and it is fundamentally important in the study of static capillary surfaces. It's a statement of normal stress balance for static fluids meeting at an interface, where the interface is treated as a surface (zero thickness): \begin \Delta p &= -\gamma \nabla \cdot \hat n \\ &= -2\gamma H_f \\ &= -\gamma \left(\frac + \frac\right) \end where \Delta p is the Laplace pressure, the pressure difference across the fluid interface (the exterior pressure minus the interior pressure), \gamma is the surface tension (or wall tension), \hat n is the unit norm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Soap Bubbles
A soap bubble is an extremely thin film of soap or detergent and water enclosing air that forms a hollow sphere with an iridescent surface. Soap bubbles usually last for only a few seconds before bursting, either on their own or on contact with another object. They are often used for children's enjoyment, but they are also used in artistic performances. Assembling many bubbles results in foam. When light shines onto a bubble it appears to change colour. Unlike those seen in a rainbow, which arise from differential refraction, the colours seen in a soap bubble arise from light wave interference, reflecting off the front and back surfaces of the thin soap film. Depending on the thickness of the film, different colours interfere constructively and destructively. Mathematics Soap bubbles are physical examples of the complex mathematical problem of minimal surface. They will assume the shape of least surface area possible containing a given volume. A true minimal surface i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Differential Geometry
Discrete differential geometry is the study of discrete counterparts of notions in differential geometry. Instead of smooth curves and surfaces, there are polygons, meshes, and simplicial complexes. It is used in the study of computer graphics, geometry processing and topological combinatorics. See also *Discrete Laplace operator *Discrete exterior calculus *Discrete Morse theory *Topological combinatorics *Spectral shape analysis * Abstract differential geometry *Analysis on fractals *Discrete calculus Discrete calculus or the calculus of discrete functions, is the mathematical study of ''incremental'' change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations. The word ''ca ... ReferencesDiscrete differential geometry Forum* * * Alexander I. Bobenko, Yuri B. Suris (2008), "Discrete Differential Geometry", American Mathematical Society, Differential geometry Simplicial sets {{differential-geomet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ulrich Pinkall
Ulrich Pinkall (born 1955) is a German mathematician, specializing in differential geometry and computer graphics. Pinkall studied mathematics at the University of Freiburg with a Diplom in 1979 and a doctorate in 1982 with thesis ''Dupin'sche Hyperflächen'' (Dupin's hypersurfaces) under the supervision of Martin Barner. Pinkall was then a research assistant in Freiburg until 1984 and from 1984 to 1986 at the Max Planck Institute for Mathematics in Bonn. In 1985 he completed his habilitation in Bonn with thesis ''Totale Absolutkrümmung immersierter Flächen'' (Total absolute curvature of immersed surfaces). Since 1986 he is professor at TU Berlin. In 1985 he received the Otto Hahn Medal of the Max Planck Society. In 1986 he received a ''Heisenberg-Stipendium'' from the Deutsche Forschungsgemeinschaft (DFG). From 1992 to 2003 he was a speaker of the Sonderforschungsbereich (SFB) 288 (differential geometry and quantum physics). In 1998 he was an Invited Speaker with talk ''Quate ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weierstrass–Enneper Parameterization
In mathematics, the Weierstrass–Enneper parameterization of minimal surfaces is a classical piece of differential geometry. Alfred Enneper and Karl Weierstrass studied minimal surfaces as far back as 1863. Let f and g be functions on either the entire complex plane or the unit disk, where g is meromorphic and f is analytic, such that wherever g has a pole of order m, f has a zero of order 2m (or equivalently, such that the product f g^2 is holomorphic), and let c_1,c_2,c_3 be constants. Then the surface with coordinates (x_1, x_2, x_3) is minimal, where the x_k are defined using the real part of a complex integral, as follows: \begin x_k(\zeta) &= \Re \left\ + c_k , \qquad k=1,2,3 \\ \varphi_1 &= f(1-g^2)/2 \\ \varphi_2 &= \mathbf f(1+g^2)/2 \\ \varphi_3 &= fg \end The converse is also true: every nonplanar minimal surface defined over a simply connected domain can be given a parametrization of this type. For example, Enneper's surface has , . Parametric surface of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harmonic Map
In the mathematical field of differential geometry, a smooth map between Riemannian manifolds is called harmonic if its coordinate representatives satisfy a certain nonlinear partial differential equation. This partial differential equation for a mapping also arises as the Euler-Lagrange equation of a functional called the Dirichlet energy. As such, the theory of harmonic maps contains both the theory of unit-speed geodesics in Riemannian geometry and the theory of harmonic functions. Informally, the Dirichlet energy of a mapping from a Riemannian manifold to a Riemannian manifold can be thought of as the total amount that stretches in allocating each of its elements to a point of . For instance, an unstretched rubber band and a smooth stone can both be naturally viewed as Riemannian manifolds. Any way of stretching the rubber band over the stone can be viewed as a mapping between these manifolds, and the total tension involved is represented by the Dirichlet energy. Harm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
