Curvature Totally Explained

Everything about Curvature totally explained

In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, 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 at each point in a differential manifold. This article deals primarily with the first concept. The primordial example of extrinsic curvature is that of a circle, which has curvature equal to the inverse 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.
In a plane, this is a scalar quantity, but in three or more dimensions it's described by a curvature vector that takes into account the direction of the bend as well as its sharpness. 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.
The remainder of this article discusses, from a mathematical perspective, some geometric examples of curvature: the curvature of a curve embedded in a plane and the curvature of a surface in Euclidean space. See the links below for further reading.

One dimension in two dimensions: Curvature of plane curves

For a plane curve C, the mathematical definition of curvature uses a parametric representation of C with respect to the arc length parametrization. It can be computed given any regular parametrization by a more complicated formula given below. Let γ(s) be a regular parametric curve, where s is the arc length, or natural parameter. This determines the unit tangent vector T, the unit normal vector N, the curvature κ(s), the signed curvature k(s) and the radius of curvature at each point:
ť

T(s)=gamma'(s),quad T'(s)=k(s)N(s),quad kappa(s) = |gamma

(s)| = left|k(s) ight|, quad R(s)=frac. The integral of the Gaussian curvature over the whole surface is closely related to the surface's Euler characteristic; see the Gauss-Bonnet theorem.
   The discrete analog of curvature, corresponding to curvature being concentrated at a point and particularly useful for polyhedra, is the (angular) defect; the analog for the Gauss-Bonnet theorem is Descartes' theorem on total angular defect.
   Because curvature can be defined without reference to an embedding space, it isn't necessary that a surface be embedded in a higher dimensional space in order to be curved. Such an intrinsically curved two-dimensional surface is a simple example of a Riemannian manifold.
   The mean curvature is equal to the sum of the principal curvatures, k₁+k₂, over 2. It has the dimension of 1/length. Mean curvature is closely related to the first variation of surface area, in particular a minimal surface such as a soap film, has mean curvature zero and a soap bubble has constant mean curvature. Unlike Gauss curvature, the mean curvature is extrinsic and depends on the embedding, for instance, a cylinder and a plane are locally isometric but the mean curvature of a plane is zero while that of a cylinder is nonzero.
Three dimensions: Curvature of space
By extension of the former argument, a space of three or more dimensions can be intrinsically curved; the full mathematical description is described at curvature of Riemannian manifolds. Again, the curved space may or may not be conceived as being embedded in a higher-dimensional space. In recent physics jargon, the embedding space is known as the bulk and the embedded space as a p-brane where p is the number of dimensions; thus a surface (membrane) is a 2-brane; normal space is a 3-brane etc.
   After the discovery of the intrinsic definition of curvature, which is closely connected with non-Euclidean geometry, many mathematicians and scientists questioned whether ordinary physical space might be curved, although the success of Euclidean geometry up to that time meant that the radius of curvature must be astronomically large. In the theory of general relativity, which describes gravity and cosmology, the idea is slightly generalised to the "curvature of space-time"; in relativity theory space-time is a pseudo-Riemannian manifold. Once a time coordinate is defined, the three-dimensional space corresponding to a particular time is generally a curved Riemannian manifold; but since the time coordinate choice is largely arbitrary, it's the underlying space-time curvature that's physically significant.
   Although an arbitrarily-curved space is very complex to describe, the curvature of a space which is locally isotropic and homogeneous is described by a single Gaussian curvature, as for a surface; mathematically these are strong conditions, but they correspond to reasonable physical assumptions (all points and all directions are indistinguishable). A positive curvature corresponds to the inverse square radius of curvature; an example is a sphere or hypersphere. An example of negatively curved space is hyperbolic geometry. A space or space-time without curvature (formally, with zero curvature) is called flat. For example, Euclidean space is an example of a flat space, and Minkowski space is an example of a flat space-time. There are other examples of flat geometries in both settings, though. A torus or a cylinder can both be given flat metrics, but differ in their topology. Other topologies are also possible for curved space. See also shape of the universe.

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