## Lissajous Curves.

Many of this blog’s entries seem … random and unconnected. This is another one, despite it being quite connected to some research I’m presently conducting. This week, we discuss Lissajous curves.

We’ll see the formulas, and how to select “nice” parameters.

The Lissajous parametric curve is given by

$x_t=A \sin(2 \pi \alpha t + \delta)$

$y_t=B \cos(2 \pi \beta t)$

Where

• $t$ varies from 0 to 1, with $2 \pi$ needed for this to happen,
• $A$ controls x-axis scaling,
• $B$ controls y-axis scaling,
• $\alpha$ gives the x-frequency,
• $\delta$ gives the phase of the x-frequencies,
• $\beta$ gives the y-frequency.

If…

• $\displaystyle\frac{\alpha}{\beta}$ is rational, the curve on $t$ from 0 to 1 closes,
• $\alpha$ and $\beta$ are integers and relatively prime, you get the maximum amount of wiggle for the curve.
• $\delta$ is well-chosen, you can either have the curves pass all very near to each other, or very far, even have a crossing at the origin.

For ill-chosen $\delta$, you can have a “catastrophe”, where the curve backs up onto itself giving it “horns”:

To understand how this happens, we must have a look at how the sine and cosine behave relative to $t$, from 0 to 1:

Lo! Sine and cosine reach -1 at the same time, at $t\approx{0.5}$, and are complementary (+1 and -1) at $t=0$. Whenever the sine and cosine line up exactly (both at +1 or both at -1) or are in opposition (one at +1 and the other -1), you get a catastrophe. You must therefore choose $\alpha$, $\beta$, and $\delta$ in order to avoid having these co-occurrences. If you choose these wisely, you can get a curve that covers all of the surface evenly. With $\alpha=11$, $\beta=10$ and $\delta=\frac{\pi}{8}$, you get the very beautiful curve: