# 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 ${\ displaystyle r = \ cos (5 \ varphi) + c, \ c = 0 {,} 1,0 {,} 3}$
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 . ${\ displaystyle a}$