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Henneberg Surface
In differential geometry, the Henneberg surface is a non-orientable minimal surface named after Lebrecht Henneberg. It has parametric equation :\begin x(u,v) &= 2\cos(v)\sinh(u) - (2/3)\cos(3v)\sinh(3u)\\ y(u,v) &= 2\sin(v)\sinh(u) + (2/3)\sin(3v)\sinh(3u)\\ z(u,v) &= 2\cos(2v)\cosh(2u) \end and can be expressed as an order-15 algebraic surface. It can be viewed as an immersion of a punctured projective plane. Up until 1981 it was the only known non-orientable minimal surface. The surface contains a semicubical parabola ("Neile's parabola") and can be derived from solving the corresponding Björling problem In differential geometry, the Björling problem is the problem of finding a minimal surface passing through a given curve with prescribed normal (or tangent planes). The problem was posed and solved by Swedish mathematician Emanuel Gabriel Björl ....Kai-Wing Fung, Minimal Surfaces as Isotropic Curves in C3: Associated minimal surfaces and the Björling's problem. MIT BA T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Henneberg Surface
In differential geometry, the Henneberg surface is a non-orientable minimal surface named after Lebrecht Henneberg. It has parametric equation :\begin x(u,v) &= 2\cos(v)\sinh(u) - (2/3)\cos(3v)\sinh(3u)\\ y(u,v) &= 2\sin(v)\sinh(u) + (2/3)\sin(3v)\sinh(3u)\\ z(u,v) &= 2\cos(2v)\cosh(2u) \end and can be expressed as an order-15 algebraic surface. It can be viewed as an immersion of a punctured projective plane. Up until 1981 it was the only known non-orientable minimal surface. The surface contains a semicubical parabola ("Neile's parabola") and can be derived from solving the corresponding Björling problem In differential geometry, the Björling problem is the problem of finding a minimal surface passing through a given curve with prescribed normal (or tangent planes). The problem was posed and solved by Swedish mathematician Emanuel Gabriel Björl ....Kai-Wing Fung, Minimal Surfaces as Isotropic Curves in C3: Associated minimal surfaces and the Björling's problem. MIT BA T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry as far back as antiquity. It also relates to astronomy, the geodesy of the Earth, and later the study of hyperbolic geometry by Lobachevsky. The simplest examples of smooth spaces are the plane and space curves and surfaces in the three-dimensional Euclidean space, and the study of these shapes formed the basis for development of modern differential geometry during the 18th and 19th centuries. Since the late 19th century, differential geometry has grown into a field concerned more generally with geometric structures on differentiable manifolds. A geometric structure is one which defines some notion of size, distance, shape, volume, or other rigidifying structu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Non-orientable
In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space is orientable if such a consistent definition exists. In this case, there are two possible definitions, and a choice between them is an orientation of the space. Real vector spaces, Euclidean spaces, and spheres are orientable. A space is non-orientable if "clockwise" is changed into "counterclockwise" after running through some loops in it, and coming back to the starting point. This means that a geometric shape, such as , that moves continuously along such a loop is changed into its own mirror image . A Möbius strip is an example of a non-orientable space. Various equivalent formulations of orientability can be given, depending on the desired application and level of generality. Formulations applicable to general topological manifolds ofte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minimal Surface
In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below). The term "minimal surface" is used because these surfaces originally arose as surfaces that minimized total surface area subject to some constraint. Physical models of area-minimizing minimal surfaces can be made by dipping a wire frame into a soap solution, forming a soap film, which is a minimal surface whose boundary is the wire frame. However, the term is used for more general surfaces that may self-intersect or do not have constraints. For a given constraint there may also exist several minimal surfaces with different areas (for example, see minimal surface of revolution): the standard definitions only relate to a local optimum, not a global optimum. Definitions Minimal surfaces can be defined in several equivalent ways in R3. The fact that they are equivalent serves to demonstrate how minimal surface theory lies at the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ernst Lebrecht Henneberg
Ernst Lebrecht Henneberg (27 September 1850 – 29 April 1933) was a German professor of mechanics and mathematics. Life Ernst Lebrecht Henneberg was born in Wolfenbüttel in 1850 to Heinrich Henneberg and Sophie Rimpau. From 1870 until 1876 he studied mathematics in Zürich, Heidelberg, and Berlin, receiving his doctorate in 1875 from Hermann Schwarz while in Zürich. After living in Zürich from 1876 until 1878, he became an associate professor for descriptive and synthetic geometry and graphic statics at TU Darmstat. Not soon after in 1879, he became a professor of mechanics at the university. From 1887 to 1890 Henneberg was the dean of the Electrical Engineering school at TU Darmstat. In 1888, Henneburg was elected as a member of the Leopoldina. By 1890, he was one of the founders of the German Association of Mathematicians. From 1890 to 1891 Henneberg had become the dean of the Mathematics and Sciences school, while also acting as the rector Rector (Latin for the member ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Immersion (mathematics)
In mathematics, an immersion is a differentiable function between differentiable manifolds whose differential (or pushforward) is everywhere injective. Explicitly, is an immersion if :D_pf : T_p M \to T_N\, is an injective function at every point ''p'' of ''M'' (where ''TpX'' denotes the tangent space of a manifold ''X'' at a point ''p'' in ''X''). Equivalently, ''f'' is an immersion if its derivative has constant rank equal to the dimension of ''M'': :\operatorname\,D_p f = \dim M. The function ''f'' itself need not be injective, only its derivative must be. A related concept is that of an embedding. A smooth embedding is an injective immersion that is also a topological embedding, so that ''M'' is diffeomorphic to its image in ''N''. An immersion is precisely a local embedding – that is, for any point there is a neighbourhood, , of ''x'' such that is an embedding, and conversely a local embedding is an immersion. For infinite dimensional manifolds, this is sometimes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Plane
In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that do not intersect. A projective plane can be thought of as an ordinary plane equipped with additional "points at infinity" where parallel lines intersect. Thus ''any'' two distinct lines in a projective plane intersect at exactly one point. Renaissance artists, in developing the techniques of drawing in perspective, laid the groundwork for this mathematical topic. The archetypical example is the real projective plane, also known as the extended Euclidean plane. This example, in slightly different guises, is important in algebraic geometry, topology and projective geometry where it may be denoted variously by , RP2, or P2(R), among other notations. There are many other projective planes, both infinite, such as the complex projective plane, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semicubical Parabola
In mathematics, a cuspidal cubic or semicubical parabola is an algebraic plane curve that has an implicit equation of the form : y^2 - a^2 x^3 = 0 (with ) in some Cartesian coordinate system. Solving for leads to the ''explicit form'' : y = \pm a x^, which imply that every real point satisfies . The exponent explains the term ''semicubical parabola''. (A parabola can be described by the equation .) Solving the implicit equation for yields a second ''explicit form'' :x = \left(\frac\right)^. The parametric equation : \quad x = t^2, \quad y = a t^3 can also be deduced from the implicit equation by putting t = \frac. . The semicubical parabolas have a cuspidal singularity; hence the name of ''cuspidal cubic''. The arc length of the curve was calculated by the English mathematician William Neile and published in 1657 (see section History). Properties of semicubical parabolas Similarity Any semicubical parabola (t^2,at^3) is similar to the ''semicubical unit parabola'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Björling Problem
In differential geometry, the Björling problem is the problem of finding a minimal surface passing through a given curve with prescribed normal (or tangent planes). The problem was posed and solved by Swedish mathematician Emanuel Gabriel Björling, with further refinement by Hermann Schwarz. The problem can be solved by extending the surface from the curve using complex analytic continuation. If c(s) is a real analytic curve in \mathbb^3 defined over an interval ''I'', with c'(s)\neq 0 and a vector field n(s) along ''c'' such that , , n(t), , =1 and c'(t)\cdot n(t)=0, then the following surface is minimal: :X(u,v) = \Re \left ( c(w) - i \int_^w n(w)\times c'(w) \, dw \right) where w = u+iv \in \Omega, u_0\in I, and I \subset \Omega is a simply connected domain where the interval is included and the power series expansions of c(s) and n(s) are convergent. A classic example is Catalan's minimal surface, which passes through a cycloid curve. Applying the method to a semicubical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minimal Surfaces
In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below). The term "minimal surface" is used because these surfaces originally arose as surfaces that minimized total surface area subject to some constraint. Physical models of area-minimizing minimal surfaces can be made by dipping a wire frame into a soap solution, forming a soap film, which is a minimal surface whose boundary is the wire frame. However, the term is used for more general surfaces that may self-intersect or do not have constraints. For a given constraint there may also exist several minimal surfaces with different areas (for example, see minimal surface of revolution): the standard definitions only relate to a local optimum, not a global optimum. Definitions Minimal surfaces can be defined in several equivalent ways in R3. The fact that they are equivalent serves to demonstrate how minimal surface theory lies at the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
