In differential geometry , the radius of curvature , r , is the reciprocal of curvature . For a curve , it is equal to the radius of the circular arc that best approximates the curve at that point. For surfaces , the radius of curvature is the radius of a circle that best fits a normal section or combination .
Definition
In the case of a space curve , the radius of curvature is the length of the curvature vector .
In the case of a plane curve , then r is the absolute value of
{\displaystyle R\equiv \left|{\frac {ds}{d\varphi }}\right|={\frac {1}{\kappa }},}where s is the arc length from a fixed point on the curve, is the tangent angle and is the curvature .
Formula
2D. In
If the curve is given as y ( x ) in Cartesian coordinates , then the radius of curvature is (assuming that the curve is differentiable on the order of 2):
{\displaystyle R=\left|{\frac {\left(1+y'^{\,2}\right)^{\frac {3}{2}}}{y''}}\right|,\qquad {{where}}\quad y'={\frac {dy}{dx}},\quad y''={\frac {d^{2}y}{dx^{2}}},}and | Jade | Indicates the absolute value of z .
If the curve is given parametrically by the x ( t ) and y ( t ) functions , then the radius of curvature is
{\displaystyle R=\left|{\frac {ds}{d\varphi }}\right|=\left|{\frac {\left({{\dot {x}}^{2}+{\dot {y}}^{2}}\right)^{\frac {3}{2}}}{{\dot {x}}{\ddot {y}}-{\dot {y}}{\ddot {x}}}}\right|,\qquad {{where}}\quad {\dot {x}}={\frac {dx}{dt}},\quad {\ddot {x}}={\frac {d^{2}x}{dt^{2}}},\quad {\dot {y}}={\frac {dy}{dt}},\quad {\ddot {y}}={\frac {d^{2}y}{dt^{2}}}.}Presumably, this result can be interpreted as
{\displaystyle R={\frac {\left|\mathbf {v} \right|^{3}}{\left|\mathbf {v} \times \mathbf {\dot {v}} \right|}},\qquad {{where}}\quad \left|\mathbf {v} \right|={\big |}({\dot {x}},{\dot {y}}){\big |}=R{\frac {d\varphi }{dt}}.}in n dimensions
If : → n is a parametrized curve in n then the radius of curvature at each point of the curve, : → , is given by
{\displaystyle \rho ={\frac {\left|{\boldsymbol {\gamma }}'\right|^{3}}{\sqrt {\left|{\boldsymbol {\gamma }}'\right|^{2}\,\left|{\boldsymbol {\gamma }}''\right|^{2}-\left({\boldsymbol {\gamma }}'\cdot {\boldsymbol {\gamma }}''\right)^{2}}}}}.As a special case, if f ( t ) is a function from to , then the radius of curvature of its graph , ( t ) = ( t , f ( t ) ) , is
{\displaystyle \rho (t)={\frac {\left|1+f'^{\,2}(t)\right|^{\frac {3}{2}}}{\left|f''(t)\right|}}.}Let be as above, and t ok . We want to find the radius of a parametrized circle which corresponds to in its zeroth , first, and second derivatives t . Obviously the radius will not depend on position ( t ) , only on velocity ‘( t ) and acceleration “( t ) . 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 three scalars .
‘ ( t ) | 2 , | ” ( t ) | 2 and ‘ ( t ) ” ( t ) .
The general equation for a parametrized circle in n is
{\displaystyle \mathbf {g} (u)=\mathbf {a} \cos h(u)+\mathbf {b} \sin h(u)+\mathbf {c} }where c n is the center of the circle (irrelevant since it disappears in derivatives), a , b n are perpendicular vectors of length ( that is , a a = b b = 2and a b = 0 ) , and h : → is an arbitrary function that is twice differentiable at t .
The corresponding derivative of g is
{\displaystyle {\begin{aligned}|\mathbf {g} '|^{2}&=\rho ^{2}(h')^{2}\\\mathbf {g} '\cdot \mathbf {g} ''&=\rho ^{2}h'h''\\|\mathbf {g} ''|^{2}&=\rho ^{2}\left((h')^{4}+(h'')^{2}\right)\end{aligned}}}If we now equate these derivatives of g then for the corresponding derivatives of g at t we get
{\displaystyle {\begin{aligned}|{\boldsymbol {\gamma }}'(t)|^{2}&=\rho ^{2}h'^{\,2}(t)\\{\boldsymbol {\gamma }}'(t)\cdot {\boldsymbol {\gamma }}''(t)&=\rho ^{2}h'(t)h''(t)\\|{\boldsymbol {\gamma }}''(t)|^{2}&=\rho ^{2}\left(h'^{\,4}(t)+h''^{\,2}(t)\right)\end{aligned}}}The three unknowns (in these three equations , h ‘ ( t ) and h “( t ) ) can be solved for , giving the formula for the radius of curvature:
{\displaystyle \rho (t)={\frac {\left|{\boldsymbol {\gamma }}'(t)\right|^{3}}{\sqrt {\left|{\boldsymbol {\gamma }}'(t)\right|^{2}\,\left|{\boldsymbol {\gamma }}''(t)\right|^{2}-{\big (}{\boldsymbol {\gamma }}'(t)\cdot {\boldsymbol {\gamma }}''(t){\big )}^{2}}}}}or, omitting the parameter t for readability,
{\displaystyle \rho ={\frac {\left|{\boldsymbol {\gamma }}'\right|^{3}}{\sqrt {\left|{\boldsymbol {\gamma }}'\right|^{2}\;\left|{\boldsymbol {\gamma }}''\right|^{2}-\left({\boldsymbol {\gamma }}'\cdot {\boldsymbol {\gamma }}''\right)^{2}}}}.}Example
semicircle and circle
In the plane one upper half of the radius of the semi-circle for a
{\displaystyle y={\sqrt {a^{2}-x^{2}}},\quad y'={\frac {-x}{\sqrt {a^{2}-x^{2}}}},\quad y''={\frac {-a^{2}}{\left(a^{2}-x^{2}\right)^{\frac {3}{2}}}},\quad R=|-a|=a.}For a semicircle of radius a in the lower half plane
{\displaystyle y=-{\sqrt {a^{2}-x^{2}}},\quad R=|a|=a.}One of the radius of the circle is equal to a radius of curvature a .
multiple points
In an ellipse with major axis 2 a and minor axis 2 b , the radius of curvature of any point in the vertices of the major axis is the smallest, R = B2 / a , and any point on the minor axis has the greatest radius of curvature, R = a2 / b.
Application
- See the Cesaro equation for use in differential geometry .
- for the radius of curvature of the Earth (a flat ellipsoid its approximation); See also: arc measurement
- The radius of curvature is also used in the three part equation for the bending of the beam .
- radius of curvature (optics)
- thin film technologies
- printed electronics
stress in semiconductor structures
Stresses in semiconductor structures that include evaporated thin films usually result from thermal expansion (thermal stress) during the manufacturing process. Thermal stress occurs because the deposition of the film usually occurs above room temperature. Upon cooling from the deposition temperature to room temperature, the difference in thermal expansion coefficients of the substrate and the film causes thermal stress. [4]
Internal stresses result from the microstructure created in the film as the atoms are deposited on the substrate. Tensile stress arises from microvoids (small holes, considered defects) in the thin film, because of the attractive interactions of atoms in the vacancies.
The stress buckling of wafers results in thin film semiconductor structures. The radius of curvature of the strained structure is related to the strain tensor in the structure, and can be described by a modified Stony formula. The topography of the strained structure, including the radius of curvature, can be measured using optical scanner methods. Modern scanner instruments have the ability to measure the absolute topography of the substrate and both major radii of curvature, while providing an accuracy of the order of 0.1% for radii of curvature of 90 m and more.
