Did you already complete the Lissajous Plot, Part 1?

Remember that a Lissajous curve corresponds to the superposition of two oscillations (one along the x-axis and one along the y-axis) that are described by two functions of a common parameter t:

x = fx(t) = ax * sin(wx * t + px)

y = fy(t) = ay * sin(wy * t)

The curve consists of all points (x, y) for all values of t.

If it worked, you should see a blue circle.

The Five Parameters

What do the five parameters of lissajous_plot mean?

• wx and wy are the angular frequencies of the oscillations in the x and y directions.
• px is a phase shift applied to the x oscillation.
• ax and ay are the amplitudes of the oscillations in the x and y directions; they determine the overall width and height of the resulting plot (the plot will be at most 2 * ax wide and 2 * ay high).

In Lissajous Plot, Part 1 we looked at examples where the horizontal and vertical oscillations had the same frequency (wx == wy). Now let's look at examples where those frequencies differ!

For the following discussion, assume we work with an unscaled curve (ax == 1 and ay == 1).

Half/Double the Frequency

What happens if the frequency along one axis is twice the frequency along the other axis?

Yes, you get a parabola!

Let's Tabulate!

Let's pick the ratio wx / wy == 1 / 2, and let's look at the effect of varying the phase shift.

Let's Animate!

How does this look as an animation?

Third/Triple the Frequency

Let's Tabulate!

What about a ratio of wx / wy = 3 / 1?

Well, if we invert the ratio, this simply swaps the axes.

4/5 and 5/4

Let's try with a ratio of wx / wy = 4 / 5 and wx / wy = 5 / 4.

You could describe the curves of having a certain number of horizontal and vertical "lobes". Count the number of peaks along the edges. The 4:5 curve has 5 peaks along the top edge, and 4 peaks along the left edge. The peak counts are swapped for the 5:4 curve.

Note: Depending on the ratio, you might have to change px as well to get a rotation that best shows off the lobes.

For example, a ratio of wx / wy == 5 / 3 is tricky at px == 0:

Let's animate this to convince ourselves that there indeed are 5 lobes on the left edge and 3 lobes on the top edge.

Let's get Colorful!

The most powerful parametric plot function we have in the plot library is parametric_color_thickness_plot. Like the parametric_plot function used by lissajous_plot, this also expects the two functions (fx(t) and fy(t)) and a range of values for t on which to plot dots. However, additionally, you also can provide a function to determine the dot color and the dot size.

Let's use that function to build a lissajous_color_thickness_plot function!

Let's Animate!

Ok, this looks neat, but let's animate this by shifting the phase.