Lissajous Pattern

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A Lissajous curve in mathematics, also known as Lissajous Figure or Bowditch curve is the graphical representation of the following system of parametric equations.

$x = A sin(at + δ)$
$y = B sin(bt)$

These curves/patterns are highly sensitive to the values of $a$ and $b$, particularly to the ratio $a:b$. For a ratio of $1$, and $δ = \frac{π}{2} radians$, the figure is a circle. Similarly, for $a = b$ and $δ = 0$, the figure is a line.
Another simple Lissajous figure is the parabola $(\frac{b}{a} = 2, δ = \frac{π}{4})$.

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Fun Fact

The title sequence by John Whitney for Alfred Hitchcock's 1958 film Vertigo is based on Lissajous figures.

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Last update: December 25, 2021