Lissajous curves are created by two simple harmonic motions — combinations of sine functions and/or cosine functions.
The curves are named after Jules-Antoine Lissajous, who studied them in 1857. They are sometimes called Bowditch curves after Nathaniel Bowditch, who discovered them (independently) in 1815 while studying a compound’s pendulum movement .
Lissajous curves are useful for studying the mechanical problem of small oscillations with two degrees of freedom: the system of differential equations x′′ = –y(t), y′′(t) = –n2x(t), where n > 0,
Lissajous Curve Equations
The parametric equations are combinations of sines and cosines:
- sin(nt + a), cos(kt + b) or
- sin(nt + a), sin(kt + b) or
- cos(nt + a), cos(kt + b) or
It’s common for the curves to be described by equations where a + b = 0. For example,
- x = cos(nθ), sin(k&theta)
You can play around with various values for n and k with this Desmos calculator page (which I used to create the top image).
Properties of the Lissajous Curve
The constants n and k determine the curve’s size.
One oscillation determines the x-coordinate and the other the y-coordinate.
When the ratio of the two oscillations’ frequencies is rational number (i.e., a/b) the curve is a closed curve. Different shapes are created with different ratios . When the frequencies are equal, the curve is a line segment (if the difference in phases is a multiple of π) or an ellipse.
There are two types of Lissajous curves :
- Type I curves are smooth curves.
- Type II have sharp ends.
Lissajous curves are Chebyshev polynomials , under certain conditions:
- n = 1,
- k = a natural number, and
Despite these straightforward properties, predicting the shape of the curve is challenging and may require some experimentation to achieve a certain shape .
 L.R. Ingersoll Physics Museum. Lissajous Curve. Retrieved February 28, 2022 from: https://www.physics.wisc.edu/ingersollmuseum/exhibits/waves/lissajous/
 MacTutor. Lissajous Curves. Retrieved February 28, 2022 from: https://mathshistory.st-andrews.ac.uk/Curves/Lissajous/
 Project: Lissajous Figures. Retrieved February 28, 2022 from: http://mathserver.neu.edu/~bridger/U170/Lissajous/Lissajous.htm
 Merino, J. (2003). Lissajous Figures and Chebyshev Polynomials. The College Mathematics Journal.
 Seggern, D. (1994). PHB practical handbook of curve design and generation. CRC Press.