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

References

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