Cesàro equation

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.

Log-aesthetic curves
The family of log-aesthetic curves is determined in the general ($$\alpha \ne 0$$) case by the following intrinsic equation:

$$R(s)^\alpha = c_0s + c_1$$

This is equivalent to the following explicit formula for curvature:

$$\kappa(s) = (c_0s+c_1)^{-1/\alpha}$$

Further, the $$c_1$$ constant above represents simple re-parametrization of the arc length parameter, while $$c_0$$ is equivalent to uniform scaling, so log-aesthetic curves are fully characterized by the $$\alpha$$ parameter.

In the special case of $$\alpha=0$$, the log-aesthetic curve becomes Nielsen's spiral, with the following Cesàro equation (where $$a$$ is a uniform scaling parameter):

$$\kappa(s) = \frac{e^{\frac{s}{a}}}{a}$$

A number of well known curves are instances of the log-aesthetic curve family. These include circle ($$\alpha = \infty$$), Euler spiral ($$\alpha = -1$$), Logarithmic spiral ($$\alpha = 1$$), and Circle involute ($$\alpha = 2$$).

Examples
Some curves have a particularly simple representation by a Cesàro equation. Some examples are:
 * Line: $$\kappa = 0$$.
 * Circle: $$\kappa = \frac{1}{\alpha}$$, where $α$ is the radius.
 * Logarithmic spiral: $$\kappa=\frac{C}{s}$$, where $C$ is a constant.
 * Circle involute: $$\kappa=\frac{C}{\sqrt s}$$, where $C$ is a constant.
 * Euler 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 $φ = f&thinsp;(s)$ then the Cesàro equation is $κ = f&thinsp;′(s)$.