Cesàro equation

Template:Short description In geometry, the Cesàro equation of a plane curve is an equation relating the curvature (Template:Mvar) at a point of the curve to the arc length (Template:Mvar) from the start of the curve to the given point. It may also be given as an equation relating the radius of curvature (Template:Mvar) to arc length. (These are equivalent because Template:Math.) 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:

$R(s{)}^{\alpha }=c_{0}s+c_1$

This is equivalent to the following explicit formula for curvature:

$\kappa (s)=(c_{0}s+c_1{)}^{-1/\alpha }$

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 ($\alpha =\mathrm{\infty }$), 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 Template:Mvar is the radius.
• Logarithmic spiral: $\kappa =\frac{C}{s}$, where Template:Mvar is a constant.
• Circle involute: $\kappa =\frac{C}{\sqrt{s}}$, where Template:Mvar is a constant.
• Euler spiral: $\kappa =Cs$, where Template:Mvar 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 Template:Math then the Cesàro equation is Template:Math.

References

• Template:Cite book
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External links

• Template:MathWorld
• Template:MathWorld
• Curvature Curves at 2dcurves.com.