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...

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...

Friedrich Gauss in 1827, that concerns the curvature of surfaces. The theorem says that Gaussian curvature can be determined entirely by measuring angles...

geometry of surfaces, the scalar curvature is twice the Gaussian curvature, and completely characterizes the curvature of a surface. In higher dimensions...

surfaces, surfaces with a constant negative Gaussian curvature. Saddle surfaces have negative Gaussian curvature in at least some regions, where they locally...

concepts investigated is the Gaussian curvature, first studied in depth by Carl Friedrich Gauss, who showed that curvature was an intrinsic property of...

component. The Gaussian curvature coincides with the sectional curvature of the surface. It is also exactly half the scalar curvature of the 2-manifold...

→ ±∞. The curvature is often expressed in terms of its reciprocal, R, the radius of curvature; for a fundamental Gaussian beam the curvature at position...

mean curvature. Other such immersed surfaces as minimal surfaces have constant mean curvature. The sphere has constant positive Gaussian curvature. Gaussian...

}\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...

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...

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...

introduced an abstract and rigorous way to define curvature for these manifolds, now known as the Riemann curvature tensor. Similar notions have found applications...

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...

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...

geometry of constant positive, negative, or zero Gaussian curvature. The cases of negative and zero curvature form geodesically complete surfaces, which means...

) 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...

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...

when the Gaussian curvature is negative (or zero). There are two asymptotic directions through every point with negative Gaussian curvature, bisected...

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...

Ricci curvature. In the case of two-dimensional manifolds, negativity of the Ricci curvature is synonymous with negativity of the Gaussian curvature, which...

surfaces have negative Gaussian curvature which distinguish them from convex/elliptical surfaces which have positive Gaussian curvature. A classical third-order...

hyperbolic hyperboloid. It is a connected surface, which has a negative Gaussian curvature at every point. This implies near every point the intersection of...

disk is not necessarily a planar surface. They can be used to define Gaussian curvature. Whittemore, J. K. (1901). "A Note on Geodesic Circles". The Annals...

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...

hypersurface of four-dimensional Euclidean space which is complete and has Gaussian curvature negative and bounded away from zero. Previous examples of such surfaces...

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...

invariants such as the first and second fundamental forms, Gaussian, mean, and principal curvatures can all be computed from a given parametrization. The simplest...

mean curvature. The surfaces of unit constant mean curvature in hyperbolic space are called Bryant surfaces. Gaussian curvature Mean curvature flow Inverse...

case. Note that these surfaces are generally different from constant Gaussian curvature surfaces, with the important exception of the sphere. In 1841 Delaunay...