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Parabolic Line
In differential geometry, a smooth surface in three dimensions has a parabolic point when the Gaussian curvature is zero. Typically such points lie on a curve called the parabolic line which separates the surface into regions of positive and negative Gaussian curvature. Points on the parabolic line give rise to folds on the Gauss map: where a ridge crosses a parabolic line there is a cusp of the Gauss map.Ian R. Porteous (2001) ''Geometric Differentiation'', Chapter 11 Ridges and Ribs, pp 182–97, Cambridge University Press Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press in the world. It is also the King's Printer. Cambr ... . References Differential geometry of surfaces Surfaces {{differential-geometry-stub ... [...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 str ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Surface (differential Geometry)
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 the ... [...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 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 vector that is at right angles to the surface; planes containing the n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gauss Map
In differential geometry, the Gauss map (named after Carl F. Gauss) maps a surface in Euclidean space R3 to the unit sphere ''S''2. Namely, given a surface ''X'' lying in R3, the Gauss map is a continuous map ''N'': ''X'' → ''S''2 such that ''N''(''p'') is a unit vector orthogonal to ''X'' at ''p'', namely a normal vector to ''X'' at ''p''. The Gauss map can be defined (globally) if and only if the surface is orientable, in which case its degree is half the Euler characteristic. The Gauss map can always be defined locally (i.e. on a small piece of the surface). The Jacobian determinant of the Gauss map is equal to Gaussian curvature, and the differential of the Gauss map is called the shape operator. Gauss first wrote a draft on the topic in 1825 and published in 1827. There is also a Gauss map for a link, which computes linking number. Generalizations The Gauss map can be defined for hypersurfaces in R''n'' as a map from a hypersurface to the unit sphere ''S''''n'' & ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ridge (differential Geometry)
In differential geometry, a smooth surface in three dimensions has a ridge point when a line of curvature has a local maximum or minimum of principal curvature. The set of ridge points form curves on the surface called ridges. The ridges of a given surface fall into two families, typically designated ''red'' and ''blue'', depending on which of the two principal curvatures has an extremum. At umbilical points the colour of a ridge will change from red to blue. There are two main cases: one has three ridge lines passing through the umbilic, and the other has one line passing through it. Ridge lines correspond to cuspidal edges on the focal surface. See also *Ridge detection In image processing, ridge detection is the attempt, via software, to locate ridges in an image, defined as curves whose points are local maxima of the function, akin to geographical ridges. For a function of ''N'' variables, its ridges are a s ... References * Differential geometry of surfaces Surf ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ian R
Ian or Iain is a name of Scottish Gaelic origin, derived from the Hebrew given name (Yohanan, ') and corresponding to the English name John. The spelling Ian is an Anglicization of the Scottish Gaelic forename ''Iain''. It is a popular name in Scotland, where it originated, as well as other English-speaking countries. The name has fallen out of the top 100 male baby names in the United Kingdom, having peaked in popularity as one of the top 10 names throughout the 1960s. In 1900, Ian was the 180th most popular male baby name in England and Wales. , the name has been in the top 100 in the United States every year since 1982, peaking at 65 in 2003. Other Gaelic forms of "John" include "Seonaidh" ("Johnny" from Lowland Scots), "Seon" (from English), "Seathan", and "Seán" and "Eoin" (from Irish). Its Welsh counterpart is Ioan, its Cornish equivalent is Yowan and Breton equivalent is Yann. Notable people named Ian As a first name (alphabetical by family name) * Ian Agol (bo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press in the world. It is also the King's Printer. Cambridge University Press is a department of the University of Cambridge and is both an academic and educational publisher. It became part of Cambridge University Press & Assessment, following a merger with Cambridge Assessment in 2021. With a global sales presence, publishing hubs, and offices in more than 40 Country, countries, it publishes over 50,000 titles by authors from over 100 countries. Its publishing includes more than 380 academic journals, monographs, reference works, school and university textbooks, and English language teaching and learning publications. It also publishes Bibles, runs a bookshop in Cambridge, sells through Amazon, and has a conference venues business in Cambridge at the Pitt Building and the Sir Geoffrey Cass Spo ... [...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] |
