If A is the major axis of an ellipsoid and B and C are the other two axes, the radius of curvature in the ab plane at the end of the axis Af, and the difference in pressure along the A and B axes is Af.

For example, it follows that any closed oriented Riemannian surface can be C < sup > 1 </ sup > isometrically embedded into an arbitrarily small ε-ball in Euclidean 3-space ( there is no such C < sup > 2 </ sup >- embedding since from the formula for the Gauss curvature an extremal point of such an embedding would have curvature ≥ ε < sup >- 2 </ sup >).

If the vertical temperature gradient is + 12. 9 ° C per 100 meters ( where the positive sign means temperature gets hotter as one goes higher ), then horizontal light rays will just follow the curvature of the Earth, and the horizon will appear flat.

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.

As in the case of curves in two dimensions, the curvature of a regular space curve C in three dimensions ( and higher ) is the magnitude of the acceleration of a particle moving with unit speed along a curve.

The curvature of C at P is given by the limit

An intrinsic definition of the Gaussian curvature at a point P is the following: imagine an ant which is tied to P with a short thread of length r. She runs around P while the thread is completely stretched and measures the length C ( r ) of one complete trip around P. If the surface were flat, she would find C ( r ) = 2πr.

On curved surfaces, the formula for C ( r ) will be different, and the Gaussian curvature K at the point P can be computed by the Bertrand – Diquet – Puiseux theorem as

If the curvature of a surface of a lens is C and the index of refraction is n, the optical power is φ = ( n − 1 ) C. If both surfaces of the lens are curved, consider their curvatures as positive toward the lens and add them.

The magnitude of the centripetal force on an object of mass m moving at tangential speed v along a path with radius of curvature r is:

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 ).

Looking at the image above, one might wonder whether adequate account has been taken of the difference in curvature between ρ ( s ) and ρ ( s + ds ) in computing the arc length as ds = ρ ( s ) dθ.

He secured the feathers at their midpoints with string and at their bases with wax, and gave the whole a gentle curvature like the wings of a bird.

The acceleration at position r is equal to the curvature of the hyperbola at fixed r, and like the curvature of the nested circles in polar coordinates, it is equal to 1 / r.

Although the theory of special relativity forbids objects to have a relative velocity greater than light speed, and general relativity reduces to special relativity in a local sense ( in small regions of spacetime where curvature is negligible ), general relativity does allow the space between distant objects to expand in such a way that they have a " recession velocity " which exceeds the speed of light, and it is thought that galaxies which are at a distance of more than about 14 billion light-years from us today have a recession velocity which is faster than light.

When a ship is at the horizon, its lower part is obscured due to the curvature of the Earth.

This process is self-regulating since the atoms that are at positions of high local curvature, such as adatoms or ledge atoms, are removed preferentially.

The radius of curvature of the ellipsoid varies with latitude, being the longest at the pole and the shortest at the equator as is the nautical mile.

Even this process, they argue, is hindered by the curvature at the base of the hair.

As a consequence of this curvature, it is not a given in general relativity that inertial objects moving at a particular rate with respect to each other will continue to do so.

Kissaki have a curved profile, and smooth three-dimensional curvature across their surface towards the edge — though they are bounded by a straight line called the yokote and have crisp definition at all their edges.

The double curvature concrete arch dam was constructed between 1955 and 1959 by Impresit of Italy at a cost of $ 135, 000, 000 for the first stage with only the Kariba South power cavern.

#: The mean curvature is the average of the two principal curvatures and as these are constant at all points of the sphere then so is the mean curvature.

In this spacetime, it is possible to come up with coordinate systems such that if you pick a hypersurface of constant time ( a set of points that all have the same time coordinate, such that every point on the surface has a space-like separation, giving what is called a ' space-like surface ') and draw an " embedding diagram " depicting the curvature of space at that time, the embedding diagram will look like a tube connecting the two exterior regions, known as an " Einstein – Rosen bridge ".

To collect all the necessary information, the crystal must be rotated step-by-step through 180 °, with an image recorded at every step ; actually, slightly more than 180 ° is required to cover reciprocal space, due to the curvature of the Ewald sphere.

Centripetal force ( from Latin centrum " center " and petere " to seek ") is a force that makes a body follow a curved path: its direction is always orthogonal to the velocity of the body, toward the fixed point of the instantaneous center of curvature of the path.

The curvature of the surface near the tip causes a natural magnification — ions are repelled in a direction roughly perpendicular to the surface ( a " point projection " effect ).

The point on the axis of symmetry that intersects the parabola is called the " vertex ", and it is the point where the curvature is greatest.

For relatively short distances-where the earth's curvature may be neglected, the run is the difference in distance from a fixed point measured along a level, horizontal line, or in other words,

* Another reason for using the Cassegrain design is to increase the focal length of the antenna, to improve the field of view Parabolic reflectors used in dish antennas have a large curvature and short focal length, to locate the focal point near the mouth of the dish, to reduce the length of the supports required to hold the feed structure or secondary reflector.

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

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.

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 Riemannian manifold.

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

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.

Just as the pressure exerted by surface tension in a spherical drop is Af and the pressure exerted by surface tension on a cylindrical shape is Af, the pressure exerted by any curved surface is Af, where **yg is the interfacial tension and Af and Af are the two radii of curvature.

The curvature of this bluntness is, in the case of the Carboloy knife employed in the Hesiometer, determined by the grain sizes of the polished grit and the tungsten carbide crystals cemented together in the knife body and is in the order of 0.1 to 0.2 mil..

An electrostatic system suffers generally from image plane curvature leading to defocusing in the peripheral image region if a flat viewing screen ( or interstage coupler ) is utilized, while a magnetic system requires accurate adjustment of the solenoid, which is heavy and bulky.

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.

From this it follows that correctness of drawing depends solely upon the principal rays ; and is independent of the sharpness or curvature of the image field.

Trans., 1830, 3, p. 1 ) is fulfilled in all systems which are symmetrical with respect to their diaphragm ( briefly named symmetrical or holosymmetrical objectives ), or which consist of two like, but different-sized, components, placed from the diaphragm in the ratio of their size, and presenting the same curvature to it ( hemisymmetrical objectives ); in these systems tan w ' / tan w

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.

However, the explanation of this curvature involves centrifugal force for all observers with the exception of a truly stationary observer, who finds the curvature is consistent with the rate of rotation of the water as they observe it, with no need for an additional centrifugal force.

The curvature of the fretboard is measured by the fretboard radius, which is the radius of a hypothetical circle of which the fretboard's surface constitutes a segment.

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

In his Two New Sciences ( 1638 ), Galileo says that a hanging cord is an approximate parabola, and he correctly observes that this approximation improves as the curvature gets smaller and is almost exact when the elevation is less than 45 °.

The radius of curvature is then