Figure 1: Rosettes
Figure 2: Rosettes
Figure 3: Rosette
Figure 4: Rosette:
Figure 5: Rosettes
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,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/05b375cf34c1dd5779bfd2531bddfc3a50682afc)
can be described, d. H. the associated parametric representation is
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,
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.
If
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is the circle with the equation ,![{\ displaystyle (x-0 {,} 5) ^ {2} + y ^ {2} = 0 {,} 25}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9e4948f92b2d3a2bb2f27dce5e689219e6540fde)
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is a quadrifolium (4-leaf rosette),
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is a trifolium (3-leaf rosette),
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there is an 8-petalled rosette,
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there is a 5-petalled rosette.
For
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the rosette is straight- leaved.![2n](https://wikimedia.org/api/rest_v1/media/math/render/svg/134afa8ff09fdddd24b06f289e92e3a045092bd1)
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the rosette is odd - leaved.![n](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
Note: Using the sine function instead of the cosine function only rotates the rosette.
- Generalizations
- If one allows for rational values, closed curves also result (see Fig. 2).
- The curves are not closed for irrational values of (see Fig. 4).
![n](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
- Adding to a constant: results in rosettes with large and small petals (see Fig. 3).
![r](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538)
![r = a \ cos (n \ varphi) + \ color {magenta} {c}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4faef1332e5d80db98365c6f92653d450e10d4e0)
Note: The Foucault pendulum describes an open rosette curve.
Area
A rosette has the area
![{\ 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 }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8582f6bd7f2e8a2cf27c63936d0580aab109883e)
if n is even, and
![{\ 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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a2b7564f331c9e00a1e2471c57648617bf34a09e)
if n is odd.
So there is a simple relationship with the area of the surrounding circle with radius .
![a](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc)
Web links
Individual evidence
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↑ Pêndulo de Foucault ( Portuguese WP)