The mathematical concept of curvature is also defined in much more general contexts. [9] Many of these generalizations emphasize various aspects of curvature as understood in the lower dimensions. The same circle can also be defined by the implicit equation F(x, y) = 0 with F(x, y) = x2 + y2 – r2. Then, the curvature formula in this case gives these examples of sentences that are automatically selected from various online information sources to reflect the current use of the word “curvature.” The opinions expressed in the examples do not represent the opinion of Merriam-Webster or its editors. Send us your feedback. Gaussian curvature is an intrinsic property of the surface, which means that it does not depend on the particular embedding of the surface; Intuitively, this means that ants living on the surface could determine the Gaussian curvature. For example, an ant living on a sphere could measure the sum of the inner angles of a triangle and determine that it was greater than 180 degrees, meaning that the space it inhabited had a positive curvature. On the other hand, an ant living on a cylinder would not detect such a deviation from Euclidean geometry; In particular, the ant could not detect that the two surfaces have different average curvatures (see below), which is a purely extrinsic type of curvature. An intrinsic definition of the Gaussian curvature at a point P is as follows: Imagine an ant bound to P with a short wire of length r. It rotates around P while the wire is completely stretched and measures the length C(r) of a complete journey around P.
If the surface were flat, the ant would find C(r) = 2πr. On curved surfaces, the formula of C(r) will be different, and the Gaussian curvature K at the point P can be calculated by the Bertrand-Diguet-Puiseux theorem because the same parabola can also be defined by the implicit equation F(x, y) = 0 with F(x, y) = ax2 + bx + c – y. Since Fy = –1 and Fyy = Fxy = 0, you get exactly the same value for the curvature (unsigned). However, the signed curvature does not make sense here because –F(x, y) = 0 is an implicit equation valid for the same parabola that gives the opposite sign for the curvature. The main curvatures are the eigenvalues of the shape operator, the main directions of curvature are its eigenvectors, the Gaussian curvature is its determinant, and the mean curvature is half of its trace. where is the tangential angle and length of the arc. As can easily be seen in the definition, the curvature therefore has units of the inverse distance. The derivative of the above equation can be found using the identity, where the Gaussian is the curvature, is the mean curvature and det is the determinant. In terms of parameterizing arc length, the first Frenet-Serret formula is essentially the tangent, curvature and normal vector together describe the behavior of a second-order curve near a point. In three dimensions, the behavior of a third-order curve is described by a related torsional term that measures the extent to which a curve tends to move like a spiral path in space. Torsion and curvature are related by Frenet-Serret formulas (in three dimensions) and their generalization (in higher dimensions).
It has the sign of a for all values of x. This means that if a > 0, the concavence is directed upwards everywhere; If a < 0, the concavity is directed downwards; For a = 0, the curvature is zero everywhere, which confirms that the parabola degenerates in line in this case. Another generalization of curvature is due to the ability to compare a curved space with another space that has a constant curvature. Often this is done with triangles in the rooms. The concept of triangle makes sense in metric spaces, and cat(k) spaces emerge from it. The curvature can be evaluated along normal surface sections, similar to the curves § on the above surfaces (see for example the radius of curvature of the earth). By extending the first argument, a space of three or more dimensions can be inherently curved. Curvature is intrinsic in the sense that it is a property defined at any point in space, rather than a property defined relative to a larger space that contains it.
In general, a curved space may or may not be designed to be incorporated into a higher-dimensional environmental space; Otherwise, its curvature can only be defined intrinsically. Britannica.com: Encyclopedia article on curvature The curvature of a two-dimensional curve refers to the radius of curvature of the oscillating circle of the curve. Consider a circle parametrically defined by The signed curvature is not defined because it depends on an orientation of the curve that is not provided by the implicit equation. In addition, changing from F to –F changes the counter sign instead of the curve if the absolute value is omitted from the previous formula. Formally, the Gaussian curvature depends only on the Riemann metric of the surface. This is Gauss`s famous Theorema Egregium, which he found while studying geographical surveying and cartography. A common parameterization of a circle of radius r is γ(t) = (r cos t, r sin t). The formula for curvature yields To be significant, the definition of curvature and its various characterizations require that the curve close to P be continuously differentiable to have a tangent that varies continuously; it also requires that the curve at P be doubly differentiable to ensure the presence of the boundaries involved and the derivation of T(s). Another general generalization of curvature results from the study of parallel transport on a surface.
For example, if a vector is moved around a loop on the surface of a parallel sphere throughout motion, the final position of the vector may not be the same as the initial position of the vector. This phenomenon is called holonomy. [10] Various generalizations capture in abstract form this idea of curvature as a measure of holonomy; see form of curvature. A closely related concept of curvature comes from gauge theory in physics, where the curvature represents a field and a vector potential for the field is a quantity that usually depends on the path: it can change when an observer moves around a loop. As with two-dimensional curves, the curvature of a regular space curve C in three dimensions (and more) is the amplitude of the acceleration of a particle moving along a curve at unit velocity. So, if γ(s) is the parameterization of the arc length of C, then the unitary barntial vector T(s) is given by a point in the curve, where Fx = Fy = 0 is a singular point, which means that the curve at that point is not differentiable and therefore the curvature is not defined (most often the point is either a crossing point, or a summit). The curvature at a point on a surface assumes a variety of values when the plane varies from normal. As varied, it reaches a minimum and a maximum (which are in vertical directions), which are known as main curvatures. As Coxeter (1969, pp. 352-353) shows, the curvature of a differentiable curve was originally defined by osculating circles. In this shot, Augustin-Louis Cauchy showed that the point of curvature is the intersection of two normal lines infinitely narrow to the curve. [3] Be γ(t) = (x(t), y(t)) a correct parametric representation of a twice-differentiable plane curve.