Cesàro equation

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 a particularly 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 = Cs, 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 $\kappa = f'(s)\!$.

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. 

External links

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