# Cesàro equation

In geometry, the Cesàro equation of a plane curve is an equation relating the curvature ($\kappa$ ) at a point of the curve to the arc length ($s$ ) from the start of the curve to the given point. It may also be given as an equation relating the radius of curvature ($R$ ) to arc length. (These are equivalent because $R=1/\kappa$ .) Two congruent curves will have the same Cesàro equation. Cesàro equations are named after Ernesto Cesàro.

## Examples

Some curves have a particularly simple representation by a Cesàro equation. Some examples are:

• Line: $\kappa =0$ .
• Circle: $\kappa =1/\alpha$ , where $\alpha$ is the radius.
• Logarithmic spiral: $\kappa =C/s$ , where $C$ is a constant.
• Circle involute: $\kappa =C/{\sqrt {s}}$ , where $C$ is a constant.
• Cornu spiral: $\kappa =Cs$ , where $C$ is a constant.
• Catenary: $\kappa ={\frac {a}{s^{2}+a^{2}}}$ .

## Related parameterizations

The Cesàro equation of a curve is related to its Whewell equation in the following way. If the Whewell equation is $\varphi =f(s)\!$ then the Cesàro equation is $\kappa =f'(s)\!$ .