Curvature of space | Wikipedia audio article
Channel: wikipedia tts
Date: 2018-12-30

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Curvature


00:01:35 1 Curvature of plane curves
00:04:55 1.1 Precise definition
00:07:43 1.2 Signed curvature
00:08:20 1.3 Local expressions
00:11:33 1.4 Curvature of a graph
00:14:21 1.5 Implicit curve
00:16:35 1.6 Examples
00:16:45 2 Curvature of space curves
00:20:56 2.1 Local expressions
00:23:35 2.2 Curvature from arc and chord length
00:26:34 3 Curvature of curves on surfaces
00:28:03 3.1 Principal curvature
00:30:13 4 Curvature of surfaces
00:30:49 4.1 Normal sections
00:30:59 4.2 Gaussian curvature
00:31:18 4.3 Mean curvature
00:34:41 4.4 Second fundamental form
00:35:29 4.5 Shape operator
00:37:37 5 Curvature of space
00:38:24 6 Generalizations
00:40:50 7 See also
00:43:38 8 Notes



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SUMMARY
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In mathematics, curvature is any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object such as a surface deviates from being a flat plane, or a curve from being straight as in the case of a line, but this is defined in different ways depending on the context. There is a key distinction between extrinsic curvature, which is defined for objects embedded in another space (usually a Euclidean space) – in a way that relates to the radius of curvature of circles that touch the object – and intrinsic curvature, which is defined in terms of the lengths of curves within a Riemannian manifold.
This article deals primarily with extrinsic curvature. Its canonical example is that of a circle, which has a curvature equal to the reciprocal of its radius everywhere. Smaller circles bend more sharply, and hence have higher curvature. The curvature of a smooth curve is defined as the curvature of its osculating circle at each point.
Curvature is normally a scalar quantity, but one may also define a curvature vector that takes into account the direction of the bend in addition to its magnitude. The curvature of more complex objects (such as surfaces or even curved n-dimensional spaces) is described by more complex objects from linear algebra, such as the general Riemann curvature tensor.
This article sketches the mathematical framework which describes the curvature of a curve embedded in a plane and the curvature of a surface in Euclidean space.