Rosette (curve)

Figure 1: Rosettes ${\displaystyle r=\cos(n\varphi ),\ n=2,3,4,5}$

Figure 2: Rosettes ${\displaystyle r=\cos(n\varphi ),\ n={\tfrac {1}{2}},{\tfrac {1}{3}},{\tfrac {2}{3}},{\tfrac {3}{5}}}$

Figure 3: Rosette ${\displaystyle r=\cos(5\varphi )+c,\ c=0{,}1,0{,}3}$

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  Foucault's pendulum

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  Figure 4: Rosette: ${\displaystyle n=\pi ,\ 0\leq \varphi \leq 40\pi }$

Figure 5: Rosettes ${\displaystyle r=\cos(\varphi \cdot n/d)}$

In geometry, a rosette is a plane curve that is expressed in polar coordinates by an equation

${\displaystyle r=a\cos(n\varphi )\ ,\ n=1,2,3,\dots ,\;a>0,}$

can be described, d. H. the associated parametric representation is

${\displaystyle x=a\cos(n\varphi )\;\cos(\varphi )}$,

${\displaystyle y=a\cos(n\varphi )\;\sin(\varphi )}$.

If

${\displaystyle n=1}$is the circle with the equation , ${\displaystyle (x-0{,}5)^{2}+y^{2}=0{,}25}$

${\displaystyle n=2}$is a quadrifolium (4-leaf rosette),

${\displaystyle n=3}$is a trifolium (3-leaf rosette),

${\displaystyle n=4}$ there is an 8-petalled rosette,

${\displaystyle n=5}$ there is a 5-petalled rosette.

For

${\displaystyle n}$ the rosette is straight- leaved. ${\displaystyle 2n}$

${\displaystyle n}$ the rosette is odd - leaved. ${\displaystyle n}$

Note: Using the sine function instead of the cosine function only rotates the rosette.

Generalizations

1. If one allows for rational values, closed curves also result (see Fig. 2). ${\displaystyle n}$
2. The curves are not closed for irrational values ​​of (see Fig. 4). ${\displaystyle n}$
3. Adding to a constant: results in rosettes with large and small petals (see Fig. 3). ${\displaystyle r}$ ${\displaystyle r=a\cos(n\varphi )+\color {magenta}{c}}$

Note: The Foucault pendulum describes an open rosette curve.

Area

A rosette has the area ${\displaystyle r=a\cos(n\varphi )}$

${\displaystyle {\frac {1}{2}}\int _{0}^{2\pi }(a\cos(n\varphi ))^{2}\,d\varphi ={\frac {a^{2}}{2}}\left(\pi +{\frac {\sin(4n\pi )}{4n}}\right)={\frac {\pi a^{2}}{2}}}$

if n is even, and

${\displaystyle {\frac {1}{2}}\int _{0}^{\pi }(a\cos(n\varphi ))^{2}\,d\varphi ={\frac {a^{2}}{2}}\left({\frac {\pi }{2}}+{\frac {\sin(2n\pi )}{4n}}\right)={\frac {\pi a^{2}}{4}}}$

if n is odd.

So there is a simple relationship with the area of ​​the surrounding circle with radius . ${\displaystyle a}$

Web links

• Mathworld: Rose

Individual evidence

1. ↑ Pêndulo de Foucault ( Portuguese WP)