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Asymptotic Curve
In the differential geometry of surfaces, an asymptotic curve is a curve always tangent to an asymptotic direction of the surface (where they exist). It is sometimes called an asymptotic line, although it need not be a line. Definitions An asymptotic direction is one in which the normal curvature is zero. Which is to say: for a point on an asymptotic curve, take the plane which bears both the curve's tangent and the surface's normal at that point. The curve of intersection of the plane and the surface will have zero curvature at that point. Asymptotic directions can only occur when the Gaussian curvature is negative (or zero). There will be two asymptotic directions through every point with negative Gaussian curvature, bisected by the principal directions. If the surface is minimal, the asymptotic directions are orthogonal to one another. Related notions The direction of the asymptotic direction are the same as the asymptotes of the hyperbola of the Dupin indicatrix. A rela ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Geometry Of Surfaces
In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric. Surfaces have been extensively studied from various perspectives: ''extrinsically'', relating to their embedding in Euclidean space and ''intrinsically'', reflecting their properties determined solely by the distance within the surface as measured along curves on the surface. One of the fundamental concepts investigated is the Gaussian curvature, first studied in depth by Carl Friedrich Gauss, who showed that curvature was an intrinsic property of a surface, independent of its isometric embedding in Euclidean space. Surfaces naturally arise as graphs of functions of a pair of variables, and sometimes appear in parametric form or as loci associated to space curves. An important role in their study has been played by Lie groups (in the spirit of the Erlangen program), namely the symmetry groups of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Asymptote
In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related contexts, an asymptote of a curve is a line which is tangent to the curve at a point at infinity. The word asymptote is derived from the Greek ἀσύμπτωτος (''asumptōtos'') which means "not falling together", from ἀ priv. + σύν "together" + πτωτ-ός "fallen". The term was introduced by Apollonius of Perga in his work on conic sections, but in contrast to its modern meaning, he used it to mean any line that does not intersect the given curve. There are three kinds of asymptotes: ''horizontal'', ''vertical'' and ''oblique''. For curves given by the graph of a function , horizontal asymptotes are horizontal lines that the graph of the function approaches as ''x'' tends to Vertical asymptotes are vertical lines near which the fun ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Surfaces
A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. Surface or surfaces may also refer to: Mathematics *Surface (mathematics), a generalization of a plane which needs not be flat *Surface (differential geometry), a differentiable two-dimensional manifold *Surface (topology), a two-dimensional manifold * Algebraic surface, an algebraic variety of dimension two *Coordinate surfaces *Fractal surface, generated using a stochastic algorithm *Polyhedral surface * Surface area *Surface integral Arts and entertainment * Surface (band), an American R&B and pop trio ** ''Surface'' (Surface album), 1986 *Surfaces (band), American musical duo * ''Surface'' (Circle album), 1998 * "Surface" (Aero Chord song), 2014 * ''Surface'' (2005 TV series), an American science fiction show, 2005–2006 * ''Surface'' (2022 TV series), an American psychological thriller miniseries that began streaming in 2022 *'' The Surface'', an American film, 2 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Geometry Of Surfaces
In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric. Surfaces have been extensively studied from various perspectives: ''extrinsically'', relating to their embedding in Euclidean space and ''intrinsically'', reflecting their properties determined solely by the distance within the surface as measured along curves on the surface. One of the fundamental concepts investigated is the Gaussian curvature, first studied in depth by Carl Friedrich Gauss, who showed that curvature was an intrinsic property of a surface, independent of its isometric embedding in Euclidean space. Surfaces naturally arise as graphs of functions of a pair of variables, and sometimes appear in parametric form or as loci associated to space curves. An important role in their study has been played by Lie groups (in the spirit of the Erlangen program), namely the symmetry groups of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Curves
A curve is a geometrical object in mathematics. Curve(s) may also refer to: Arts, entertainment, and media Music * Curve (band), an English alternative rock music group * ''Curve'' (album), a 2012 album by Our Lady Peace * "Curve" (song), a 2017 song by Gucci Mane featuring The Weeknd * ''Curve'', a 2001 album by Doc Walker * "Curve", a song by John Petrucci from ''Suspended Animation'', 2005 * "Curve", a song by Cam'ron from the album '' Crime Pays'', 2009 Periodicals * ''Curve'' (design magazine), an industrial design magazine * ''Curve'' (magazine), a U.S. lesbian magazine Other uses in arts, entertainment, and media * ''Curve'' (film), a 2015 film * BBC Two 'Curve' idents, various animations based around a curve motif Brands and enterprises *Curve (payment card), a payment card that aggregates multiple payment cards * Curve (theatre), a theatre in Leicester, United Kingdom * Curve, fragrance by Liz Claiborne * BlackBerry Curve, a series of phones from Research in Motio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
French Language
French ( or ) is a Romance language of the Indo-European family. It descended from the Vulgar Latin of the Roman Empire, as did all Romance languages. French evolved from Gallo-Romance, the Latin spoken in Gaul, and more specifically in Northern Gaul. Its closest relatives are the other langues d'oïl—languages historically spoken in northern France and in southern Belgium, which French ( Francien) largely supplanted. French was also influenced by native Celtic languages of Northern Roman Gaul like Gallia Belgica and by the ( Germanic) Frankish language of the post-Roman Frankish invaders. Today, owing to France's past overseas expansion, there are numerous French-based creole languages, most notably Haitian Creole. A French-speaking person or nation may be referred to as Francophone in both English and French. French is an official language in 29 countries across multiple continents, most of which are members of the ''Organisation internationale de la Francophonie'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Principal Curvature
In differential geometry, the two principal curvatures at a given point of a surface are the maximum and minimum values of the curvature as expressed by the eigenvalues of the shape operator at that point. They measure how the surface bends by different amounts in different directions at that point. Discussion At each point ''p'' of a differentiable surface in 3-dimensional Euclidean space one may choose a unit '' normal vector''. A '' normal plane'' at ''p'' is one that contains the normal vector, and will therefore also contain a unique direction tangent to the surface and cut the surface in a plane curve, called normal section. This curve will in general have different curvatures for different normal planes at ''p''. The principal curvatures at ''p'', denoted ''k''1 and ''k''2, are the maximum and minimum values of this curvature. Here the curvature of a curve is by definition the reciprocal of the radius of the osculating circle. The curvature is taken to be positive i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dupin Indicatrix
In differential geometry, the Dupin indicatrix is a method for characterising the local shape of a surface. Draw a plane parallel to the tangent plane and a small distance away from it. Consider the intersection of the surface with this plane. The shape of the intersection is related to the Gaussian curvature. The Dupin indicatrix is the result of the limiting process as the plane approaches the tangent plane. The indicatrix was invented by Charles Dupin. For elliptical points where the Gaussian curvature is positive the intersection will either be empty or form a closed curve. In the limit this curve will form an ellipse aligned with the principal directions. For hyperbolic points, where the Gaussian curvature is negative, the intersection will form a hyperbola. Two different hyperbolas will be formed on either side of the tangent plane. These hyperbolas share the same axis and asymptotes. The directions of the asymptotes are the same as the asymptotic directions. See also * E ... [...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] |
Curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (geometry), point. This is the definition that appeared more than 2000 years ago in Euclid's Elements, Euclid's ''Elements'': "The [curved] line is […] the first species of quantity, which has only one dimension, namely length, without any width nor depth, and is nothing else than the flow or run of the point which […] will leave from its imaginary moving some vestige in length, exempt of any width." This definition of a curve has been formalized in modern mathematics as: ''A curve is the image (mathematics), image of an interval (mathematics), interval to a topological space by a continuous function''. In some contexts, the function that defines the curve is called a ''parametrization'', and the curve is a parametric curve. In this artic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Principal Curvature
In differential geometry, the two principal curvatures at a given point of a surface are the maximum and minimum values of the curvature as expressed by the eigenvalues of the shape operator at that point. They measure how the surface bends by different amounts in different directions at that point. Discussion At each point ''p'' of a differentiable surface in 3-dimensional Euclidean space one may choose a unit '' normal vector''. A '' normal plane'' at ''p'' is one that contains the normal vector, and will therefore also contain a unique direction tangent to the surface and cut the surface in a plane curve, called normal section. This curve will in general have different curvatures for different normal planes at ''p''. The principal curvatures at ''p'', denoted ''k''1 and ''k''2, are the maximum and minimum values of this curvature. Here the curvature of a curve is by definition the reciprocal of the radius of the osculating circle. The curvature is taken to be positive i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gaussian Curvature
In differential geometry, the Gaussian curvature or Gauss curvature of a surface at a point is the product of the principal curvatures, and , at the given point: K = \kappa_1 \kappa_2. The Gaussian radius of curvature is the reciprocal of . For example, a sphere of radius has Gaussian curvature everywhere, and a flat plane and a cylinder have Gaussian curvature zero everywhere. The Gaussian curvature can also be negative, as in the case of a hyperboloid or the inside of a torus. Gaussian curvature is an ''intrinsic'' measure of curvature, depending only on distances that are measured “within” or along the surface, not on the way it is isometrically embedding, embedded in Euclidean space. This is the content of the ''Theorema egregium''. Gaussian curvature is named after Carl Friedrich Gauss, who published the ''Theorema egregium'' in 1827. Informal definition At any point on a surface, we can find a Normal (geometry), normal vector that is at right angles to the sur ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
