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Roman Surface
In mathematics, the Roman surface or Steiner surface is a self-intersecting mapping of the real projective plane into three-dimensional space, with an unusually high degree of symmetry. This mapping is not an immersion of the projective plane; however, the figure resulting from removing six singular points is one. Its name arises because it was discovered by Jakob Steiner when he was in Rome in 1844. The simplest construction is as the image of a sphere centered at the origin under the map f(x,y,z)=(yz,xz,xy). This gives an implicit formula of : x^2 y^2 + y^2 z^2 + z^2 x^2 - r^2 x y z = 0. \, Also, taking a parametrization of the sphere in terms of longitude () and latitude (), gives parametric equations for the Roman surface as follows: :x=r^ \cos \theta \cos \varphi \sin \varphi :y=r^ \sin \theta \cos \varphi \sin \varphi :z=r^ \cos \theta \sin \theta \cos^ \varphi The origin is a triple point, and each of the -, -, and -planes are tangential to the surface there. The ot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Steiner's Roman Surface ...
Steiner may refer to: People * Steiner (surname) *Steiner Brothers, a professional wrestling tag team * Steiner Brothers (tap-dancing trio), a 1950s–1960s Canadian act Places * Steiner, Michigan, U.S. * Steiner, Mississippi, U.S. Math and science * Steiner's theorem, or parallel axis theorem *Steiner tree *Poncelet–Steiner theorem * Steiner surface *Steiner system, a type of block design * Steiner point (other) Other uses * Steiner House, in Vienna, Austria *Steiner Studios, a film and television production studio in New York City *Franz Steiner Verlag, a German publisher See also * Army Detachment Steiner, a temporary German military unit during the 1945 Battle of Berlin during World War II *Waldorf education Waldorf education, also known as Steiner education, is based on the educational philosophy of Rudolf Steiner, the founder of anthroposophy. Its educational style is holistic, intended to develop pupils' intellectual, artistic, and practical sk ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Linear Projection
In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself (an endomorphism) such that P\circ P=P. That is, whenever P is applied twice to any vector, it gives the same result as if it were applied once (i.e. P is idempotent). It leaves its image unchanged. This definition of "projection" formalizes and generalizes the idea of graphical projection. One can also consider the effect of a projection on a geometrical object by examining the effect of the projection on points in the object. Definitions A projection on a vector space V is a linear operator P\colon V \to V such that P^2 = P. When V has an inner product and is complete, i.e. when V is a Hilbert space, the concept of orthogonality can be used. A projection P on a Hilbert space V is called an orthogonal projection if it satisfies \langle P \mathbf x, \mathbf y \rangle = \langle \mathbf x, P \mathbf y \rangle for all \mathbf x, \mathbf y \in V. A projection on a Hi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Light Bulb
Electric light is an artificial light source powered by electricity. Electric Light may also refer to: * Light fixture, a decorative enclosure for an electric light source * ''Electric Light'' (album), a 2018 album by James Bay * Electric Light (poetry) ''Electric Light'' (Faber and Faber, 2001, ) is a poetry collection by Seamus Heaney, who received the 1995 Nobel Prize in Literature. The collection explores childhood, nature, and poetry itself. Part one presents translations and adaptations, o ..., a poetry collection by Irish poet Seamus Heaney, 2001 * "Electric Light" (song), a 2008 song by Infernal {{disambig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Orchid
Orchids are plants that belong to the family Orchidaceae (), a diverse and widespread group of flowering plants with blooms that are often colourful and fragrant. Orchids are cosmopolitan plants that are found in almost every habitat on Earth except glaciers. The world's richest diversity of orchid genera and species is in the tropics. Orchidaceae is one of the two largest families of flowering plants, the other being the Asteraceae. It contains about 28,000 currently accepted species in 702 genera. The Orchidaceae family encompasses about 6–11% of all species of seed plants. The largest genera are '' Bulbophyllum'' (2,000 species), '' Epidendrum'' (1,500 species), '' Dendrobium'' (1,400 species) and '' Pleurothallis'' (1,000 species). It also includes '' Vanilla'' (the genus of the vanilla plant), the type genus '' Orchis'', and many commonly cultivated plants such as '' Phalaenopsis'' and '' Cattleya''. Moreover, since the introduction of tropical species into cu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Origin (mathematics)
In mathematics, the origin of a Euclidean space is a special point, usually denoted by the letter ''O'', used as a fixed point of reference for the geometry of the surrounding space. In physical problems, the choice of origin is often arbitrary, meaning any choice of origin will ultimately give the same answer. This allows one to pick an origin point that makes the mathematics as simple as possible, often by taking advantage of some kind of geometric symmetry. Cartesian coordinates In a Cartesian coordinate system, the origin is the point where the axes of the system intersect.. The origin divides each of these axes into two halves, a positive and a negative semiaxis. Points can then be located with reference to the origin by giving their numerical coordinates—that is, the positions of their projections along each axis, either in the positive or negative direction. The coordinates of the origin are always all zero, for example (0,0) in two dimensions and (0,0,0) in three. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Paraboloid
In geometry, a paraboloid is a quadric surface that has exactly one axial symmetry, axis of symmetry and no central symmetry, center of symmetry. The term "paraboloid" is derived from parabola, which refers to a conic section that has a similar property of symmetry. Every plane section of a paraboloid made by a plane Parallel (geometry)#A line and a plane, parallel to the axis of symmetry is a parabola. The paraboloid is hyperbolic if every other plane section is either a hyperbola, or two crossing lines (in the case of a section by a tangent plane). The paraboloid is elliptic if every other nonempty plane section is either an ellipse, or a single point (in the case of a section by a tangent plane). A paraboloid is either elliptic or hyperbolic. Equivalently, a paraboloid may be defined as a quadric surface that is not a cylinder, and has an implicit surface, implicit equation whose part of degree two may be factored over the complex numbers into two different linear factors. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Homeomorphism
In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are the mappings that preserve all the topological properties of a given space. Two spaces with a homeomorphism between them are called homeomorphic, and from a topological viewpoint they are the same. Very roughly speaking, a topological space is a geometric object, and a homeomorphism results from a continuous deformation of the object into a new shape. Thus, a square and a circle are homeomorphic to each other, but a sphere and a torus are not. However, this description can be misleading. Some continuous deformations do not produce homeomorphisms, such as the deformation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
