Parametrizing a rose-like curve

Question

What is a parametrization for the following curve?

Maybe I can slightly alter the following parametrization in polar coordinates: $(\sin(3\theta),\theta)$ for $\theta\in [0,\pi]$ which gives:

to obtain the parametrization I'm looking for?

Answer 1

What you're looking for is a trefoil knot which Wikipedia states that the parametrization is as follows:

$$x=\sin t + 2\sin 2t$$ $$y=\cos t - 2\cos 2t$$

Where $0 \leq t \leq 2\pi$. You can flip the graph upside down by multiplying both the $x$ and $y$ by $-1$, and you scale it by changing the coefficients of the trigonometric functions.

Judging by the picture in your question, this parametrization fits well, thanks WaveX!

$$x=-1.5\sin t - 2\sin 2t$$ $$y=-1.5\cos t + 2\cos 2t$$