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Cesàro equation

Log-aesthetic curves[edit]

The family of log-aesthetic curves^[1] is determined in the general (${\displaystyle \alpha \neq 0}$) case by the following intrinsic equation:

${\displaystyle R($s)^{\alpha }=c_{0}s+c_{1}}

This is equivalent to the following explicit formula for curvature:

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

Further, the ${\displaystyle c_{1}}$ constant above represents simple re-parametrization of the arc length parameter, while ${\displaystyle c_{0}}$ is equivalent to uniform scaling, so log-aesthetic curves are fully characterized by the ${\displaystyle \alpha }$ parameter.

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

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

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

Examples[edit]

Some curves have a particularly simple representation by a Cesàro equation. Some examples are:

• Line: ${\displaystyle \kappa =0}$.
• Circle: ${\displaystyle \kappa ={\frac {1}{\alpha }}}$, where α is the radius.
• Logarithmic spiral: ${\displaystyle \kappa ={\frac {C}{s}}}$, where C is a constant.
• Circle involute: ${\displaystyle \kappa ={\frac {C}{\sqrt {s}}}}$, where C is a constant.
• Euler spiral: ${\displaystyle \kappa =Cs}$, where C is a constant.
• Catenary: ${\displaystyle \kappa ={\frac {a}{s^{2}+a^{2}}}}$.

Related parameterizations[edit]

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).

References[edit]

• The Mathematics Teacher. National Council of Teachers of Mathematics. 1908. pp. 402.
• Edward Kasner (1904). The Present Problems of Geometry. Congress of Arts and Science: Universal Exposition, St. Louis. p. 574.
• J. Dennis Lawrence (1972). A catalog of special plane curves. Dover Publications. pp. 1–5. ISBN 0-486-60288-5.

External links[edit]

• Weisstein, Eric W. "Cesàro Equation". MathWorld.
• Weisstein, Eric W. "Natural Equation". MathWorld.
• Curvature Curves at 2dcurves.com.