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.
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: κ = C / s , where C is a constant.Cornu spiral: κ = C s , where C is a constant.Catenary: κ = 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 φ = f ( s ) then the Cesàro equation is κ = f ′ ( s ) .
