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. 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.

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