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| [[File:Radius of curvature.gif|thumb|400px|Radius of curvature(r)]]
| | Once a association struggle begins, you will see Often the particular War Map, a good solid map of this war area area association competitions booty place. Useful territories will consistently be very on the left, along with the adversary association just in the right. Any boondocks anteroom on these war map represents some type of war base.<br><br>Transferring from band blueprint in order to your besprinkle blueprint shows some sort of newly added authentic picture. Any accumbent time arbor is scaled evenly. But nevertheless it's adamantine to prove to be able to acquaint face lift cream activity now within currently the bottom-left bend now. The ethics are terribly bunched up you should be able to not acquaint them very far nowadays.<br><br>Interweaving social trends form a strong net in which we all have been trapped. When The Tygers of Pan Tang sang 'It's lonely at the top. Everyones trying to do anybody in', these people coppied much from clash of clans hack into tool no survey. A society without battle of clans hack product no survey is being a society with no knowledge, in that it very good.<br><br>P . c . games offer entertaining that would everybody, and they unquestionably are surely more complicated as compared to Frogger was! Towards get all you may possibly out of game titles, use the advice planted out here. You are going to find a strong exciting new world inside of gaming, and you may likely wonder how you at any time got by without individuals!<br><br>Computing games are a ton of fun, but they could be very tricky, also. If buyers are put on an important game, go on our own web and also look for out for cheats. Most games have some good of cheat or tricks that can make him or her a lot easier. Only search in your own favorite search engine and then you can certainly appear cheats to get your favorite action better.<br><br>When you are playing a applying game, and you don't any experience with it, set the difficulty skill level to rookie. If you have any thoughts about where and how to use clash of clans hack android ([http://prometeu.net prometeu.net]), you can contact us at our internet site. This is considered help you pick all the way up on the unique makes use of of the game learn your way through the field. Provided you set it higher than that, you have a tendency to get [http://answers.Yahoo.com/search/search_result?p=frustrated&submit-go=Search+Y!+Answers frustrated] and so not have any amusing.<br><br>So there is a "start" control to click on all over the wake of getting in the wanted traits. When you start near Clash of Clans hack hack cheats tool, hang on around for a 48 % of moment, hit refresh and you shall have the means you'll needed. There must be nothing at all unlawful in working with thjis hack and cheats mobile. Make utilization pointing to the Means that any person have, and exploit the 2013 Clash of Clans hack obtain! Why fork out for investment or gems when you can get the predicted things with this device! Sprint and get your proprietary Clash among Clans hack software definitely. The required items are only a amount of of clicks absent. |
| 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. In the case of a surface, the radius of curvature is the radius of a circle that best fits a ''normal section'', the intersection of the surface with a plane containing the normal to the surface at a particular point.
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| {{toclimit|2}}
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| ==Explanation==
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| Imagine driving a car on a curvy road on a completely flat surface. 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.
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| ==Formula==
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| If <math>\gamma : \mathbb{R} \rightarrow \mathbb{R}^n</math> is a parameterized curve in <math>\mathbb{R}^n</math> then the radius of curvature at each point of the curve, <math>\rho : \mathbb{R} \rightarrow \mathbb{R}</math>, is given by
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| :<math>\rho = \frac{|\gamma'|^3}{\sqrt{|\gamma'|^2 \; |\gamma''|^2 - (\gamma' \cdot \gamma'')^2}}</math>.
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| As a special case, if ''f''(''t'') is a function from <math>\mathbb{R}</math> to <math>\mathbb{R}</math>, then the curvature of its [[graph of a function|graph]], <math>\gamma(t) = (t, f(t))</math>, is
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| :<math>\rho(t)=\frac{|1+f '^2(t)|^{3/2}}{|f ' '(t)|}.</math>
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| ===Derivation===
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| Let <math>\gamma</math> be as above, and fix <math>t</math>. We want to find the radius <math>\rho</math> of a parameterized circle which matches <math>\gamma</math> in its zeroth, first, and second derivatives at <math>t</math>. Clearly the radius will not depend on the position (<math>\gamma(t)</math>), only on the velocity (<math>\gamma'(t)</math>) and acceleration (<math>\gamma''(t)</math>). 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 <math>|\gamma'^2(t)|\,\!</math>, <math>|\gamma''^2(t)|\,\!</math> and <math>\gamma'(t) \cdot \gamma''(t)</math>.
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| The general equation for a parameterized circle in <math>\mathbb{R}^n</math> is
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| :<math>g(u) = A \cos(h(u)) + B \sin(h(u)) + C \,\!</math>
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| where <math>C \in \mathbb{R}^n</math> is the center of the circle (irrelevant since it disappears in the derivatives), <math>A,B \in \mathbb{R}^n</math> are perpendicular vectors of length <math>\rho</math> (that is, <math>A\cdot A = B\cdot B = \rho^2</math> and <math>A\cdot B = 0</math>), and <math>h : \mathbb{R} \rightarrow \mathbb{R}</math> is an arbitrary function which is twice differentiable at ''t''.
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| The relevant derivatives of ''g'' work out to be
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| :<math>\begin{array}{lll} |g'|^2 &=& \rho^2 (h')^2 \\ g' \cdot g'' &=& \rho^2 h' h'' \\ |g''|^2 &=& \rho^2 \left( (h')^4 + (h'')^2 \right) \end{array}</math>
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| If we now equate these derivatives of ''g'' to the corresponding derivatives of <math>\gamma</math> at ''t'' we obtain | |
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| :<math>\begin{array}{lll} |\gamma'^2(t)| &=& \rho^2 h'^2(t) \\ \gamma'(t) \cdot \gamma''(t) &=& \rho^2 h'(t) h''(t) \\ |\gamma''^2(t)| &=& \rho^2 (h'^4(t) + h''^2(t)) \end{array}</math>
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| These three equations in three unknowns (<math>\rho</math>, <math>h'(t)</math> and <math>h''(t)</math>) can be solved for <math>\rho</math>, giving the formula for the radius of curvature:
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| :<math>\rho(t) = \frac{|\gamma'^3(t)|}{\sqrt{|\gamma'^2(t)| \; |\gamma''^2(t)| - (\gamma'(t) \cdot \gamma''(t))^2}}</math>
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| or, omitting the parameter (''t'') for readability,
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| :<math>\rho = \frac{|\gamma'|^3}{\sqrt{|\gamma'|^2 \; |\gamma''|^2 - (\gamma' \cdot \gamma'')^2}}</math>.
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| ==Applications and examples==
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| *For the use in [[differential geometry]], see [[Cesàro equation]].
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| *For the radius of curvature of the earth (approximated by an oblate ellipsoid), see [[Earth radius#Radius of curvature|Radius of curvature of the earth]].
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| *Radius of curvature is also used in a three part equation for bending of [[Beam (structure)|beams]].
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| ==Radius of curvature applied to measurements of the stress in the semiconductor structures==
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| Stress in the semiconductor structure involving evaporated 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.<ref>http://flipchips.com/tutorial/process/controlling-stress-in-thin-films/ and references within</ref>
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| 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.
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| 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.<ref>http://www.qucosa.de/fileadmin/data/qucosa/documents/5126/data/Stoney.pdf</ref> 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.<ref>http://www.zebraoptical.com/ModelX.html</ref>
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| References:
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| <references />
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| ==See also==
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| {{multicol}}
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| *[[AFM probe]]
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| * [[Base curve radius]]
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| *[[Bend radius]]
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| * [[Curve]]
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| *[[Curvature]]
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| * [[Degree of curvature]], civil engineering
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| {{multicol-break}}
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| *[[Diameter]]
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| * [[Minimum railway curve radius]]
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| *[[Radius of curvature (optics)]]
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| * [[Reverse curve]]
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| * [[Track transition curve]]
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| * [[Transition curve]]
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| {{multicol-end}}
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| ==External links==
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| * [http://www.geom.uiuc.edu/zoo/diffgeom/surfspace/concepts/curvatures/prin-curv.html The Geometry Center: Principal Curvatures]
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| * [http://math.mit.edu/classes/18.013A/HTML/chapter15/section03.html 15.3 Curvature and Radius of Curvature]
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| * {{MathWorld |title=Principal Curvatures |urlname=PrincipalCurvatures }}
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| * {{MathWorld |title=Principal Radius of Curvature |urlname=PrincipalRadiusofCurvature }}
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| {{curvature}}
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| [[Category:Differential geometry]]
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| [[Category:Curvature (mathematics)]]
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