Roses (curves)

Planar curves whose equations in polar coordinates have the form

where and are constants. If is a rational number, then a rose is an algebraic curve of even order.

Figure: r082610a

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

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

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 .

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)