Lissajous curves can be understood as the result of smoothly interpolating between -1 and 1 at independent frequencies on both the x and y axis. Using sine and cosine functions generates typical circular Lissajous configurations.
The ratio shows the frequency at which the x and y axis rotate. For example, 1:2 means the x axis makes one rotation in the same time that it takes for the y axis to make two rotations. The second value, sigma, shows the offset of the y axis. By changing sigma, the same ratio can produce different results.
However, this suggests that other interpolations are possible. What would be the result of using another function to interpolate between values? The most obvious starting point is to look at the result of linear interpolation.
As you might expect, the sides of each lobe become straight lines connecting the extreme points at the edges of each diagram. However, these points have shifted to reflect the new interpolation function. In the final diagram, for example, the inflection points at the top and bottom are evenly spaced, whereas the equivalent Lissajous curve has the outer inflection points closer to the edges.
What about exponential easing? The diagrams below use exponential easing to the power of 2.
And to the power of 3…
And to the power of 10…
What about inverse powers, such as 1 / 2?
And 1 / 100?