# 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

and

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

and

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

and so

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

and thus

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

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

and