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Principal curvature - Wikipedia
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.
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Radius of curvature - Wikipedia
In differential geometry, the radius of curvature, R, is the reciprocal of the curvature. For a curve, it equals the radius of the circular arc which best approximates the curve at that point. For surfaces, the radius of curvature is …
Mean curvature - Wikipedia
In mathematics, the mean curvature of a surface is an extrinsic measure of curvature that comes from differential geometry and that locally describes the curvature of an embedded surface in …
Gaussian curvature - Wikipedia
In differential geometry, 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 κ2, at the given point: For example, a sphere of radius r …
Total curvature - Wikipedia
The total curvature of a closed curve is always an integer multiple of 2 π, where N is called the index of the curve or turning number – it is the winding number of the unit tangent vector about …
Center of curvature - Wikipedia
In geometry, the center of curvature of a curve is a point located at a distance from the curve curvature is zero. The osculating circle to the curve is centered at the centre of curvature.
Curvature of Riemannian manifolds - Wikipedia
The curvature of a Riemannian manifold can be described in various ways; the most standard one is the curvature tensor, given in terms of a Levi-Civita connection (or covariant differentiation) …
Riemann curvature tensor - Wikipedia
In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most …
Curvature | Riemannian, Non-Euclidean & Manifolds
Oct 15, 2024 · Curvature, in mathematics, the rate of change of direction of a curve with respect to distance along the curve. At every point on a circle, the curvature is the reciprocal of the radius; for other curves (and straight lines, …
Curved space - Wikipedia
Curved space often refers to a spatial geometry which is not "flat", where a flat space has zero curvature, as described by Euclidean geometry. [1] Curved spaces can generally be described by Riemannian geometry, though some …
General relativity - Wikipedia
General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the …
Curvature -- from Wolfram MathWorld
3 days ago · In general, there are two important types of curvature: extrinsic curvature and intrinsic curvature. The extrinsic curvature of curves in two- and three-space was the first type …
Curvature - Encyclopedia of Mathematics
Jun 5, 2020 · The curvature is one of the fundamental concepts in modern differential geometry. Restrictions on the curvature usually yield meaningful information about an object. For …
Mean Curvature -- from Wolfram MathWorld
Nov 21, 2024 · Wente (1985, 1986, 1987) found a nonspherical finite surface with constant mean curvature, consisting of a self- intersecting three-lobed toroidal surface. A family of such …
Curvature Definition (Illustrated Mathematics Dictionary) - Math is …
Illustrated definition of Curvature: How curved a line or surface is. How much a curve varies from being straight or flat.
Calculating the curvature of a surface - Mathematics Stack …
In 'alternate formulas' on wikipedia you can see how to derive the Gaussian curvature of any surface given by a local parametrization: en.wikipedia.org/wiki/Gaussian_curvature. Basically …
Curvature - SEG Wiki
Oct 10, 2017 · Geometrically, curvature (k) is defined as the radius of a circle that is tangent to a curve. Mathematically it can be represented as k= 1/r, where k is the curvature, and r is the …
How was curvature originally defined and calculated?
Sep 16, 2021 · Curvature was originally defined as a property of the two classical Greek curves, the line and the circle. It was noted that lines do not curve amd that every point on a circle …
Curvature | Math Wiki - Fandom
The curvature, represented by κ {\displaystyle \kappa} , of a smooth (that is, with no cusps or sharp corners) function r → ( t ) {\displaystyle \vec r(t)} is a measure of how fast the direction of …
calculus - Radius of Curvature - Mathematics Stack Exchange
It's easier to start with curvature. Given a curve γ: t ↦ z(t) = (x(t), y(t)) (a <t <b) the vector ˙z(t) = (˙x(t), ˙y(t)) points for each t in the direction of the forward tangent vector at z(t). Curvature is …
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