# Rhodonea Curves (Roses)

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

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.