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Planar curves whose equations in polar coordinates have the form
 
Planar curves whose equations in polar coordinates have the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082610/r0826101.png" /></td> </tr></table>
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$$\rho=a\sin k\phi,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082610/r0826102.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082610/r0826103.png" /> are constants. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082610/r0826104.png" /> is a rational number, then a rose is an [[Algebraic curve|algebraic curve]] of even order.
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where $a$ and $k$ are constants. If $k=m/n$ is a rational number, then a rose is an [[Algebraic curve|algebraic curve]] of even order.
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/r082610a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/r082610a.gif" />
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Figure: r082610a
 
Figure: r082610a
  
The order of a rose is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082610/r0826105.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082610/r0826106.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082610/r0826107.png" /> are odd, and to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082610/r0826108.png" /> if either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082610/r0826109.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082610/r08261010.png" /> is even. The entire curve is situated inside the circle of radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082610/r08261011.png" /> and consists of congruent parts, called petals (see Fig.). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082610/r08261012.png" /> is an integer, then the rose consists of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082610/r08261013.png" /> petals for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082610/r08261014.png" /> odd and of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082610/r08261015.png" /> petals for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082610/r08261016.png" /> even. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082610/r08261017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082610/r08261018.png" /> are relatively prime, then the rose consists of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082610/r08261019.png" /> petals for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082610/r08261020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082610/r08261021.png" /> odd, and of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082610/r08261022.png" /> petals when either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082610/r08261023.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082610/r08261024.png" /> is even.
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The order of a rose is equal to $m+n$ if $m$ and $n$ are odd, and to $2(m+n)$ if either $m$ or $n$ is even. The entire curve is situated inside the circle of radius $a$ and consists of congruent parts, called petals (see Fig.). If $k$ is an integer, then the rose consists of $k$ petals for $k$ odd and of $2k$ petals for $k$ even. If $k=m/n$ and $m,n$ are relatively prime, then the rose consists of $m$ petals for $m$ and $n$ odd, and of $2m$ petals when either $m$ or $n$ is even.
  
When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082610/r08261025.png" /> is irrational there are infinitely many petals. Roses belong to the family of cycloidal curves (cf. [[Cycloidal curve|Cycloidal curve]]). They are hypocycloids if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082610/r08261026.png" />, and epicycloids if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082610/r08261027.png" />.
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When $k$ is irrational there are infinitely many petals. Roses belong to the family of cycloidal curves (cf. [[Cycloidal curve|Cycloidal curve]]). They are hypocycloids if $k>1$, and epicycloids if $k<1$.
  
 
Roses are also related to the family of cycloidal curves by the fact that they are pedals of epi- and hypocycloids with respect to the centre of their fixed circle.
 
Roses are also related to the family of cycloidal curves by the fact that they are pedals of epi- and hypocycloids with respect to the centre of their fixed circle.
  
The arc length of a rose is given by an elliptic integral of the second kind. The area of one petal is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082610/r08261028.png" />.
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The arc length of a rose is given by an elliptic integral of the second kind. The area of one petal is $S=\pi a^2/4k$.
  
 
Roses are also called curves of Guido Grandi, who was the first to describe them in 1728.
 
Roses are also called curves of Guido Grandi, who was the first to describe them in 1728.

Latest revision as of 19:58, 16 April 2014

Planar curves whose equations in polar coordinates have the form

$$\rho=a\sin k\phi,$$

where $a$ and $k$ are constants. If $k=m/n$ is a rational number, then a rose is an algebraic curve of even order.

Figure: r082610a

The order of a rose is equal to $m+n$ if $m$ and $n$ are odd, and to $2(m+n)$ if either $m$ or $n$ is even. The entire curve is situated inside the circle of radius $a$ and consists of congruent parts, called petals (see Fig.). If $k$ is an integer, then the rose consists of $k$ petals for $k$ odd and of $2k$ petals for $k$ even. If $k=m/n$ and $m,n$ are relatively prime, then the rose consists of $m$ petals for $m$ and $n$ odd, and of $2m$ petals when either $m$ or $n$ is even.

When $k$ is irrational there are infinitely many petals. Roses belong to the family of cycloidal curves (cf. Cycloidal curve). They are hypocycloids if $k>1$, and epicycloids if $k<1$.

Roses are also related to the family of cycloidal curves by the fact that they are pedals of epi- and hypocycloids with respect to the centre of their fixed circle.

The arc length of a rose is given by an elliptic integral of the second kind. The area of one petal is $S=\pi a^2/4k$.

Roses are also called curves of Guido Grandi, who was the first to describe them in 1728.

References

[1] A.A. Savelov, "Planar curves" , Moscow (1960) (In Russian)


Comments

These curves are also called rhodoneas, cf. [a1].

References

[a1] J.D. Lawrence, "A catalog of special plane curves" , Dover, reprint (1972)
[a2] F. Gomes Teixeira, "Traité des courbes" , 1–3 , Chelsea, reprint (1971)
How to Cite This Entry:
Roses (curves). Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Roses_(curves)&oldid=16994
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article