Curvature Definition

The Latin word curvatūra came to our language as curvature. The concept refers to the curved condition (bent or crooked). The idea of ​​curvature is also used with respect to the deviation that a curved line has with respect to a straight line.

For example: “The criminals tried to take advantage of the curvature of the wall to hide, but they were discovered”, “Poor body posture can cause, in the long term, the curvature of the spine ”, “The curvature of the screen surprised the public”.

If someone talks about the curvature of a television, to cite one case, it refers to the fact that its screen is not straight. The curvature of a cell ( mobile ) phone, meanwhile, is linked to its curved edges. In these cases, the curvature can represent both an aesthetic and a functional aspect, or a fusion of both. Regardless of the purpose of this feature in a home appliance, electronic device, or automobile, among other products, fashion trends make it inevitable that its lifespan will be limited, so sooner or later curvature is replaced by sharp edges, and vice versa.

In the field of geometry and mathematics, curvature can be the magnitude or the number that measures this quality. It is, in this framework, the amount that a geometric object deviates from a line or a plane.

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The notion of curvature of spacetime derives from the theory of general relativity, which posits that gravity is an effect of the curved geometry that spacetime has. According to this theory, bodies in a gravitational field follow a curved path in space. The curvature of spacetime is measured according to the so-called curvature tensor or Riemann tensor.

Curvature displacement, on the other hand, is a theory that a vehicle could travel faster than the speed of light from a distortion that creates a greater curvature in space-time.

There is a quantity called the radius of curvature that is used to measure the curvature of an object belonging to geometry as if it were a surface, a curved line or, in more general terms, a differentiable manifold found in a Euclidean space.

If we take an object or a curved line as a reference, its radius of curvature is a geometric magnitude that we can define at each of its points, and it is equivalent to the inverse of the absolute value of the curvature at all of them. We must not forget that curvature is the alteration that crosses the direction of the tangent vector to a given curve as we move along it.

One of the measurements that we can make on a given surface is the Gaussian curvature, a number belonging to the set of real numbers that represents the intrinsic curvature for each of the regular points. It is possible to calculate it starting from the determinants of the two fundamental forms of the surface.

The first fundamental form of the surface is a 2-covariant tensor that presents symmetry and is defined in the space tangent to each of its points; it is the metric tensor (that is, of rank 2, used for the definition of concepts such as volume, angle and distance) that induces the Euclidean metric on the surface. The second, on the other hand, is the projection of the covariant derivative that is carried out on the normal vector to the surface, and is induced by the first fundamental form.

In general, the Gaussian curvature is different at each point on the surface and is related to its principal curvatures. The sphere is a special case of surface, since in all its points it presents the same curvature.