Cesàro equation
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. 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 case by the following intrinsic equation:This is equivalent to the following explicit formula for curvature:
Further, the constant above represents simple re-parametrization of the arc length parameter, while is equivalent to uniform scaling, so log-aesthetic curves are fully characterized by the parameter.
In the special case of, the log-aesthetic curve becomes Nielsen's spiral, with the following Cesàro equation :
A number of well known curves are instances of the log-aesthetic curve family. These include circle, Euler spiral, Logarithmic spiral, and Circle involute.
Examples
Some curves have a particularly simple representation by a Cesàro equation. Some examples are:- Line:.
- Circle:, where is the radius.
- Logarithmic spiral:, where is a constant.
- Circle involute:, where is a constant.
- Euler spiral:, where is a constant.
- Catenary:, where is a constant.