**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 [1].

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*) = –*n*^{2}*x*(*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(
*n*θ), 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 [2]. 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 [4]:

- Type I curves are smooth curves.
- Type II have sharp ends.

Lissajous curves are Chebyshev polynomials [5], 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 [6].

## References

[1] L.R. Ingersoll Physics Museum. Lissajous Curve. Retrieved February 28, 2022 from: https://www.physics.wisc.edu/ingersollmuseum/exhibits/waves/lissajous/

[2] MacTutor. Lissajous Curves. Retrieved February 28, 2022 from: https://mathshistory.st-andrews.ac.uk/Curves/Lissajous/

[3] Krishnavedala,

[4] Project: Lissajous Figures. Retrieved February 28, 2022 from: http://mathserver.neu.edu/~bridger/U170/Lissajous/Lissajous.htm

[5] Merino, J. (2003). Lissajous Figures and Chebyshev Polynomials. The College Mathematics Journal.

[6] Seggern, D. (1994). PHB practical handbook of curve design and generation. CRC Press.