Rhodonea Curves
Rhodonea Curves (also known as Rose Curves) are plane curves described by the polar equations:
$$r = a \cos \left( k \theta \right)$$
$$r = a \sin \left( k \theta \right)$$
or in Cartesian coordinates using parametric equations:
$$x = \cos \left( kt \right) \cos \left( t \right)$$
$$y = \cos \left( kt \right) \sin \left( t \right)$$
where:
- \(a\) is a parameter that controls the size of the curve,
- \(k\) is a real number that determines the number of petals,
- \(\theta\) is the angle in polar coordinates.
Properties
- Number of Petals
- If \(k\) is an odd integer, the curve has \(k\) petals.
- If \(k\) is an even integer, the curve has \(2k\) petals.
- If \(k\) is rational, its shape depends on the fraction in its simplest form.
- If \(k\) is irrational, the curve never fully closes.
- Symmetry
- Curves generated with cosine are symmetric with respect to the horizontal axis.
- Curves generated with sine are symmetric with respect to the origin.
- Special Cases
- For \(k = 1\), the curve is a circunference.
- For \(k = 2,3,4,\ldots\) the classic flower-like shapes appear.
Interactive chart
Move sliders \(d\) and \(n\) (natural numbers) for a \(k = \frac{d}{n}\) (angular frequency), to change the rhodonea curve. Press the "Start" button to watch the animation, this will move the \(t\) (time) slider. Press the "Stop" button to interrupt the animation and press the "Reset" button to reboot the starting values.
$$x = \cos \left(\frac{d}{n} t \right) \cos \left( t \right)$$
$$y =\cos \left(\frac{d}{n} t \right) \sin \left(t \right) $$