Formulas of Curvature and Radius of Curvature

The commonly used results and formulas of curvature and radius of curvature are as shown below:

1. Curvature {\rm K} and radius of curvature \rho for a Cartesian curve is

{\rm K} = \frac{{\left| {\frac{{{d^2}y}}{{d{x^2}}}} \right|}}{{{{\left[ {1 + {{\left( {\frac{{dy}}{{dx}}} \right)}^2}} \right]}^{3/2}}}}


\rho = \frac{{{{\left[ {1 + {{\left( {\frac{{dy}}{{dx}}} \right)}^2}} \right]}^{3/2}}}}{{\left| {\frac{{{d^2}y}}{{d{x^2}}}} \right|}} = \frac{1}{{\rm K}}

2. If the equation of the curve is given by the implicit relation f\left( {x,y} \right) = 0, then

{\rm K} = \frac{{\left| { - {{\left( {{f_y}} \right)}^2}{f_{xx}} + 2{f_x}{f_y}{f_{xy}} - {{\left( {{f_x}} \right)}^2}{f_{yy}}} \right|}}{{{{\left[ {{{\left( {{f_x}} \right)}^2} + {{\left( {{f_y}} \right)}^2}} \right]}^{3/2}}}}


\rho = \frac{1}{{\rm K}}

3. If the curve is defined by parametric equations x = f\left( t \right) and y = f\left( t \right) then

{\rm K} = \frac{{\left| {f'\left( t \right)g''\left( t \right) - g'\left( t \right)f''\left( t \right)} \right|}}{{{{\left[ {{{f'}^2}\left( t \right) + {{g'}^2}\left( t \right)} \right]}^{3/2}}}}

and so  

\rho = \frac{1}{{\rm K}} = \frac{{{{\left[ {{{f'}^2}\left( t \right) + {{g'}^2}\left( t \right)} \right]}^{3/2}}}}{{\left| {f'\left( t \right)g''\left( t \right) - g'\left( t \right)f''\left( t \right)} \right|}}

4. For the curve r = f\left( \theta \right) i.e., the curve in polar coordinates

{\rm K} = \frac{{\left| {{r^2} + 2{{\left( {\frac{{dr}}{{d\theta }}} \right)}^2} - r\frac{{{d^2}r}}{{d{\theta ^2}}}} \right|}}{{{{\left[ {{r^2} + {{\left( {\frac{{dr}}{{d\theta }}} \right)}^2}} \right]}^{3/2}}}}

and thus

\rho = \frac{1}{{\rm K}}\frac{{{{\left[ {{r^2} + {{\left( {\frac{{dr}}{{d\theta }}} \right)}^2}} \right]}^{3/2}}}}{{\left| {{r^2} + 2{{\left( {\frac{{dr}}{{d\theta }}} \right)}^2} - r\frac{{{d^2}r}}{{d{\theta ^2}}}} \right|}}

5. For the pedal curve r = f\left( p \right) then,

\rho = r\frac{{dr}}{{dp}}

6. If \left( {\alpha ,\beta } \right) are the coordinates of the center of curvature of the curve y = f\left( x \right) at \left( {{x_1},{y_1}} \right) then

\alpha = {x_1} - \frac{{\frac{{dy}}{{dx}}\left[ {1 + {{\left( {\frac{{dy}}{{dx}}} \right)}^2}} \right]}}{{\frac{{{d^2}y}}{{d{x^2}}}}}


\beta = {y_1} + \frac{{1 + {{\left( {\frac{{dy}}{{dx}}} \right)}^2}}}{{\frac{{{d^2}y}}{{d{x^2}}}}}