In geometry, the Cesàro equation of a plane curve is an equation relating the curvature (κ) 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/κ.) 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: ${\displaystyle \kappa ={\frac {1}{\alpha }}}$, where α is the radius.
• Logarithmic spiral: ${\displaystyle \kappa ={\frac {C}{s}}}$, where C is a constant.
• Circle involute: ${\displaystyle \kappa ={\frac {C}{\sqrt {s}}}}$, where C is a constant.
• Cornu spiral: ${\displaystyle \kappa =Cs}$, where 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 φ = f (s) then the Cesàro equation is κ = 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.