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The distance from the center of a circle or sphere to its surface is its radius. For other curved lines or surfaces, the radius of curvature at a given point is the radius of a circle that mathematically best fits the curve at that point.
The equivalent "surface radius" that is described by radial distances at points along a body's surface is its radius of curvature (more formally, the radius of curvature of a curve at a point is the radius of the osculating circle at that point). With a sphere, the radius of curvature equals the radius {thus, radius of curvature is sometimes used as a synonym for radius). With an oblate ellipsoid (or, more properly, an oblate spheroid), however, not only does it differ from the radius, but it varies, depending on the direction being faced. The extremes are known as the principal radii of curvature.
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Imagine driving a car on a curvy road on a completely flat plain (so that the geographic plain is a geometric plane). At any one point along the way, lock the steering wheel in its position, so that the car thereafter follows a perfect circle. The car will, of course, deviate from the road, unless the road is also a perfect circle. The radius of that circle the car makes is the radius of curvature of the curvy road at the point at which the steering wheel was locked. The more sharply curved the road is at the point you locked the steering wheel, the smaller the radius of curvature.
If is a parameterized curve in then the radius of curvature at each point of the curve, , is given by
As a special case, if f(t) is a function from to , then the curvature of its graph, , is
Let be as above, and fix . We want to find the radius of a parameterized circle which matches in its zeroth, first, and second derivatives at . Clearly the radius will not depend on the position (), only on the velocity () and acceleration (). There are only three independent scalars that can be obtained from two vectors v and w, namely v·v, v·w, and w·w. Thus the radius of curvature must be a function of the three scalars , and .
The general equation for a parameterized circle in is
where is the center of the circle (irrelevant since it disappears in the derivatives), are perpendicular vectors of length (that is, and ), and is an arbitrary function which is twice differentiable at t.
The relevant derivatives of g work out to be
If we now equate these derivatives of g to the corresponding derivatives of at t we obtain
These three equations in three unknowns (, and ) can be solved for , giving the formula for the radius of curvature:
or, omitting the parameter (t) for readability,
The radius extremes of an oblate spheroid are the equatorial radius, or semimajor axis, a, and the polar radius, or semiminor axis, b. The "ellipticalness" of any ellipsoid, like any ellipse, is measured in different ways (e.g., eccentricity and flattening), any and all of which are trigonometric functions of its angular eccentricity, :
The primary parameter utilized in identifying a point's vertical position is its latitude, . A latitude can be expressed either directly or from the arcsine of a trigonometric product, the arguments (i.e., a function's "input") of the factors being the arc path (which defines, and is the azimuth at the equator of, a given great circle, or its elliptical counterpart) and the transverse colatitude, which is a corresponding, vertical latitude ring that defines a point along an arc path/great circle. The relationship can be remembered by the terms' initial letter, LAT:
Therefore, along a northsouth arc path (which equals 0°), the primary quadrant form of latitude equals the transverse colatitude's at a given point. As most introductory discussions of curvature and their radius identify position in terms of latitude, this article will too, with only the added inclusion of a "0" placeholder for more advanced discussions where the arc path is actively utilized: There are two types of latitude commonly employed in these discussions, the planetographic (or planetodetic; for Earth, the customized terms are "geographic" and "geodetic"), or , and reduced latitudes, or :
The calculation of elliptic quantities usually involves different elliptic integrals, the most basic unit integrands being , its inverse, , and its complement, :
A simple, if crude, definition of a circle is "a curved line bent in equal proportions, where its endpoints meet". Curvature, then, is the state and degree of deviation from a straight line—i.e., an "arced line". There are different interpretations of curvature, depending on such things as the planular angle the given arc is dividing and the direction being faced at the surface's point. What is concerned with here is normal curvature, where "normal" refers to orthogonality, or perpendicularity. There are two principal curvatures identified, a maximum, κ_{1}, and a minimum, κ_{2}.
The spot of least curvature on an oblate spheroid is at the poles, where the principal curvatures converge (as there is only one facing direction——towards the planetographic equator!) and the surface is most flattened.
A curvature's radius, RoC, is simply its reciprocal:
Therefore, there are two principal radii of curvature: A vertical, corresponding to κ_{1}, and a horizontal, corresponding to κ_{2}. Most introductions to the principal radii of curvature provide explanations independent to their curvature counterparts, focusing more on positioning and angle, rather than shape and contortion.
In fact, S is the function composition of M:
or, conversely and more appropriately, where :
One of Euler's many formulas gives the radius of curvature of the ellipse crosssectioned by a vertical plane in some direction other than northsouth or eastwest, in the normal section:
where is the geodetic azimuth of the line at the point: North equals zero, east equals 90 degrees. At the pole M = N, but at any other point M is the minimum radius of curvature of all the possible vertical crosssections through that point, while N is the maximum.
Euler's radius of curvature is similar to, yet distinctive from, the more inherent radius of curvature of the great elliptic arc^{[1]}, which is the geocentric radius of curvature in the great elliptic section, or arcradius, or . It is G that is the radius of curvature of the differential rectangle on an ellipsoid^{[2]}.
Radius of curvature is also used in a three part equation for bending of beams.
Stress in the semiconductor structure involving evaproated thin films usually results from the thermal expansion (thermal stress) during the manufacturing process. Thermal stress occurs because film depositions are usually made above room temperature. Upon cooling from the deposition temperature to room temperature, the difference in the thermal expansion coefficients of the substrate and the film cause thermal stress. ^{[3]}
Intrinsic stress results from the microstructure created in the film as atoms are deposited on the substrate. Tensile stress results from microvoids in the thin film, because of the attractive interaction of atoms across the voids.
The stress in thin film semiconductor structures results in the buckling of the wafers. The radius of the curvature of the stressed structure is related to stress tensor in the structure, and can be described by modified Stoney formula ^{[4]}. The topography of the stressed structure including radii of curvature can be measured using optical scanner methods. The modern scanner tools have capability to measure full topography of the substrate and to measure both principal radii of curvature, while providing the accuracy of the order of 0.1% for radii of curvature of 90 m and more ^{[5]}.
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