Graphing Calculator Drawing: Lissajous Curves

by Justin Skycak on May 11, 2019

Lissajous curves use sine functions to create interesting patterns in the plane.

This post is part of the book Graphing Calculator Drawing Exercises. Suggested citation: Skycak, J. (2019). Graphing Calculator Drawing: Lissajous Curves. In Graphing Calculator Drawing Exercises. https://justinmath.com/graphing-calculator-drawing-lissajous-curves/

Setup. Navigate to https://www.desmos.com/calculator. Be sure to sign in so that you can save your graph.

Demonstration - Lissajous Curves. Lissajous curves take the form

$\begin{align*} x &= \sin(t) \\[5pt] y &= \sin(at+b) \end{align*}$

for some values of $a$ and $b.$ Observe the graph as you type each of the following Lissajous plot inputs, with $0 \leq t \leq 100.$

$\begin{align*} &\left( \sin(t), \sin(t+1) \right) \\[10pt] &\left( \sin(t), \sin(t+2) \right) \\[10pt] &\left( \sin(t), \sin(t+3) \right) \\[10pt] &\left( \sin(t), \sin(2t+1) \right) \\[10pt] &\left( \sin(t), \sin(3t+1) \right) \\[10pt] &\left( \sin(t), \sin(4t+1) \right) \\[10pt] &\left( \sin(t), \sin(5t+1) \right) \\[10pt] &\left( \sin(t), \sin(1.1t+1) \right) \\[10pt] &\left( \sin(t), \sin(1.2t+1) \right) \\[10pt] &\left( \sin(t), \sin(1.3t+1) \right) \\[10pt] &\left( \sin(t), \sin(1.4t+1) \right) \\[10pt] &\left( \sin(t), \sin(1.5t+1) \right) \end{align*}$

Exercise. Attempt to reproduce the graphs below by setting $a=1$ and varying the $b$ parameter in the Lissajous curve equations. You may have to play with the parameter a bit to get a sense of what it controls.

Exercise. Attempt to reproduce the graphs below by setting $b=\frac{\pi}{2}$ and varying the $a$ parameter in the Lissajous curve equations. You may have to play with the parameter a bit to get a sense of what it controls.

Challenge. Attempt to reproduce the Lissajous graphs below by setting $b=1$ and varying $a.$