View source for Roses (curves)

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>

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.

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/r082610a.gif" />

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.

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" />.

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" />.

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

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.A. Savelov,   "Planar curves" , Moscow  (1960)  (In Russian)</TD></TR></table>



====Comments====
These curves are also called rhodoneas, cf. [[#References|[a1]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.D. Lawrence,   "A catalog of special plane curves" , Dover, reprint  (1972)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  F. Gomes Teixeira,   "Traité des courbes" , '''1–3''' , Chelsea, reprint  (1971)</TD></TR></table>