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In [[geometry]], the '''radius of curvature''', ''R'', of a curve at a point is a measure of the [[radius]] of the [[circular arc]] which best approximates the curve at that point. It is the inverse of the [[curvature]].
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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
 
: <math>\frac{ds}{d\varphi} = \frac{1}{\kappa},</math>
 
where ''s'' is the arc length from a fixed point on the curve, ''φ'' is the [[tangential angle]] and <math>\scriptstyle\kappa</math> is the [[curvature#Curvature|curvature]].
 
If the curve is given in [[Cartesian coordinates]] as ''y(x)'', then the radius of curvature is (assuming the curve is differentiable up to order 2):
 
: <math>R =\left| \frac { \left(1 + y'^{\,2}\right)^{3/2}}{y''}\right|,  
\qquad\mbox{where}\quad
y' = \frac{dy}{dx},\quad y'' = \frac{d^2y}{dx^2},</math> 
and | ''z'' | denotes the absolute value of ''z''.
 
If the curve is given [[parametric equation|parametrically]] by functions  ''x(t)''  and  ''y(t)'', then the radius of curvature is
:<math>R = \;\left|\frac{ds}{d\varphi}\right| \;= \;\left|\frac {\big({\dot{x}^2 + \dot{y}^2}\big)^{3/2}}{\dot {x}\ddot{y} - \dot{y}\ddot{x}}\right|,
\qquad\mbox{where}\quad
\dot{x} = \frac{dx}{dt},\quad\ddot{x} = \frac{d^2x}{dt^2},\quad
\dot{y} = \frac{dy}{dt},\quad\ddot{y} = \frac{d^2y}{dt^2}.</math>
 
Heuristically, this result can be interpreted as
:<math> R =  \frac{\left|\mathbf{v}\right|^3}{\left| \mathbf{v}  \times \mathbf{ \dot v} \right|},
\qquad\mbox{where}\quad
\left| \mathbf{v} \right| = \left| (\dot x, \dot y) \right| = R \frac{d\varphi}{dt}.</math>
 
==Examples==
===Semicircles and circles===
 
For a [[semi-circle]] of radius ''a''  in the upper half-plane
: <math>
y=\sqrt{a^2-x^2}, \quad
y'=\frac{-x}{\sqrt{a^2-x^2}}, \quad
y''=\frac{-a^2}{(a^2-x^2)^{3/2}},\quad
R=|-a| =a.
</math>
 
[[Image:Ellipse evolute.svg||right|thumb|240px|An ellipse (red) and its [[evolute]] (blue). The dots are the vertices of the ellipse, at the points of greatest and least curvature.]]
 
For a semi-circle  of radius ''a''  in the lower half-plane
: <math>
y=-\sqrt{a^2-x^2}, \quad
R=|a|=a.
</math>
The [[circle]] of radius ''a'' has a radius of curvature equal to ''a''.
 
===Ellipses===
 
In an [[ellipse]] with major axis 2''a'' and minor axis 2''b'', the [[Vertex (curve)|vertices]] on the major axis have the smallest radius of curvature of any points <math>\left( R = \frac{b^2}{a} \right) </math>, and the vertices on the minor axis have the largest radius of curvature of any points <math>\left( R = \frac{a^2}{b} \right) </math>.
 
== See also ==
 
* [[Osculating circle]]
 
==References==
*{{cite book |title = Differential Geometry of Curves and Surfaces|first = Manfredo|last = do Carmo| id = ISBN 0-13-212589-7 | year = 1976}}
 
[[Category:Curvature (mathematics)]]
[[Category:Curves]]
[[Category:Integral calculus]]
[[Category:Multivariable calculus]]
[[Category:Theoretical physics]]

Latest revision as of 18:48, 4 December 2014

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