In differential geometry, the Gaussian curvature or Gauss curvatureΚ of a smooth surface in three-dimensional space at a point is the product of the principal curvatures, κ1 and κ2, at the given point:
The Gaussian radius of curvature is the reciprocal of Κ.
For example, a sphere of radius r has Gaussian curvature 1/r2 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.
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the Gaussiancurvature or Gauss curvature Κ of a smooth surface in three-dimensional space at a point is the product of the principal curvatures, κ1 and...
surfaces have zero Gaussiancurvature (see below). In contrast to curves, which do not have intrinsic curvature, but do have extrinsic curvature (they only have...
→ ±∞. The curvature is often expressed in terms of its reciprocal, R, the radius of curvature; for a fundamental Gaussian beam the curvature at position...
concepts investigated is the Gaussiancurvature, first studied in depth by Carl Friedrich Gauss, who showed that curvature was an intrinsic property of...
Friedrich Gauss in 1827, that concerns the curvature of surfaces. The theorem says that Gaussiancurvature can be determined entirely by measuring angles...
surfaces, surfaces with a constant negative Gaussiancurvature. Saddle surfaces have negative Gaussiancurvature in at least some regions, where they locally...
geometry of surfaces, the scalar curvature is twice the Gaussiancurvature, and completely characterizes the curvature of a surface. In higher dimensions...
mean curvature. Other such immersed surfaces as minimal surfaces have constant mean curvature. The sphere has constant positive Gaussiancurvature. Gaussian...
}\sin v\,du\,dv=2\pi {\Big [}{-\cos v}{\Big ]}_{0}^{\pi }=4\pi } The Gaussiancurvature of a surface is given by K = det I I p det I p = L N − M 2 E G − F...
the Gaussiancurvature and a, b, c and d take values either 1 or 2. The Riemann tensor has only one functionally independent component. The Gaussian curvature...
) If the metric is normalized to have Gaussiancurvature −1, then the horocycle is a curve of geodesic curvature 1 at every point. Every horocycle is the...
curvatures is the Gaussiancurvature, K, and the average (k1 + k2)/2 is the mean curvature, H. If at least one of the principal curvatures is zero at every...
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...
a point p of the manifold. It can be defined geometrically as the Gaussiancurvature of the surface which has the plane σp as a tangent plane at p, obtained...
mean curvature. The surfaces of unit constant mean curvature in hyperbolic space are called Bryant surfaces. Gaussiancurvature Mean curvature flow Inverse...
introduced an abstract and rigorous way to define curvature for these manifolds, now known as the Riemann curvature tensor. Similar notions have found applications...
Ricci curvature. In the case of two-dimensional manifolds, negativity of the Ricci curvature is synonymous with negativity of the Gaussiancurvature, which...
ds={\frac {2|dz|}{1-|z|^{2}}}} . In terms of the (constant and negative) Gaussiancurvature K of a hyperbolic plane, a unit of absolute length corresponds to...
constant negative Gaussiancurvature. A pseudosphere of radius R is a surface in R 3 {\displaystyle \mathbb {R} ^{3}} having curvature −1/R2 at each point...
invariants such as the first and second fundamental forms, Gaussian, mean, and principal curvatures can all be computed from a given parametrization. The simplest...
conformally equivalent to one that has constant Gaussiancurvature. In the case of a torus, the constant curvature must be zero. Then one defines the "moduli...
hyperbolic hyperboloid. It is a connected surface, which has a negative Gaussiancurvature at every point. This implies near every point the intersection of...
the isothermal coordinates ( u , v ) {\displaystyle (u,v)} , the Gaussiancurvature takes the simpler form K = − 1 2 e − ρ ( ∂ 2 ρ ∂ u 2 + ∂ 2 ρ ∂ v 2...
the Gaussiancurvature to the Euler characteristic. Here the Gaussiancurvature is concentrated at the vertices: on the faces and edges the Gaussian curvature...
surfaces have negative Gaussiancurvature which distinguish them from convex/elliptical surfaces which have positive Gaussiancurvature. A classical third-order...