Cesàro curve

The Cesàro curve is a strictly self-similar fractal that was described by Ernesto Cesàro around 1905 . It represents a generalization of the well-known Koch curve . As there, the initiator is also the unit distance, but the base angle of the isosceles triangle enclosed by the curve , which is θ = 60 ° in the Koch curve, is variable in the range of θ = 0 ° to θ = 90 °. The Cesàro curve thus results as a family of curves with the parameter θ.

Different Cesàro curves

Ten different Cesàro curves from θ = 0 ° to θ = 90 ° in steps of 10 °

Depending on the parameter θ, very different curves result. The unit distance is obtained for θ = 0 °, since there is no increase in length. With increasing θ the curve appears rougher and more rugged, as its fractal dimension increases from 1 at θ = 0 ° to 2 at 90 °, where the curve finally fills an isosceles triangle with an area 1/4. In this case it is therefore a fractal fill curve .

The fractal dimension can be determined using the following formula:

${\displaystyle D_{Ces{\grave {a}}ro}(\theta )={\frac {\log \,4}{\log \,(2(1+\cos \theta ))}}\,\,}$

The area below the Cesàro curve

The area "below" the curve (i.e. between the curve and initiator) results as a function of a row over the parameter : ${\displaystyle \theta }$

${\displaystyle A_{Ces{\grave {a}}ro}\,(\theta )=\sin \,\theta \cdot \cos \,\theta \cdot \sum _{n=0}^{\infty }4^{n}\left(\left({\frac {1}{4(1+\cos \theta )^{2}}}\right)^{n+1}\right)={\frac {\sin(\theta )}{4(2+\cos(\theta ))}};\,0^{\circ }\leq \theta \leq 90^{\circ }}$

The area of increased at up to at to. ${\displaystyle A=0}$ ${\displaystyle \theta =0^{\circ }}$ ${\displaystyle A\approx 0{,}125}$ ${\displaystyle \theta =90^{\circ }}$

Individual evidence

1. ↑ Fundamentals of fractal geometry with iterated function systems (IFS) , A. Jablonski, ( Online )

literature

• Benoît B. Mandelbrot : The fractal geometry of nature , Birkhäuser, ISBN 3-7643-2646-8