Cesàro equation Information & Cesàro equation Links at HealthHaven.com

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

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

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

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 $\kappa = f'(s)\!$ .

[edit] References

The Mathematics Teacher. National Council of Teachers of Mathematics. 1908. pp. 402.
Edward Kasner (1904). The Present Problems of Geometry. Congress of Arts and Science: Universal Exposition, St. Louis. pp. 574.
J. Dennis Lawrence (1972). A catalog of special plane curves. Dover Publications. pp. 1-5. ISBN 0-486-60288-5.

[edit] External links

Weisstein, Eric W., "Cesàro Equation" from MathWorld.
Weisstein, Eric W., "Natural Equation" from MathWorld.
Curvature Curves at 2dcurves.com.