In geometry, the Cesàro equation of a plane curve is an equation relating the curvature at a point of the curve to the arc length from the start of the curve to the given point. It may also be given as an equation relating the radius of curvature to arc length. (These are equivalent because .) Two congruent curves will have the same Cesàro equation. Cesàro equations are named after Ernesto Cesàro.
The family of log-aesthetic curves[1] is determined in the general (
\alpha\ne0
R(s)\alpha=c0s+c1
This is equivalent to the following explicit formula for curvature:
\kappa(s)=(c0s+c
| -1/\alpha | |
| 1) |
Further, the
c1
c0
\alpha
In the special case of
\alpha=0
a
\kappa(s)=
| ||||||||
| a |
A number of well known curves are instances of the log-aesthetic curve family. These include circle (
\alpha=infty
\alpha=-1
\alpha=1
\alpha=2
Some curves have a particularly simple representation by a Cesàro equation. Some examples are:
\kappa=0
\kappa=
| 1 | |
| \alpha |
| \kappa= | C |
| s |
| \kappa= | C |
| \sqrts |
\kappa=Cs
| \kappa= | a |
| s2+a2 |
The Cesàro equation of a curve is related to its Whewell equation in the following way: if the Whewell equation is then the Cesàro equation is .