Cesàro equation

In geometry, the Cesàro equation of a plane curve is an equation relating the curvature (${\displaystyle \kappa }$) at a point of the curve to the arc length (${\displaystyle s}$) from the start of the curve to the given point. It may also be given as an equation relating the radius of curvature (${\displaystyle R}$) to arc length. (These are equivalent because ${\displaystyle R=1/\kappa }$.) 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: ${\displaystyle \kappa =0}$.
• Circle: ${\displaystyle \kappa =1/\alpha }$, where ${\displaystyle \alpha }$ is the radius.
• Logarithmic spiral: ${\displaystyle \kappa =C/s}$, where ${\displaystyle C}$ is a constant.
• Circle involute: ${\displaystyle \kappa =C/{\sqrt {s}}}$, where ${\displaystyle C}$ is a constant.
• Cornu spiral: ${\displaystyle \kappa =Cs}$, where ${\displaystyle C}$ is a constant.
• Catenary: ${\displaystyle \kappa ={\frac {a}{s^{2}+a^{2}}}}$.

Related parameterizations

The Cesàro equation of a curve is related to its Whewell equation in the following way. If the Whewell equation is ${\displaystyle \varphi =f(s)\!}$

then the Cesàro equation is

${\displaystyle \kappa =f'(s)\!}$.

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