the Gaussian curvature or Gauss curvature Κ of a smooth surface in three-dimensional space at a point is the product of the principal curvatures, κ1 and...
19 KB (2,612 words) - 22:21, 7 August 2024
surfaces have zero Gaussian curvature (see below). In contrast to curves, which do not have intrinsic curvature, but do have extrinsic curvature (they only have...
44 KB (6,460 words) - 20:41, 28 August 2024
Friedrich Gauss in 1827, that concerns the curvature of surfaces. The theorem says that Gaussian curvature can be determined entirely by measuring angles...
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surfaces, surfaces with a constant negative Gaussian curvature. Saddle surfaces have negative Gaussian curvature in at least some regions, where they locally...
56 KB (7,019 words) - 21:44, 8 August 2024
concepts investigated is the Gaussian curvature, first studied in depth by Carl Friedrich Gauss, who showed that curvature was an intrinsic property of...
128 KB (17,447 words) - 02:21, 18 August 2024
→ ±∞. The curvature is often expressed in terms of its reciprocal, R, the radius of curvature; for a fundamental Gaussian beam the curvature at position...
47 KB (6,953 words) - 18:59, 11 August 2024
geometry of surfaces, the scalar curvature is twice the Gaussian curvature, and completely characterizes the curvature of a surface. In higher dimensions...
35 KB (5,029 words) - 23:36, 30 May 2024
mean curvature. Other such immersed surfaces as minimal surfaces have constant mean curvature. The sphere has constant positive Gaussian curvature. Gaussian...
41 KB (5,331 words) - 01:06, 31 August 2024
the Gaussian curvature and a, b, c and d take values either 1 or 2. The Riemann tensor has only one functionally independent component. The Gaussian curvature...
19 KB (2,911 words) - 19:07, 3 September 2024
ds={\frac {2|dz|}{1-|z|^{2}}}} . In terms of the (constant and negative) Gaussian curvature K of a hyperbolic plane, a unit of absolute length corresponds to...
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First fundamental form (section Gaussian curvature)
}\sin v\,du\,dv=2\pi {\Big [}{-\cos v}{\Big ]}_{0}^{\pi }=4\pi } The Gaussian curvature of a surface is given by K = det I I p det I p = L N − M 2 E G − F...
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a point p of the manifold. It can be defined geometrically as the Gaussian curvature of the surface which has the plane σp as a tangent plane at p, obtained...
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introduced an abstract and rigorous way to define curvature for these manifolds, now known as the Riemann curvature tensor. Similar notions have found applications...
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curvatures is the Gaussian curvature, K, and the average (k1 + k2)/2 is the mean curvature, H. If at least one of the principal curvatures is zero at every...
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hypersurface of four-dimensional Euclidean space which is complete and has Gaussian curvature negative and bounded away from zero. Previous examples of such surfaces...
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Möbius strip (section Surfaces of constant curvature)
geometry of constant positive, negative, or zero Gaussian curvature. The cases of negative and zero curvature form geodesically complete surfaces, which means...
88 KB (9,611 words) - 15:13, 22 August 2024
Ricci curvature. In the case of two-dimensional manifolds, negativity of the Ricci curvature is synonymous with negativity of the Gaussian curvature, which...
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Earth radius (redirect from Earth radius of curvature)
combine the principal radii of curvature above in a non-directional manner. The Earth's Gaussian radius of curvature at latitude φ is: R a ( φ ) = 1...
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point of view – corresponds to a developable surface, which has null Gaussian curvature, therefore it can be flattened to a planar surface with no distortion...
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Riemannian manifold with boundary ∂M. Let K be the Gaussian curvature of M, and let kg be the geodesic curvature of ∂M. Then ∫ M K d A + ∫ ∂ M k g d s = 2 π...
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Horocycle (section Standardized Gaussian curvature)
) If the metric is normalized to have Gaussian curvature −1, then the horocycle is a curve of geodesic curvature 1 at every point. Every horocycle is the...
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Pseudosphere (category Surfaces of revolution of constant negative curvature)
constant negative Gaussian curvature. A pseudosphere of radius R is a surface in R 3 {\displaystyle \mathbb {R} ^{3}} having curvature −1/R2 at each point...
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when the Gaussian curvature is negative (or zero). There are two asymptotic directions through every point with negative Gaussian curvature, bisected...
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Hyperboloid (section Curvature)
hyperbolic hyperboloid. It is a connected surface, which has a negative Gaussian curvature at every point. This implies near every point the intersection of...
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surfaces have negative Gaussian curvature which distinguish them from convex/elliptical surfaces which have positive Gaussian curvature. A classical third-order...
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conformally equivalent to one that has constant Gaussian curvature. In the case of a torus, the constant curvature must be zero. Then one defines the "moduli...
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between points. The unit disk U with the Poincaré metric has negative Gaussian curvature −1. In 1938, Lars Ahlfors generalised the lemma to maps from the unit...
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Paraboloid (section Curvature)
v)=\left(u,v,{\frac {u^{2}}{a^{2}}}+{\frac {v^{2}}{b^{2}}}\right)} has Gaussian curvature K ( u , v ) = 4 a 2 b 2 ( 1 + 4 u 2 a 4 + 4 v 2 b 4 ) 2 {\displaystyle...
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List of things named after Carl Friedrich Gauss (redirect from Gaussian)
Gauss's lemma in Riemannian geometry Gauss map in differential geometry Gaussian curvature, defined in his Theorema egregium Gauss circle problem Gauss–Kuzmin–Wirsing...
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Parametric surface (section Curvature)
invariants such as the first and second fundamental forms, Gaussian, mean, and principal curvatures can all be computed from a given parametrization. The simplest...
14 KB (2,323 words) - 18:38, 4 February 2024