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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[edit]

The family of log-aesthetic curves^[1] is determined in the general ( $\alpha \neq 0$ ) 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 ( $\alpha =\infty$ ), Euler spiral ( $\alpha =-1$ ), Logarithmic spiral ( $\alpha =1$ ), and Circle involute ( $\alpha =2$ ).

Examples[edit]

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[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]

^ Miura, K. T. (2006). "A General Equation of Aesthetic Curves and its Self-Affinity". Computer-Aided Design and Applications. 3 (1–4): 457–464. doi:10.1080/16864360.2006.10738484.

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.

[1] Miura, K. T. (2006). "A General Equation of Aesthetic Curves and its Self-Affinity". Computer-Aided Design and Applications. 3 (1–4): 457–464. doi:10.1080/16864360.2006.10738484.

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Log-aesthetic curves[edit]

Examples[edit]

Related parameterizations[edit]

References[edit]

External links[edit]

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