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# Cesàro equation

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 Title: Cesàro equation Author: World Heritage Encyclopedia Language: English Subject: Collection: Publisher: World Heritage Encyclopedia Publication Date:

### 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 equation's 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: $\kappa=\frac\left\{a\right\}\left\{s^2+a^2\right\}$.

## Related parameterizations

The Cesàro equation of a curve is related to its Whewell equation in the following way, if the Whewell equation is $\varphi = f\left(s\right)\!$ then the Cesàro equation is $\kappa = f\text{'}\left(s\right)\!$.