Rose (mathematics) - General Overview

General Overview

Up to similarity, these curves can all be expressed by a polar equation of the form

or, alternatively, as a pair of Cartesianparametric equations of the form

If k is an integer, the curve will be rose shaped with

• 2k petals if k is even, and
• k petals if k is odd.

When k is even, the entire graph of the rose will be traced out exactly once when the value of θ changes from 0 to 2π. When k is odd, this will happen on the interval between 0 and π. (More generally, this will happen on any interval of length 2π for k even, and π for k odd.)

If k ends in 1/2 (ex: 0.5, 2.5), the curve will be rose shaped with 4k petals.

If k ends in 1/6 or 5/6 and is greater than 1 (ex: 1.16666667, 2.8333333), the curve will be rose shaped with 12k petals.

If k ends in 1/3 and is greater than 1 (ex: 1.333334, 2.333334), the curve will be rose shaped and will have:

• 3k petals if k is even, and
• 6k petals if k is odd.

If k ends in 2/3 and is greater than 1 (ex: 1.666667, 2.666667), the curve will be rose shaped and will have:

• 6k petals if k is even, and
• 3k petals if k is odd.

If k is rational, then the curve is closed and has finite length. If k is irrational, then it is not closed and has infinite length. Furthermore, the graph of the rose in this case forms a dense set (i.e., it comes arbitrarily close to every point in the unit disk).

Since

for all, the curves given by the polar equations

and

are identical except for a rotation of π/2k radians.

Rhodonea curves were named by the Italian mathematician Guido Grandi between the year 1723 and 1728.