Rhodonea Curves (Roses)

An interactive demonstration of Rhodonea curves, also known as Roses.

Maths Geometry Polar plot parametric

A rhodonea curve is a graph of the following polar equation:

\[ r = A cos (k \theta) \]

where \( k = \frac{m}{n} \). The polar equation can also be written as two Cartesian parametric equations:

\begin{align} x &= A cos (kt) cos(t) \\ y &= A cos (kt) sin(t) \end{align}

The shape of the graph is strongly dependent on the value of \( k \), and the values of \( m \) and \( n \) can give clues as to what the shape will be. For example, if \( k \) is an integer, the graph will look like a classic flower and have \( 2k \) petals if \( k \) is even, and \( k \) petals if \( k \) is odd.

If \( k \) is half an integer, such as 1.5, 2.5 and so on, the graph will \( 4k \) petals but they will overlap which does not occur for integer \( k \). If \( m = n \) then we will get a perfect circle.

The user interface above allows you to adjust the values of \( m \) and \( n \) so you can see how they affect the graph and see if you can figure out the patterns.