The radius of curvature of the path is ρ as found from the rate of rotation of the tangent to the curve with respect to arc length, and is the radius of the osculating circle at position s. The unit circle on the left shows the rotation of the unit vectors with s.

A center of curvature is defined at each position s located a distance ρ ( the radius of curvature ) from the curve on a line along the normal u < sub > n </ sub > ( s ).

The points on the two branches that are closest to each other are called the vertices ; they are the points where the curve has its smallest radius of curvature.

* Radius of curvature, a measure of how gently a curve bends

The curvature of the curve of intersection is the sectional curvature.

The intersection of a plane containing the normal with the surface will form a curve called a normal section and the curvature of this curve is the normal curvature.

If q is the product of that curvature with the circle's radius, signed positive for epi-and negative for hypo -, then the curve: evolute similitude ratio is 1 + 2q.

* Vertex ( curve ), a local extreme point of curvature

Since Earth is round, if their downward bending curve is about the same as the curvature of the Earth, light rays can travel large distances, perhaps from beyond the horizon.

In other words, if a planet, M, follows an elliptical orbit around a focus F and has a point P where PM is tangent to the curve and FPM is a right angle so that FP is the perpendicular to the tangent, then the centripetal force at point P is proportional to F * M /( R *( F * P )< sup > 3 </ sup >) where R is the radius of the curvature at M. Johann Bernoulli proved this formula in 1710.

The curvature of a smooth curve is defined as the curvature of its osculating circle at each point.

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.

Cauchy defined the centre of curvature C as the intersection point of two infinitely close normals to the curve, the radius of curvature as the distance from the point to C, and the curvature itself as the inverse of the radius of curvature.

Given any curve C and a point P on it, there is a unique circle or line which most closely approximates the curve near P, the osculating circle at P. The curvature of C at P is then defined to be the curvature of that circle or line.

The relative forces can be calculated from the various radii of curvature if we assume: ( A ) The surface tension is uniform on the surface of the drop.

An expanding universe generally has a cosmological horizon which, by analogy with the more familiar horizon caused by the curvature of the Earth's surface, marks the boundary of the part of the universe that an observer can see.

Informally, the theorem says that the curvature of a surface can be determined entirely by measuring angles and distances on the surface.

From a qualitative standpoint, the path can be approximated by an arc of a circle for a limited time, and for the limited time a particular radius of curvature applies, the centrifugal and Euler forces can be analyzed on the basis of circular motion with that radius.

For instance, volume and Riemannian curvature are invariants that can distinguish different geometric structures on the same smooth manifold — that is, one can smoothly " flatten out " certain manifolds, but it might require distorting the space and affecting the curvature or volume.

In general relativity, in any region small enough for the curvature of spacetime to be negligible one can find a set of inertial frames that approximately describe that region.

However, the general theory reduces to the special theory over sufficiently small regions of spacetime, where curvature effects become less important and the earlier inertial frame arguments can come back into play.

Extra curvature can be introduced into knitted garments without seams, as in the heel of a sock ; the effect of darts, flares, etc.

Results of scientific experiments published in 1976 involving Kirlian photography of living tissue ( human finger tips ) showed that most of the variations in corona discharge streamer length, density, curvature and color can be accounted for by the moisture content on the surface of and within the living tissue.

The idea is to try to improve this metric ; for example, if the metric can be improved enough so that it has constant curvature, then it must be the 3-sphere.

Furthermore, for mathematicians, according to this formula the gluon color field can be represented by a SU ( 3 )- Lie algebra-valued " curvature "- 2-form where is a " vector potential "- 1-form corresponding to G and is the ( antisymmetric ) " wedge product " of this algebra, producing the " structure constants " f < sup > abc </ sup >.

Just as the curvature of the earth's surface is not noticeable in everyday life, the curvature of spacetime can be neglected on small scales, so that locally, special relativity is a valid approximation to general relativity.

Hence, bending a surface will not alter the Gaussian curvature and other surfaces with constant positive Gaussian curvature can be obtained by cutting a small slit in the sphere and bending it.

Experimentation has led to boards with rocker, or upward curvature, which makes for a more buttery board and can improve float in deep powder.

This metric has only two undetermined parameters: an overall length scale R that can vary with time, and a curvature index k that can be only 0, 1 or − 1, corresponding to flat Euclidean geometry, or spaces of positive or negative curvature.

First and most importantly, the length scale R of the universe can remain constant only if the universe is perfectly isotropic with positive curvature ( k = 1 ) and has one precise value of density everywhere, as first noted by Albert Einstein.

Paper exhibits zero Gaussian curvature at all points on its surface, and only folds naturally along lines of zero curvature.

Unlike the Riemann curvature tensor or the Ricci tensor, which both can be naturally defined for any affine connection, the scalar curvature requires a metric of some kind.

An anti de Sitter space, in contrast, is a general relativity-like spacetime, where in the absence of matter or energy, the curvature of spacetime is naturally hyperbolic.

Such scale breaking terms arise more naturally in a constraint formalism, the so-called BF scheme, in which the gauge curvature is denoted by F. In the case of gravity, it departs from the meta-linear group SL ( 5, R ) in four dimensions, thus generalizing ( Anti -) de Sitter gauge theories of gravity.

As such, a line element is then naturally a function of the metric, and can be related to the curvature tensor.

The principle is simple: when passing between the poles of a magnet, a monoenergetic beam of ions of naturally occurring uranium splits into several streams according to their momentum, one per isotope, each characterized by a particular radius of curvature ; collecting cups at the ends of the semicircular trajectories catch the homogeneous streams.

Quaternion-Kähler manifolds divide naturally into those with positive and negative Ricci curvature.

It is thought to be the earliest painting to show the curvature of the Earth from a great height.

If the above errors be eliminated, the two astigmatic surfaces united, and a sharp image obtained with a wide aperture — there remains the necessity to correct the curvature of the image surface, especially when the image is to be received upon a plane surface, e. g. in photography.

The exact splitting algorithm is implementation dependent, only the flatness criteria must be respected to reach the necessary precision and to avoid non-monotonic local changes of curvature.

That is, curvature does not depend on how the surface might be embedded in 3-dimensional space or 2-dimensional space.

Taken as a physical description of space, postulate 2 ( extending a line ) asserts that space does not have holes or boundaries ( in other words, space is homogeneous and unbounded ); postulate 4 ( equality of right angles ) says that space is isotropic and figures may be moved to any location while maintaining congruence ; and postulate 5 ( the parallel postulate ) that space is flat ( has no intrinsic curvature ).

Both distortions would need to create a very strong curvature in a highly localized region of space-time and their gravity fields would be immense.

Fine frets, much flatter, allow a very low string-action but require other conditions, such as curvature of the neck, to be well-maintained to prevent buzz.

Two interesting experiments were devised by Newton to demonstrate how these forces could be discovered, thereby revealing to an observer that they were not in an inertial frame: the example of the tension in the cord linking two spheres rotating about their center of gravity, and the example of the curvature of the surface of water in a rotating bucket.

with R < sub > AB </ sub > the five-dimensional Ricci curvature, may be re-expressed so that in four dimensions, these solutions satisfy Einstein's equations

They are usually shaped to fit in a roughly oval, not circular, frame ; the optical centers are placed over the eyeballs ; their curvature may not be axially symmetric to correct for astigmatism.

The scale statement may be taken as exact when the region mapped is small enough for the curvature of the Earth to be neglected, for example in a town planner's city map.

Over larger regions where the curvature cannot be ignored we must use map projections from the curved surface of the Earth ( sphere or ellipsoid ) to the plane.

A peculiarity of Pellucidar's geography is that due to the concave curvature of its surface there is no horizon ; the further distant something is, the higher it appears to be, until it is finally lost in the atmospheric haze.