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In [[geometry]], the '''Cesàro equation''' of a [[plane curve]] is an [[equation]] relating the [[curvature]] (<math>\kappa</math>) at a point of the curve to the [[arc length]] (<math>s</math>) from the start of the curve to the given point. It may also be given as an equation relating the [[Radius of curvature (mathematics)|radius of curvature]] (<math>R</math>) to [[arc length]]. (These are equivalent because <math>R = 1/\kappa</math>.) Two [[congruence (geometry)|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 (geometry)|Line]]: <math>\kappa = 0</math>.
* [[Circle]]: <math>\kappa = 1/\alpha</math>, where <math>\alpha</math> is the radius.
* [[Logarithmic spiral]]: <math>\kappa=C/s</math>, where <math>C</math> is a constant.
* [[Involute|Circle involute]]: <math>\kappa=C/\sqrt s</math>, where <math>C</math> is a constant.
* [[Cornu spiral]]: <math>\kappa=Cs</math>, where <math>C</math> is a constant.
* [[Catenary]]: <math>\kappa=\frac{a}{s^2+a^2}</math>.

==Related parameterizations==
The Cesàro equation of a curve is related to its [[Whewell equation]] in the following way, if the Whewell equation is 
<math>\varphi = f(s)\!</math>
then the Cesàro equation is
<math>\kappa = f'(s)\!</math>.

==References==
* {{cite book | title=The Mathematics Teacher | year=1908 | publisher=National Council of Teachers of Mathematics | pages=402 }}
* {{cite book | author=Edward Kasner | title=The Present Problems of Geometry | publisher=Congress of Arts and Science: Universal Exposition, St. Louis | year=1904 | pages=574 }}
* {{cite book | author=J. Dennis Lawrence | title=A catalog of special plane curves | publisher=Dover Publications | year=1972 | isbn=0-486-60288-5 | pages=1–5 }}

==External links==
* {{MathWorld | title=Cesàro Equation | urlname=CesaroEquation }}
* {{MathWorld | title=Natural Equation | urlname=NaturalEquation }}
* [http://www.2dcurves.com/derived/curvature.html#curvature Curvature Curves] at 2dcurves.com.

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[[Category:Curves]]